8:00am  10:00am 
M04: Theoretical and computational studies of PDEs driven by random processes
8:00am  8:30am 
8:00am  8:30am 
Nguyen, Phuong: The Stampacchia Maximum Principle For Stochastic Partial Differential Equations Forced By Levy Noise (Abstract)
In this work, we investigate the existence of positive (martingale and pathwise)
solutions of stochastic partial differential equations (SPDEs) driven by a Levy noise. The proof relies on the use of truncation, following the Stampacchia approach to maximum principle. Among the applications, the positivity and boundedness results for the solutions of some biological systems and reaction diffusion equations are provided under suitable hypotheses, as well as some comparison theorems.

Zoom Meeting ID: 928 0586 1204

8:30am  9:00am 
8:30am  9:00am 
Surnachev, Mikhail: Green's Function Estimates For The Stationary ConvectionDiffusion Equation (Abstract)
I will disuss some recent results on estimates of Green's function of the Dirichlet problem (and fundamental solution) for the stationary convectiondiffusion equation $$\mathrm{div}\, (\mathbf{a}(x)\nabla u) + \mathbf{b}(x)\cdot \nabla u=0$$ defined in $$\Omega \subset \mathbb{R}^n$$, $$n\geq 3$$, with bounded measurable matrix $$\mathbf{a}$$ satisfying the uniform ellipticity condition $$\mathbf{a}(x)\xi \cdot \xi \geq \nu \xi^2$$, $$\nu>0$$, for all $$\xi \in \mathbb{R}^n$$ and a.e. $$x\in \Omega$$. The goal is to identify simple efficient conditions on the drift $$\mathbf{b}$$ that guarantee the existence of a fundamental solution with the Newtonian estimate, $$C_1 xy^{2n} \leq \Gamma(x,y) \leq C_2 xy^{2n}$$, which was obtained for $$\mathbf{b}=0$$ in [ref] (see also [ref]). The simplest of such conditions is to require the boundedness of $$\mathbf{b}$$ together with $$\mathbf{b}(x)\leq \psi(x)x^{1}$$, $$x>1$$, where the function $$\psi$$ is nonincreasing and satisfies the Dini condition at infinity $$\int_1^\infty \psi(s) s^{1}\, ds <\infty$$. For $$\mathbf{a}$$ with uniformly Holder continuous coefficients this result follows from [ref] (even without $$\psi$$ being monotone). This is joint work with Yurij Alkhutov.

Zoom Meeting ID: 928 0586 1204

9:00am  9:30am 
9:00am  9:30am 
Klevtsova, Yulia: On Integral Properties And The Inviscid Limit Of Stationary Measures For The Stochastic System Of The Lorenz Model Describing A Baroclinic Atmosphere (Abstract)
We consider the system of equations for the quasisolenoidal Lorenz model for a baroclinic atmosphere
\[
\frac{\partial}{\partial t}A_1 u+\nu A_2u+A_3u+B(u)=g,
\qquad
t>0,
{(1)}
\]
on the twodimensional unit sphere $$S$$ centered at the origin of the spherical polar coordinates $$(\lambda , \varphi )$$, $$\lambda \in [0, 2\pi)$$,
$$\varphi \in \left[\frac{\pi }{2}, \frac{\pi }{2}\right]$$, $$\mu = \sin \varphi $$. Here $$\nu > 0$$ is the kinematic viscosity, $$u(t,x, \omega )=\left(u_1(t,x, \omega ), u_2(t,x, \omega )\right)^{\rm T} $$ is an unknown vector function and \mbox{$$g(t, x, \omega )=\left(
g_1(t, x, \omega ),
g_2(t, x, \omega )
\right)^{\rm T}$$} is a given vector function, $$x=(\lambda , \mu)$$, $$\omega \in \Omega $$, $$(\Omega , P, F)$$ is a complete probability space,
\[
A_1=\left(\begin{array}{ll}
\Delta & 0\\
0 & \Delta + \gamma I
\end{array} \right),
A_2 = \left(
\begin{array}{ll}
\Delta ^2 & 0\\
0 & \Delta ^2
\end{array}
\right), \]
\[A_3=\left(
\begin{array}{ll}
k\Delta & 2k\Delta \\
k\Delta & (2k+k_1+\nu \gamma )\Delta + \rho I
\end{array}
\right), \]
\[ \begin{array}{c}
B(u)=(J( \Delta u_1 +2\mu ,u_1 ) + J (\Delta u_2, u_2),
J(\Delta u_2 \gamma u_2, u_1 )+ J(\Delta u_1 + 2\mu , u_2))^{\rm T}.
\end{array} \]
Also, $$\gamma,\rho, k_0, k_1\ge 0$$ are numerical parameters, $$I$$ is the identity operator, $$J(\psi,\theta)=\psi _{\lambda}\theta _{\mu}\psi _{\mu}\theta _{\lambda}$$ is the Jacobi operator and $$\Delta \psi=((1\mu ^2)\psi _\mu)_\mu+(1\mu ^2)^{1} \psi _{\lambda \lambda}$$ is the LaplaceBeltrami operator on the sphere $$S$$. A random vector function $$g = f+ \eta $$ is taken as the righthand side of (1); here $$f(x)\!=\!(f_1(x),f_2(x))^{\mathrm T}$$ and $$\eta (t, x,\omega)=(\eta_1(t,x,\omega),\eta_2(t,x,\omega))^{\mathrm T}$$ is a white noise in $$t$$. It was obtained in [1], [2] and in the present work the sufficient conditions on the righthand side and the parameters of the system (1) with white noise perturbation for existence of a unique stationary measure of Markov semigroup defined by solutions of the Cauchy problem for the system (1), for the exponential convergence of the distributions of solutions to the stationary measure as $$t \to + \infty$$ and for the existence a limiting point for any sequence of the stationary measures for this system when any sequence of the kinematic viscosity coefficients goes to zero. Several integrals over such stationary measures were estimated
by the set of the righthand side and the parameters.

Zoom Meeting ID: 928 0586 1204

9:30am  10:00am 
9:30am  10:00am 
Iyer, Gautam: Bounds On The Heat Transfer Rate Via Passive Advection (Abstract)
In heat exchangers, an incompressible fluid is heated initially and cooled at the boundary. The goal is to transfer the heat to the boundary as efficiently as possible. In this talk we study a related steady version of this problem: Consider the steady state temperature of in a fluid that is stirred, uniformly heated and cooled on the boundary. For a given large P'eclet number, how should one stir to
minimize the total heat? This problem was studied by Marcotte, Doering,
Thiffeault and Young in '18, where the authors provided many heuristics
and numerical simulations. In this talk we will show that when the
P'eclet number is large, one can always find a stirring velocity field so
that the total heat is at most $$O( \operatorname{Pe}^9 / \operatorname{Pe}^{1/2} )$$. We suspect this is optimal (up to a logarithmic correction), but are presently unable to prove a matching lower bound. This is joint work with Son Van.

Zoom Meeting ID: 928 0586 1204


8:00am  10:00am 
M06: Structure preserving techniques for nonlinear conservation equations
8:00am  8:30am 
8:00am  8:30am 
Zeng, Xianyi: A HybridVariable Discretization Method For Hyperbolic Problems (Abstract)
In this work, we extend a recently developed superconvergent hybridvariable (HV) discretization method [ref] to solve nonlinear hyperbolic conservation laws in one and two space dimensions.
Particularly, the artificial viscosity approach is adopted to capture the strong discontinuities that frequently occur in nonlinear conservation problems.
To this end, our exploration begins with the investigation of using HV methods to solve the model advectiondiffusion equation while confirming the universal superconvergence property; next, the previous analysis is utilized to construct artificial viscosities that efficiently suppress spurious oscillations.

Zoom Meeting ID: 966 1652 9039

8:30am  9:00am 
8:30am  9:00am 
Chan, Jesse: Efficient Computation Of Jacobian Matrices For Entropy Stable SummationByParts Schemes (Abstract)
Entropy stable schemes replicate an entropy inequality at the semidiscrete level. These schemes rely on an algebraic summationbyparts (SBP) structure and a technique referred to as flux differencing. We provide simple and efficient formulas for Jacobian matrices for the semidiscrete systems of ODEs produced by entropy stable discretizations. These formulas are derived based on the structure of flux differencing and derivatives of flux functions, which can be computed using automatic differentiation (AD). Numerical results demonstrate the efficiency and utility of these Jacobian formulas, which are then used in the context of twoderivative explicit timestepping schemes and implicit timestepping.

Zoom Meeting ID: 966 1652 9039

9:00am  9:30am 
9:00am  9:30am 
Dzanic, Tarik: A Riemann Difference Scheme For Shock Capturing In Discontinuous Finite Element Methods (Abstract)
We will present a novel structurepreserving numerical scheme for discontinuous finite element approximations of nonlinear hyperbolic systems. The method can be understood as a generalization of the LaxFriedrichs flux to a highorder staggered grid of flux and solution points and does not depend on any tunable parameters. Under a presented set of conditions, we will show that the method is conservative, invariant domain preserving, and satisfies the entropy condition. Numerical experiments on the Burgers' and Euler equations show the ability of the scheme to resolve discontinuities without introducing excessive spurious oscillations or dissipation.

Zoom Meeting ID: 966 1652 9039

9:30am  10:00am 
9:30am  10:00am 
Popov, Bojan: Accurate Upper Bound For The Maximum Speed Of Propagation
In The Riemann Problem (Abstract)
We will present a derivation of a guaranteed upper bound for the maximum speed of propagation in the Riemann solution for the Shallow water system and the Euler system of gas dynamics with the covolume equation of state. The novelty is that an accurate upper bound on the speed is given explicitly, hence no iterative solver is needed to compute a good estimate for the the maximum speed. The bound for the Euler system of gas dynamics is guaranteed for gasses with a heat capacity ratio $$\gamma$$ in the physical range, $$1<\gamma\le\frac53$$. This is a joint work with JeanLuc Guermond.

Zoom Meeting ID: 966 1652 9039


8:00am  10:00am 
M07: Topics in qualitative and quantitative properties of partial differential equations
8:00am  8:30am 
8:00am  8:30am 
Dong, Hongjie: Evolutionary Equations With Nonlocal Time Derivatives (Abstract)
I will discuss some recent results about fractional parabolic equations as well as fractional wave equations with Caputo time derivatives.

Zoom Meeting ID: 954 6813 3566

8:30am  9:00am 
8:30am  9:00am 
Gui, Changfeng: Propagation Acceleration In Reaction Diffusion Equations With Fractional Laplacians (Abstract)
In this paper we consider the propagation speed in a reaction diffusion system with an anomalous L'evy process diffusion, modeled by a nonlocal equation with a fractional Laplacian and a generalized KPP type monostable nonlinearity. Given a typical Heavy side initial datum, we show that the speed of interface propagation displays an algebraic rate behavior in time, in contrast to the known linear rate in the classical model of Brownian motion and the exponential rate in the KPP model with the anomalous diffusion, and depends on the sensitive balance between the anomaly of the diffusion process and the strength of monostable reaction. In particular, for the combustion model with
a fractional Laplacian $$(\Delta)^{s}$$, we show that the speed of propagation transits continuously from being linear in time, when a traveling wave solution exists for $$s \in (1/2, 1)$$, to being algebraic in time with a power reciprocal to $$2s$$, when no traveling wave solution exists for $$s \in (0, 1/2)$$.

Zoom Meeting ID: 954 6813 3566

9:00am  9:30am 
9:00am  9:30am 
Feldman, Will: Mean Curvature Flow With Positive Random Forcing In 2D (Abstract)
I will discuss some history, new results, and potential future directions in the study of the forced mean curvature flow in heterogeneous random media. I am interested in front propagation / homogenization in the depinned case. Techniques from HamiltonJacobi homogenization have been successful for "large" forcing, but near the pinning transition the level set equation loses coercivity and there are many open issues.

Zoom Meeting ID: 954 6813 3566

9:30am  10:00am 
9:30am  10:00am 
Prange, Christophe: Quantitative Regularity Vs. Concentration Near Potential Singularities In Incompressible Viscous Fluids (Abstract)
In this talk I will focus on two related aspects of the regularity theory for the threedimensional NavierStokes equations: quantitative regularity estimates on the one hand and concentration estimates for blowup solutions on the other hand. This connection enables in particular a quantification of Seregin's 2012 regularity criterion in terms of the critical $$L^3$$ norm. A counterpart of this is that we are able to give lower bounds on the blowup rate of certain critical norms near potential singularities. This talk is based on recent works [ref] in collaboration with Tobias Barker (University of Warwick).

Zoom Meeting ID: 954 6813 3566


8:00am  10:00am 
M08: Multiphase Flows in Porous Media at the Darcy Scale
8:00am  8:30am 
8:00am  8:30am 
Masson, Roland: Gradient Discretization Of TwoPhase Flows Coupled With Mechanical Deformation In Fractured Porous Media (Abstract)
We consider a twophase Darcy flow in a fractured porous medium consisting in a matrix flow coupled with a tangential flow in the fractures, described as a network of planar surfaces. This flow model is also coupled with the mechanical deformation of the matrix assuming that the fractures are open and filled by the fluids, as well as small deformations and a linear elastic constitutive law. The model is discretized using the gradient discretization method [ref], which covers a large class of conforming and non conforming schemes. This framework allows for a generic convergence analysis of the coupled model using a combination of discrete functional tools. Here, we describe the model together with its numerical discretization, and we prove the convergence of the discrete solution to a weak solution of the model [ref]. This is, to our knowledge, the first convergence result for this type of models taking into account twophase flows and the nonlinear poromechanical coupling. Previous related works consider a linear approximation obtained for a single phase flow by freezing the fracture conductivity [ref]. Numerical tests employing the TwoPoint Flux Approximation (TPFA) finite volume scheme for the flows and $$\mathbb{P}_2$$ finite elements for the mechanical deformation are also provided to illustrate the behavior of the solution to the model.

Zoom Meeting ID: 949 6350 6956

8:30am  9:00am 
8:30am  9:00am 
Arbogast, Todd: A SelfAdaptive Theta Method Using Discontinuity Aware Quadrature For Solving Hyperbolic Conservation Laws (Abstract)
We present a discontinuity aware quadrature (DAQ) rule, and use it to develop
implicit selfadaptive theta (SATh) schemes for the approximation of scalar hyperbolic
conservation laws. Our SATh schemes require the solution of a system of two equations, one
controlling the cell averages of the solution at the time levels, and the other controlling the
spacetime averages of the solution. These quantities are used within the DAQ rule to approximate
the time integral of the hyperbolic flux function accurately, even when the solution may be
discontinuous somewhere over the time interval. The result is a finite volume scheme using the theta
time stepping method, with theta defined implicitly (or selfadaptively). Two schemes are
developed, SAThup for a monotone flux function using simple upstream stabilization, and SAThLF
using the LaxFriedrichs numerical flux. DAQ is accurate to second order when there is a
discontinuity in the solution and third order when it is smooth. SAThup is unconditionally stable
provided that theta is at least 1/2, and satisfies the maximum principle and is total variation
diminishing under appropriate monotonicity and boundary conditions. General flux functions require
the SAThLF scheme, so we assess its accuracy through numerical examples in one and two space
dimensions. These results suggest that SAThLF is also stable and satisfies the maximum principle
(at least at reasonable CFL numbers). Compared to solutions of finite volume schemes using
CrankNicolson and backward Euler time stepping, SAThLF solutions often approach the accuracy of
the former but without oscillation, and they are numerically less diffuse than the later.

Zoom Meeting ID: 949 6350 6956

9:00am  9:30am 
9:00am  9:30am 
Sarraf Joshaghani, Mohammad: MaximumPrinciplePreserving VertexBased Method For Two Phase Flows In Porous Media (Abstract)
This talk presents the numerical solution of immiscible twophase flows in porous media, obtained by
a first order finite element method equipped with masslumping and flux upwinding. The unknowns are the phase pressure
and phase saturation. Recently, the theoretical convergence analysis of the method was derived in [ref]. It was also shown that
the numerical saturation satisfies a maximum principle.
Our numerical experiments confirm that the method converges optimally for manufactured solutions.
For both structured and unstructured meshes, we observe the highaccuracy
wetting saturation profile that ensures minimal numerical diffusion at the front.
Performing several examples of quarterfive spot problems in two and three dimensions,
we show that the method can easily handle heterogeneities in the permeability field.
Two distinct features that make the method appealing to reservoir simulators are:
 It respects the maximum principle by limiting the wetting phase saturation
profile to a physical upper and lowerbounds. In simulating twophase flow,
it is known that the major problem of classical formulations, when no limiting
mechanism or mitigation technique is implemented, is the lack of monotonicity of
the solution (i.e. ``overshoot'' of the solution right before and an ``undershoot``
right after the saturation front). These oscillations become significant in coarse
meshes and nonhomogeneous media.
 Similar to discontinuous Galerkin formulation, this formulation preserves
elementwise (local) mass balance. Violation of mass balance is a known problem
in continuous formulations.
To solve the discrete system that arises from this vertex method, we choose the Schur
complement preconditioning strategy due to the masslumped nature of the resulting
stiffness matrix. Using the composable solvers feature available in PETSc [ref] and the
finite element libraries available under the FEniCS Project [ref], we illustrate how to
effectively precondition this system for largescale problems.

Zoom Meeting ID: 949 6350 6956

9:30am  10:00am 
9:30am  10:00am 
Firoozabadi, Abbas: Mixed Finite Element And Discontinuous Galerkin Methods For Complex Multiphase Flow Problems And Complex Geometries In The Subsurface (Abstract)
Flow of complex hydrocarbon fluids in multiphases with capillary and Fickian diffusion and complex geometries including nonplanar fracture and heterogenous and layered rocks is among the most challenging problems. The working expression are highly nonlinear and discretization can introduce large errors. Finite elements are natural choice. The combination of physical concepts and powerful features of the combined mixed finite element for flux calculation and discontinuous Galerkin methods allow ease of accurate numerical solution of a vast group of problems in CO2 sequestration in the aquifers, fluid injection in hydrocarbon formations as well precipitation modeling. This presentation will cover some recent modeling of complex flows in the lab scale and large scale. We will also cover briefly the potential of the method in modeling of rock fracturing. The basic physical concepts that will facilitate numerical solution of complex problems will be emphasized.

Zoom Meeting ID: 949 6350 6956


8:00am  10:00am 
M11: Recent advances in the numerical approximation of geometric partial differential equations
8:00am  8:30am 
8:00am  8:30am 
Bartels, Soeren: Simulation Of Nonlinear Bending Phenomena For Plates In The Presence Of Contact (Abstract)
The bending behavior of plates is usually described using dimensionally reduced models. We propose a practical method for the numerical simulation of bilayer plate bending that is based on a nonlinear twodimensional plate model. Our method employs a discretization of the resulting energy using DKT (discrete Kirchhof triangle) elements in space and a discrete gradient flow restricted to appropriate tangent spaces for the minimization of the energy functional. Particularly, we discuss the simulation of (self)contact and the applicability of the method for simulating plates of nematic liquid crystal elastomers. The talk extends methods developed in [ref].

Zoom Meeting ID: 920 0068 6385

8:30am  9:00am 
8:30am  9:00am 
Caboussat, Alexandre: Numerical Approximation Of Orthogonal Maps With Adaptive Finite Elements. Application To Paper Folding (Abstract)
Orthogonal maps are the solutions of the socalled origami problem [ref], which consists of a system of first order fully nonlinear equations involving the gradient of the solution.
The Dirichlet problem for orthogonal maps consists in finding a vectorvalued function $$\mathbf{u} : \Omega \subset \mathbb{R}^2 \to \mathbb{R}^2$$ verifying
\begin{equation*}
\left\{ \begin{array}{ll}
\displaystyle \nabla\mathbf{u} \in \mathcal{O}(2) & \text{in}\ \Omega, \\[1mm]
\displaystyle \mathbf{u}=\mathbf{g} & \text{on} \ \partial \Omega.
\end{array}\right.
\end{equation*}
where $$\mathcal{O}(2)$$ denotes the set of orthogonal matrixvalued functions, and $$\mathbf{g}$$ is a given, sufficiently smooth, function.
The solution $$\mathbf{u}$$ is piecewise linear, with a singular set composed of straight lines representing the folding edges.
A variational approach relies on the minimization of a variational principle, which enforces the uniqueness of the solution [ref].
We present a strategy based on a splitting algorithm for the flow problem derived from the firstorder optimality conditions.
It leads to decoupling the timedependent problem into a sequence of local nonlinear problems and a global variational problem at each time step.
Within the splitting algorithm, adaptive techniques are introduced and rely on error estimate based techniques developed for the solution of linear Poisson problems [ref].
Anisotropic adaptive techniques allow to obtain refined triangulations near the folding edges while keeping the number of vertices as low as possible [ref].
Numerical experiments validate the accuracy and efficiency of the adaptive method in various situations.
Appropriate convergence properties are exhibited, and solutions with sharp edges are recovered.
Joint work with Prof. M. Picasso, D. Gourzoulidis (EPFL), Prof. R. Glowinski (University of Houston).

Zoom Meeting ID: 920 0068 6385

9:00am  9:30am 
9:00am  9:30am 
Zhiliakov, Alexander: InfSup Stability Of The Trace $$ P_2$$$$P_1$$ TaylorHood Elements For Surface Pdes (Abstract)
We present a geometrically unfitted finite element method (FEM), known as trace FEM or cut FEM, for the numerical solution of the Stokes system posed on a closed smooth surface. A trace FEM based on standard TaylorHood (continuous $$\mathbf P_2$$$$P_1$$) bulk elements is proposed. A socalled volume normal derivative stabilization, known from the literature on trace FEM, is an essential ingredient of this method. The key result is an infsup stability of the trace $$\mathbf P_2$$$$P_1$$ finite element pair, with the stability constant uniformly bounded with respect to the discretization parameter and the position of the surface in the bulk mesh. Optimal order convergence of a consistent variant of the finite element method follows from this new stability result and interpolation properties of the trace FEM. Properties of the method are illustrated with numerical examples.

Zoom Meeting ID: 920 0068 6385

9:30am  10:00am 
9:30am  10:00am 
Yushutin, Vladimir: Are Colloidal Particles Immersed In Liquid Crystals Attracted To
The Walls? (Abstract)
The answer, of course, depends on the shape of the wall. A
flat wall generally repels a colloidal particle to minimize the liquid
crystal energy by reducing the distortion of the orientation field
caused by the boundary conditions. However, there is experimental
evidence of the opposite behavior if the colloidal particle is put near
a pit of comparable size. To address this question, we aim to develop an
unfitted numerical scheme to model the motion of the particle that seeks
an optimal position near a curved wall by conducting a geometric
gradient flow along the shape gradient of the Frank's energy. In this
talk we focus on finding harmonic maps, i.e. minimizers of the Dirichlet
energy under the nonlinear, nonconvex unit length constraint, accurate
knowledge of which is required on each step of the shape optimization
procedure. Dirichlet boundary conditions are weakly imposed on an
unfitted mesh, and a second order in time technique is suggested to
conduct a discrete gradient flow to find a harmonic map on a fixed domain.

Zoom Meeting ID: 920 0068 6385


8:00am  10:00am 
M14: Spectral Theory and Mathematical Physics
8:00am  8:30am 
8:00am  8:30am 
Jauslin, Ian: An Effective Equation To Study Bose Gasses At All Densities (Abstract)
I will discuss an effective equation, which is used to study the ground state of the interacting Bose gas.
The interactions induce manybody correlations in the system, which makes it very difficult to study, be it analytically or numerically.
A very successful approach to solving this problem is Bogolubov theory, in which a series of approximations are made, after which the analysis reduces to an integrable system, which incorporates the manybody correlations.
The effective equation I will discuss is arrived at by making a very different set of approximations, and ultimately reduces to a oneparticle problem.
But, whereas Bogolubov theory is accurate only for very small densities or for large densities, but not both at once, the effective equation coincides with the manybody Bose gas at both low and at high densities.
I will show some theorems which make this statement more precise, and present numerical evidence that this effective equation is remarkably accurate for all densities, small, intermediate, and large.
That is, the analytical and numerical evidence suggest that this effective equation can capture manybody correlations in a oneparticle picture beyond what Bogolubov can accomplish.
Thus, this effective equation gives an alternative approach to study the low density behavior of the Bose gas (about which there still are many important open questions).
In addition, it opens an avenue to understand the physics of the Bose gas at intermediate densities, which, until now, were only accessible to Monte Carlo simulations.

Zoom Meeting ID: 928 1327 1650

8:30am  9:00am 
8:30am  9:00am 
Chen, Thomas: Dynamics Of A Heavy Quantum Tracer Particle In A Bose Gas (Abstract)
We consider the dynamics of a heavy quantum tracer particle coupled to a nonrelativistic boson field in $$\mathbb{R}^3$$. The pair interactions of the bosons are of meanfield type, with coupling strength proportional to $$1/N$$ where $$N$$ is the expected particle number. Assuming that the mass of the tracer particle is proportional to $$N$$, we derive generalized Hartree equations in the limit where $$N$$ tends to infinity. Moreover, we prove the global wellposedness of the associated Cauchy problem for sufficiently weak interaction potentials. This is joint work with Avy Soffer (Rutgers University), [ref].

Zoom Meeting ID: 928 1327 1650

9:00am  9:30am 
9:00am  9:30am 
Li, Wei: Infinitely Many Embedded Eigenvalues For The NeumannPoincare Operator In 3D (Abstract)
We construct a surface whose NeumannPoincar'e (NP) integral operator has infinitely many eigenvalues embedded in its essential spectrum. The surface is a sphere perturbed by smoothly attaching a conical singularity, which imparts essential spectrum.
Rotational symmetry allows a decomposition of the operator into Fourier components.
Eigenvalues of infinitely many Fourier components are constructed so that they lie within the essential spectrum of other Fourier components and thus within the essential spectrum of the full NP operator.
The proof requires the perturbation to be sufficiently small, with controlled curvature, and the conical singularity to be sufficiently flat.

Zoom Meeting ID: 928 1327 1650

9:30am  10:00am 
9:30am  10:00am 
Baskin, Dean: Propagation Of Singularities For The DiracCoulomb System (Abstract)
The Dirac equation describes the relativistic evolution of electrons
and positrons. We consider the (timedependent!) Dirac equation in
three spatial dimensions coupled to a potential with Coulombtype
singularities. We describe how singularities of solutions propagate,
leading to a diffractive effect arising from the singularities of the
Coulomb potential.

Zoom Meeting ID: 928 1327 1650


8:00am  10:00am 
M17: Mathematical and computational models for understanding emerging epidemics and evaluating intervention strategies
8:00am  8:30am 
8:00am  8:30am 
Xue, Ling: Assessing The Impact Of NonPharmaceutical Intervention Strategies For Containing Covid19 Epidemics (Abstract)
The current COVID2019 pandemic has raised serious public health concerns and severely impacted the progress of economy worldwide. To contain current outbreaks and prevent future outbreaks, it is essential to project effective mitigation strategies. In this talk, I present the networkbased models developed to examine the impact of multiple nonpharmaceutical intervention strategies.

Zoom Meeting ID: 995 5798 8257

8:30am  9:00am 
8:30am  9:00am 
Gutierrez, Juan B.: A Model For Covid19 Community Transmission Considering Asymptomatics And Mitigation (Abstract)
Asymptomatic carriers of the SARSCoV2 virus display no clinical symptoms but are known to be contagious. Recent evidence reveals that this subpopulation, as well as carriers with mild disease, are major contributor in the propagation of COVID19. In this talk, we present a traditional compartmentalized mathematical model taking into account asymptomatic carriers. Wel also present a modeling framework to account for governmentdriven mitigation strategies. This model was used to estimate projections of cases for every county in the USA.

Zoom Meeting ID: 995 5798 8257

9:00am  9:30am 
9:00am  9:30am 
Azizi, Asma: Optimizing Covid19 Awareness And Testing Strategy (Abstract)
It is essential to understand how human social distancing behavior could aid in limiting COVID19 spread, and to find a systematic testing strategy that allows to finetune human behavior to slow epidemic in the most efficient way. We will use an agent based network model for predicting the spread of COVID19 among a synthetic population. The synthetic reallife social network that is constructed based on data generated by Simfrastructure, captures demographics, activities, locations and interaction of individuals. We use this model to incorporate selfregulated social distancing of individuals as a function of their characteristic such as their number of friends, Centrality Social Distancing, and or the infection prevalence on their neighborhood, Adaptive Neighborhood Social Distancing. Then we introduce Random Test and Social Ring Awareness and Testing as a strategy of systematic testing individuals and their selected friends based on the level of friendship . This strategy utilize the information about selfregulated Adaptive Neighborhood Social Distancing to optimize infection characteristic using a limited number of test can be done per unit time. Our preliminary study, conducted on Scale Free and Small World networks, showed the efficiency of Adaptive Neighborhood Social Distancing in reducing the peak of infection, as it increases chance of reducing contact with infected cases. Random Test and Social Ring Awareness and Testing strategy defined on this selfregulated social distancing shows different impacts on networks with different structure, particularly different clustering coefficients, revealing the fact of need for implementing different testing efforts on different communities (subgraph) of the social network.

Zoom Meeting ID: 995 5798 8257

9:30am  10:00am 
9:30am  10:00am 
Qu, Zhuolin: Staged Progression Epidemic Model For The Transmission Of Invasive Nontyphoidal {Salmonella} (Ints) With Treatment (Abstract)
We develop and analyze a stageprogression compartmental model to study the emerging invasive nontyphoidal \textit{Salmonella} (iNTS) epidemic in subSaharan Africa.
iNTS bloodstream infections are often fatal, and the diverse and nonspecific clinical features of iNTS make it difficult to diagnose. We focus our study on identifying approaches that can reduce the incidence of new infections. In subSaharan Africa, transmission and mortality are correlated with the ongoing HIV epidemic and severe malnutrition. We use our model to quantify the impact that increasing antiretroviral therapy (ART) for HIV infected adults and reducing malnutrition in children would have on mortality from iNTS in the population. We consider immunocompromised subpopulations in the region with major risk factors for mortality, such as malaria and malnutrition among children and HIV infection and ART coverage in both children and adults. We parameterize the progression rates between infection stages using the branching probabilities and estimated time spent at each stage. We interpret the basic reproduction number $$\mathcal{R}_0$$ as the total contribution from an infinite infection loop produced by the asymptomatic carriers in the infection chain. The results indicate that the asymptomatic HIV+ adults without ART serve as the driving force of infection for the iNTS epidemic. We conclude that the worst disease outcome is among the pediatric population, which has the highest infection rates and death counts.
Our sensitivity analysis indicates that the most effective strategies to reduce iNTS mortality in the studied population are to improve the ART coverage among highrisk HIV+ adults and reduce malnutrition among children.

Zoom Meeting ID: 995 5798 8257


8:00am  10:00am 
M19: Nonlocal models in mathematics and computation
8:00am  8:30am 
8:00am  8:30am 
Silling, Stewart: Using Nonlocality To Predict The Rate Of Material Failure (Abstract)
All solid materials fail under sufficiently large stress.
If a large mechanical loading is applied, the process of failure is not
instantaneous, but evolves over time.
In many engineering applications, it is important to understand this time
evolution, which gives rise to rate effects in material strength.
In this work, it is shown that a nonlocal model of mechanics can give
insight into the amount of time it takes for a small crack to nucleate, which
is the first step in the process of material failure.
To study this,
the formation of a crack under dynamic loading in a peridynamic material
is investigated as an outcome of material instability.
The material model is nonlinear elastic and has a nonconvex strain energy
density function.
With this model, application of loads may take the material into a dynamically
unstable regime.
Within this regime, there may fail to be realvalued wave speeds,
resulting in the exponential growth of small perturbations in displacement.
But in peridynamics,
unlike the local theory, the growth of these unstable waveforms can
occur at a finite rate, rather than blowing up instantaneously.
By accounting for the finiteness of the rate of growth of the
various Fourier components in the initial data,
we can determine a finite time to failure of such a system.
The main result is that with a nonlocal continuum model,
material instability can be used as a tool to reproduce
realistic features of material failure [ref].

Zoom Meeting ID: 950 2884 6922

8:30am  9:00am 
8:30am  9:00am 
Oterkus, Erkan: Utilisation Of EulerLagrange Equation To Derive DualHorizon Peridynamic Equations (Abstract)
Peridynamics [ref] is a nonlocal continuum mechanics formulation where a material point can interact with other material points which are located at a finite distance with respect to each other. The equations of peridynamics are in integrodifferential equation form and it is difficult to obtain closedform solutions for these equations. The numerical solution of peridynamic equations is generally obtained by using meshless method and uniform spatial discretisation. The implementation of uniform discretisation is usually straightforward. However, this approach can increase computational time significantly for specific problems. On the other hand, nonuniform discretisation can also be utilised and different discretisation sizes can be used at different parts of the solution domain. In peridynamics, there is also a length scale parameter called ``horizon'' which defines the range of nonlocal interactions. In addition to nonuniform discretisation, variable horizon size may also be required. For such cases, a new peridynamic formulation, Dual Horizon Peridynamics [ref], was introduced so that both nonuniform discretisation and variable horizon size can be utilised. In this presentation, the derivation of Dual Horizon Peridynamics formulation by using EulerLagrange equation will be presented for statebased peridynamics. Moreover, application of boundary conditions and determination of surface correction factors will also be discussed. Finally, the capability of Dual Horizon Peridynamics formulation will be demonstrated by considering plate under tension and plate vibration problems.

Zoom Meeting ID: 950 2884 6922

9:00am  9:30am 
9:00am  9:30am 
Scabbia, Francesco: A Note On The Surface Effect In OsbPd Models (Abstract)
Peridynamics is a recently proposed continuum theory which has been devised to effectively describe fracture phenomena in solid bodies [ref]. Due to the nonlocal nature of the theory, peridynamic models exhibit an undesired stiffness fluctuation near the boundaries, which is known as \emph{surface effect} [ref]. The authors will propose an innovative method to exploit the introduction of a fictitious boundary layer in order to mitigate the surface effect and properly impose the nonlocal boundary conditions. The basic idea is that the fictitious nodes are bound to move according to the displacements of the nodes of the real body close to the boundary. In this way the neighborhoods of all boundary points are completed in a rational way and the previously missing peridynamic interactions are provided. The proposed method is verified for a 1dimensional ordinary statebased body under some constraints and loads for which the same solution of classical mechanics is expected: the numerical results recover exactly the classical solution in the whole domain, even near the boundaries.
\vspace{8pt}
\noindent U. Galvanetto and M. Zaccariotto would like to acknowledge the support they received from MIUR under the research project PRIN2017DEVISU and from University of Padua under the research projects BIRD2018 NR.183703/18.

Zoom Meeting ID: 950 2884 6922

9:30am  10:00am 
9:30am  10:00am 
Oterkus, Selda: Peridynamic Polycrystalline Ice Model (Abstract)
Arctic region has started being considered as an alternative shipping route due to its advantages such as being a shorter route. However, it also introduces additional challenges due to its harsh environment. For instance, ships must be designed to withstand ice loads if a collision between a ship and ice occurs. Although experimental studies can provide very useful information, full scale tests are very expensive for icestructure interactions. Instead, computer simulations can be utilized as alternative option. However, modelling ice as a material has also its own challenges because its material behavior depends on many different factors such as appliedstress, strainrate, temperature, grainsize, salinity, porosity and confining pressure. Moreover, microcracks occurring at the microscale may have significant influence on the macroscopic behavior. Therefore, a multiscale methodology may also be necessary especially to capture the correct physics around the icestructure interaction region. Within finite element framework, there are various techniques which are available for numerical analysis of icestructure interaction problem including cohesive zone model (CZM) and extended finite element method (XFEM). Although these are powerful techniques, they have certain limitations. As an alternative, peridynamics [ref] can be utilized. Peridynamics is a nonlocal continuum mechanics formulation which is very suitable for failure analysis of materials due its mathematical structure. Furthermore, due to its nonlocal character, it can capture the physical phenomena at multiple scales. Therefore, in this presentation, a new peridynamic polycrystalline ice model [ref] will be presented to be used for icestructure interaction analysis.

Zoom Meeting ID: 950 2884 6922


8:00am  10:00am 
M20: Dynamics of Nonlinear PDE and Applications
8:00am  8:30am 
8:00am  8:30am 
Tedeev, Anatoli: Asymptotic Properties Of Solutions To The Cauchy Problem For Degenerate Parabolic Equations With Inhomogeneous Density On Manifolds (Abstract)
We consider the Cauchy problem for doubly nonlinear degenerate parabolic equations with inhomogeneous density on noncompact Riemannian manifolds. We give a qualitative classification of the behavior of the solutions of the problem depending on the behavior of the density function at infinity and the geometry of the manifold, which is described in terms of its
isoperimetric function. We establish for the solutions properties as:
stabilization of the solution to zero for large times, finite speed of
propagation, universal bounds of the solution, blow up of the interface. Each one of these behaviors of course takes place in a suitable range of parameters, whose definition involves a universal geometrical characteristic function, depending both on the geometry of the manifold and on the asymptotics of the density at infinity.

Zoom Meeting ID: 999 2943 5357

8:30am  9:00am 
8:30am  9:00am 
Bhatta, Dambaru: Nonlinear Free Surface Condition Due To Wave Diffraction By A Pair Of Cylinders (Abstract)
In the computations of the nonlinear loads on offshore structures, the most challenging task is the computation of the free surface integral. The main contribution to this integrand is due to the nonhomogeneous term present in the free surface condition for the second order potential function. Here we derive the nonhomogeneous term involved in the free surface condition due to second order wave diffraction by a pair of cylinders. We also present computational aspect of the coefficients appearing in this expression and some computational results of the free surface term.

Zoom Meeting ID: 999 2943 5357

9:00am  9:30am 
9:00am  9:30am 
Lin, Quyuan: $$3D$$ Inviscid Primitive Equations With Rotation (Abstract)
Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). It is wellknown that the $$3D$$ viscous primitive equations are globally wellposed in Sobolev spaces. In this talk, I will discuss the illposedness in Sobolev spaces, the local wellposedness in the space of analytic functions, and finitetime blowup of solutions to the $$3D$$ inviscid PEs with rotation (Coriolis force). Moreover, I will also show, in the case of ``wellprepared" analytic initial data, the regularizing effect of the Coriolis force by providing a lower bound for the lifespan of the solutions that grows toward infinity with the rotation rate. This is a joint work with Tej Eddine Ghoul (New York University in Abu Dhabi), Slim Ibrahim (University of Victoria), and Edriss S. Titi (Texas A\&M and University of Cambridge).

Zoom Meeting ID: 999 2943 5357

9:30am  10:00am 
9:30am  10:00am 
Qiao, Zhijun (George): Recent Developments On Integrable Peakon Systems (Abstract)
This talk will introduce integrable peakon and cuspon models. Those models include the wellknown CamassaHolm (CH), the DegasperisProcesi (DP), and other new peakon equations recently developed. Some open problems will be addressed for discussion.

Zoom Meeting ID: 999 2943 5357


8:00am  10:00am 
M22: Biological Oscillations: From genes to populations
8:00am  8:30am 
8:00am  8:30am 
Karamched, Bhargav: Bacterial CellShape Modulation And Induced Population Dynamics Of Synthetic Microbial Consortia (Abstract)
Rodshaped bacteria exhibit remarkable regularity in their cell shape while also demonstrating notable plasticity in these characteristics when subjected to environmental changes. However, spatiotemporal cell shape uniformity and its selective advantage to bacteria are not well understood. Likewise, cell shape mutations and the mechanisms of cooperation in bacterial consortia of different shapes have received limited attention. Here, we present a lattice model of synthetic microbial consortia in microfluidic traps where cell aspect ratio is dynamically modulated in a simulated genetic circuit via quorum sensing signaling. Our results show that population dynamics in bacterial consortia can be altered by asserting an aspect ratio change in one of the strains. Experimental synthetic biology often relies on distributed functionality among distinct engineered strains whose spatial separation or relative population fraction is critical to maintaining desired functionality. Our simulations demonstrate controllable alteration of a population's strain fraction in a microfluidic trap while using a purely mechanical interaction. When strains are coupled through quorum sensing, we show how interesting spatiotemporal dynamics emerge, such as spatial population oscillations. The ability to control a bacterial strain's population fraction will better understanding of the role of cell shape in spatial orderings exhibited by microbial consortia in natural and synthetic settings. We also discuss a more detailed agentbased model that confirms predictions of our simple lattice model.

Zoom Meeting ID: 960 0969 3953

8:30am  9:00am 
8:30am  9:00am 
Sirlanci, Melike: A Different Approach To Model Oscillatory Blood Glucose Behavior (Abstract)
It is a longknown fact that blood glucose (BG) levels show oscillatory behavior. Hence, researchers who use the mechanistic modeling approach to describe BG dynamics have been building models to represent this phenomenon. It is obviously important for a model to represent the realistic behavior of the corresponding system, however, in many cases it is hard to use these models within computational frameworks including realworld data. For the glucoseinsulin system, there are two main reasons for this: (1) insulin is almost never measured, which requires the corresponding insulin state to be severely constrained for meaningful model identification and forecasting, and (2) the realworld data is generally sparse that makes identification even harder. In order to address this type of data constraints, which is very common in realworld settings, we propose a new modeling approach that uses a stochastic differential equation [ref]. In this approach, rather than modeling the trajectory of oscillatory BG levels, we aim to create a model that resolves its mean behavior and the amplitude of the oscillations. In addition, the model can be specialized for different settings such as type 2 diabetes mellitus (T2DM) and intensive care unit (ICU) by appropriate modifications. Finally, we present numerical results for which the model is used in T2DM and ICU settings separately to forecast future BG levels with realworld data.

Zoom Meeting ID: 960 0969 3953

9:00am  9:30am 
9:00am  9:30am 
MacLaurin, James: Synchronization Of Biochemical Oscillators (Abstract)
Biochemical systems are often characterized by low copy numbers, with discrete stochastic switching between different states. In this talk I focus on biochemical systems that exhibit oscillatory behaviors. I outline a method for understanding the phase of these discrete systems. I then study different mechanisms by which separate oscillators can synchronize their phases. I determine accurate asymptotic estimates for the Lyapunov exponent (indicating the expected rate of synchronization). The systems to be studied include pure jumpMarkov stochastic processes (which are purely discrete) and piecewise deterministic Markov processes (with both a `discrete' component, and a `continuous' component). Applications include the MorrisLecar neural oscillator and glycolytic oscillations.

Zoom Meeting ID: 960 0969 3953

9:30am  10:00am 
9:30am  10:00am 
Ott, William: DelayInduced Uncertainty In Physiological Systems (Abstract)
We introduce a novel route through which delay causes dynamical systems to lose reliability.
We precisely explain the nature of the resulting delayinduced uncertainty (DIU).
Importantly, the chaos induced by the delay is both sustained in time and observable.
Our work poses new mathematical questions at the interface of ergodic theory and (infinitedimensional) delay dynamical systems.
We show that DIU occurs in an archetypal physiological model, the Ultradian glucoseinsulin model.
This observation suggests that DIU may profoundly affect clinical medical care, including glycemic management in the intensive care unit.
DIU may be relevant throughout biomedicine because delay is ubiquitous in physiological systems.
Developing DIU detection methods and assessing the impact of DIU on data assimilation techniques will be important future research directions.

Zoom Meeting ID: 960 0969 3953


8:00am  10:00am 
M25: Applications of Algebraic Geometry
8:00am  8:30am 
8:00am  8:30am 
Sherman, Samantha: Generating Cognates For 6, 8, And 10Bar Mechanisms (Abstract)
A coupler cognate of a planar linkage is a different mechanism that has the same coupler curve. Roberts showed that there are 3 fourbar mechanisms that trace out the same coupler curve[ref]. Dijksman provided a list of cognates for sixbar mechanisms by way of intricate geometric drawings and without proof the list was complete[ref]. This talk will demonstrate how cognates can be easily understood and generated using kinematic equations. Then, we combine this with numerical algebraic geometry to give a method to produce a complete list of all coupler cognates for sixbar mechanisms. Examples on sixbar mechanisms will be shown to demonstrate the method as well as generating a cognate to eight and tenbar mechanisms.

Zoom Meeting ID: 912 9802 4361

8:30am  9:00am 
8:30am  9:00am 
Duff, Timothy: Galois Groups In 3D Reconstruction (Abstract)
In computer vision, the study of minimal problems is critical for many 3D reconstruction tasks. Solving minimal problems comes down to solving systems of polynomial equations of a very particular structure. ``Structure" may be understood in terms of the Galois/monodromy group of an associated branched cover. For classical problems such as homography estimation and fivepoint relative pose, efficient solutions exploit imprimitivity of the Galois groups; in these cases, the imprimitivity comes from the existence of certain rational deck transformations. In general, Galois groups can be computed with numerical homotopy continuation using a variety of software. I will highlight joint work with Viktor Korotynskiy, Tomas Pajdla, and Maggie Regan that studies an everexpanding zoo of minimal problems and their Galois groups, with a view towards identifying new minimal problems that may be useful in practice.

Zoom Meeting ID: 912 9802 4361

9:00am  9:30am 
9:00am  9:30am 
Regan, Margaret: Machine Learning The Discriminant Locus (Abstract)
Parameterized systems of polynomial equations arise in many applications in science and engineering with the real solutions describing, for example, equilibria of a dynamical system, linkages satisfying design constraints, and scene reconstruction in computer vision. Since different parameter values can have a different number of real solutions, the parameter space is decomposed into regions whose boundary forms the real discriminant locus. In this talk, I will discuss a novel sampling method for multidimensional parameter spaces and how it is used in various machine learning algorithms to locate the real discriminant locus as a supervised classification problem, where the classes are the number of real solutions. Examples such at the Kuramoto model will be used to show the efficacy of the methods. Finally, an application to real parameter homotopy methods will be presented. This project is joint work with Edgar Bernal, Jonathan Hauenstein, Dhagash Mehta, and Tingting Tang.

Zoom Meeting ID: 912 9802 4361

9:30am  10:00am 
9:30am  10:00am 
Walker, Elise: Numerical Homotopies From Khovanskii Bases (Abstract)
Homotopies are useful numerical methods for solving systems of polynomial equations. Embedded toric degenerations are one source for homotopy algorithms. In particular, if a projective variety has a toric degeneration, then linear sections of that variety can be optimally computed using the polyhedral homotopy. Any variety whose coordinate ring has a finite Khovanskii basis is known to have a toric degeneration [ref]. We provide embeddings for this Khovanskii toric degeneration to compute general linear sections of the variety. This is joint work with Michael Burr and Frank Sottile.

Zoom Meeting ID: 912 9802 4361


8:00am  10:00am 
M30: Elastic Imaging and Full waveform inversion for hydrocarbon exploration and production
8:00am  8:30am 
8:00am  8:30am 
Liu, Yanhua: TimeLapse Fwi For Vti Media: Methodology Of TimeLapse Elastic FullWaveform Inversion For Vti Media (Abstract)
Timelapse fullwaveform inversion (FWI) can provide highresolution information about the variations of reservoir properties during hydrocarbon production and CO2 injection. However, most existing timelapse FWI methods are limited to isotropic and, often, acoustic media. Here, we develop a timelapse FWI methodology for elastic VTI (transversely isotropic with a vertical symmetry axis) media and evaluate several strategies for applying it to synthetic surface multicomponent and pressure wavefields from a 2D graben model. Using multicomponent data improves the resolution of the estimated temporal parameter variations, although the convergence of the inversion algorithm is hindered by the multimodality of the objective function. The timelapse variations in the shearwave vertical velocity Vs0 influence the estimated changes in the other VTI parameters, especially in the Pwave normalmoveout velocity. Application of conventional isotropic FWI to timelapse data from typical VTI models leads to large errors in the inverted temporal variations of all parameters. Overall, our work demonstrates the importance of taking elasticity (in particular, the velocity Vs0) and anisotropy into account in timelapse FWI algorithms.

Zoom Meeting ID: 976 3201 3764

8:30am  9:00am 
8:30am  9:00am 
Brossier, Romain: 3D MultiParameter ViscoElastic Full Waveform Inversion: Methods And Application For A NearSurface CaseStudy (Abstract)
Full Waveform Inversion (FWI) [ref] has become one of the most popular seismic inversion techniques amongst geophysical scales from the nearsurface to the global Earth scale, through crustal exploration scales. FWI is a PDEconstrained nonlinear inverse problem, which tries to update an earth subsurface model subjected to the waveequation constraint. At the exploration scale, FWI is now an industrial standard in the velocity model building workflow, mostly under the acoustic approximation of the waveequation. For shallow seismic scale, challenges of FWI rely on usually sparse acquisitions, weak signal to noise ratio, lack of high frequency, complex wave propagation with strong elastic effects and strong attenuation [ref].
In this work, we implement and apply 3D viscoelastic FWI scheme in order to invert a unique ninecomponents seismic data. The experiment's target is the Ettlingen Line (EL), a defensive trenchline which was built by the German Troop in 1707, located at Rheinstetten, Germany. Led by Karlsruhe Institute of Technology, in collaboration with Univ. Grenoble Alpes, GFZ Potsdam, and ETH Zurich, a dense 3D seismic survey with threecomponent sources (128 shot points) and threecomponent receivers (888 receiver position), leading to 9C data, has been acquired in 2017 on the target of 30$$\times$$30 meters.
Our scheme relies, for the forward problem, on a spectral element implementation of the 3D viscoelastic waveequation [ref]. The inverse problem uses an adjointbased approach to evaluate the gradient of the misfit function [ref], lBFGS [ref] for the optimization scheme implemented in the SEISCOPE nonlinear toolbox [ref]. In addition to these rather standard schemes, specific functionalities have been implemented for the simultaneous reconstruction of P and Swave velocity models on this dataset. First, because the dataset is strongly dominated by surfacewaves (sensitive mostly to Swave velocity), a first inversion step has been focused on $$V_S$$ reconstruction with an assumed prior $$V_P/V_S$$ ratio which is taken into account in the $$V_S$$ update. In a second step, the two parameters are simultaneously and independently reconstructed, while satisfying prior bound constraints and nonlinear $$V_P/V_S$$ ratio constraint through a Dykstra algorithm [ref].
On the dataset preprocessing side, matchingfilter stratgey has been implemented to account for the fact that the dataset has been acquired as six separated subsets (during five days) with uncontrolled 4D effects. FWI has then been performed from low ([315]Hz) to high frequency ([365]Hz for the last frequency band) from a homogenous background, thanks to the rich lowfrequency content. The final model makes it possible to identify the trench structure of the nearsurface, as well as one additional trenchlike shape previouly unknown. A sensitivity study on the multicomponent interest has also been performed.

Zoom Meeting ID: 976 3201 3764

9:00am  9:30am 
9:00am  9:30am 
Aragao, Odette: Elastic FullWaveform Inversion With Petrophysical Information In A Probabilistic Approach (Abstract)
Elastic full waveform inversion (EFWI) augmented with petrophysical information defines a high standard for velocity model building, as it delivers highresolution, accurate and lithologically feasible subsurface models. The technique enhances the benefits of using an elastic wave equation over the acoustic implementation while constraining the inverted models to geologically plausible solutions. We derive elastic models of subsurface properties using EFWI and explicitly incorporate petrophysical penalties to guide models toward realistic lithology, i.e., to models consistent with the seismic data as well as with the petrophysical context in the area of study. This methodology mitigates several issues related to EFWI, as it reduces the high nonlinearity of the inverse problem, mitigates the artifacts created by interparameter crosstalk, and prevents geologically implausible earth models [ref]. We define this penalty using multiple probability density functions (PDFs) derived from petrophysical information, such as well logs, where each PDF represents a different lithology. We demonstrate that the combined FWI objective function establishes a more robust foundation for EFWI by explicitly guiding models toward plausible solutions in the specific geological context of the exploration problem, while at the same time reducing the misfit between the observed and modelled data.

Zoom Meeting ID: 976 3201 3764

9:30am  10:00am 
9:30am  10:00am 
Biswas, Reetam: TwoStep Velocity Inversion Using TransDimensional Tomography And Elastic Fwi (Abstract)
Full Waveform Inversion (FWI) has become a powerful tool to generate highresolution subsurface velocity models. FWI attempts to solve a nonlinear and nonunique inverse problem, and is traditionally based on a local optimization technique. As a result, it can easily get stuck in a local minimum. To mitigate this deleterious effect, FWI requires a good starting model, which should be close enough to the optimal model to properly converge to the global minimum. Here, we investigate a twostep approach for solving this problem. In the first step, we generate a starting model for FWI, that includes the lowwavenumber information, from firstarrival traveltime tomography of downward extrapolated streamer data. We solve the tomography problem using a transdimensional approach, based on a Bayesian framework. The number of model parameters is treated as a variable, similar to the Pwave velocity information. We use an adaptive cloud of nuclei points and Voronoi cells to represent our 2D velocity model. We use Reversible Jump Markov Chain Monte Carlo (RJMCMC) to sample models from a variable dimensional model space and obtain an optimum starting model for local elastic FWI. We also estimate uncertainty in our tomography derived model. We solve for the Eikonal equation using a shortest path method for ray tracing in tomography and we solve the elastic wave equation using a timedomain finitedifference method in FWI. To compute the gradient we used the adjoint method. We demonstrate our algorithm on a real 2D seismic streamer dataset from Axial Seamount, which is the most volcanically active site of the northeastern Pacific. We ran 17 Markov chains with different starting number of nuclei and convergence for all chains was attained in about 1000 iterations. Marginal posterior density plots of velocity models demonstrate uncertainty in the obtained starting velocity models. We then ran a local elastic FWI using the combined result from all chains. [ref]

Zoom Meeting ID: 976 3201 3764


10:00am  10:30am 
Break / discussions 
10:30am  11:30am 
Graeme Milton (University of Utah)
Untangling in time: designing time varying applied fields to reveal interior structure.
In two phase materials, each phase having a nonlocal response in time, we were surprised to discover that for appropriate driving fields the response somehow untangles at specific times, allowing one to directly infer useful information about the geometry of the material, such as the volume fractions of the phases.
The underlying mathematics, showing how the appropriate driving fields may be designed, rests on the existence of approximate, measure independent, linear relations between the values that Markov functions take at a given set of possibly complex points, not belonging to the interval [1,1] where the measure is supported. The problem is reduced to simply one of polynomial approximation of a given function on the interval [1,1]. This allows one to obtain explicit estimates of the error of the approximation, in terms of the number of points and the minimum distance of the points to the interval [1,1]. In the context of the motivating problem, the analysis also yields bounds on the response at any particular time for any driving field, and allows one to estimate the response at a given frequency using an appropriately designed driving field that effectively is turned on only for a fixed interval of time. The approximation extends directly to Markovtype functions with a positive semidefinite operator valued measure, and this has applications to determining the shape of an inclusion in a body from boundary flux measurements at a specific time, when the timedependent boundary potentials are suitably tailored. This is joint work with Ornella Mattei and Mihai Putinar.

11:30am  12:00pm 
Break, discussion with the plenary speaker

12:00pm  12:30pm 
Lunch break 
12:30pm  2:30pm 
M03: Nonlinear Waves and Applications
12:30pm  1:00pm 
12:30pm  1:00pm 
Tarfulea, Andrei: SelfGenerating Lower Bounds For The Boltzmann Equation (Abstract)
The Boltzmann equation arises in statistical physics and plasma dynamic. It models the
space and velocity distribution of the particles in a diffuse gas. The particles collide with
each other at microscopic scales, leading to a quadratic, nonlocal (in velocity), collision
operator that behaves somewhat like a fractional Laplacian. In recent years there has been
substantial progress on the regularity and continuation program for the Cauchy problem (see,
for instance, [ref]). Notably, a smooth and unique solution exists for as long as
the socalled hydrodynamic quantities remain ``under control'': the mass, energy, and entropy
densities must stay bounded above uniformly in space and the mass density must stay bounded
below uniformly in space. The last condition is crucial for smoothing since it gives the
collision operator elliptic properties in certain velocity directions.
In this work, we show that the solution to the Boltzmann equation (even starting from
initial data that contains large regions of vacuum) instantaneously fill space. That is,
the gas diffuses and spreads positive mass to every space and velocity coordinate at
any positive times. We obtain this result dynamically through barrier arguments for moving
mass through space and a De Giorgi type iteration for spreading mass to arbitrary velocities.
A consequence is that the above continuation criterion can now be weakened; it is no longer
necessary to assume that the mass density is bounded from below (or, as a side result, that
the entropy density is bounded from above) for continuation of smooth solutions; those
bounds are now available a priori.
Joint work with Christopher Henderson and Stanley Snelson.

Zoom Meeting ID: 916 3502 2832

1:00pm  1:30pm 
1:00pm  1:30pm 
Yamazaki, Kazuo: NonUniqueness In Law For TwoDimensional NavierStokes Equations With Diffusion Weaker Than A Full Laplacian (Abstract)
We study the twodimensional NavierStokes equations forced by random noise with a diffusive term generalized via a fractional Laplacian that has a positive exponent strictly less than one. Because intermittent jets are inherently threedimensional, we instead adapt the theory of intermittent form of the twodimensional stationary flows to the stochastic approach presented by Hofmanov$$\acute{\mathrm{a}}$$, Zhu $$\&$$ Zhu (2019, arXiv:1912.11841 [math.PR]) and prove its nonuniqueness in law.

Zoom Meeting ID: 916 3502 2832

1:30pm  2:00pm 
1:30pm  2:00pm 
Chong, Jacky: Dynamics Of Large Boson Systems With Attractive Interaction And A Derivation Of The Cubic Focusing Nls In $$ {R}^3$$ (Abstract)
The talk is based on a recent revision and improvement of an old work of mine (with the same title as the talk). We consider a system of $$N$$ bosons where the particles
experience a short range twobody interaction given by
$$N^{1}v_N(x)=N^{3\beta1}v(N^\beta x)$$ where $$v \in C^\infty_c(\mathbb{R})$$,
without a definite sign on $$v$$.
We extend the results of M. Grillakis and M. Machedon, Comm. Math. Phys., \textbf{324}, 601(2013) and E. Kuz, Differ. Integral Equ., \textbf{137}, 1613(2015)
regarding the secondorder correction to the meanfield evolution of systems with repulsive interaction to systems with attractive interaction for $$0<\beta<\frac{1}{2}$$.
Our extension allows for a more general set of initial data which includes \emph{coherent states}. We also provide both a
derivation of the \emph{focusing nonlinear Schr\" odinger equation} (NLS) in $$3$$D from the manybody system and its rate of convergence toward meanfield
for $$0<\beta<\frac{1}{3}$$. In particular, we give two derivations of the focusing NLS, one based on the $$N$$norm approximation
proven in the work of P. T. Nam and M. Napi'orkowski, Adv. Theor. Math. Phys., \textbf{21}, 683(2017) and the other via a method introduced in
P. Pickl, J. Stat. Phys., \textbf{140}, 76(2010). Moreover, the talk will be delivered in English.

Zoom Meeting ID: 916 3502 2832

2:00pm  2:30pm 
2:00pm  2:30pm 
Booth, Robert: LongTime Asymptotics For The Massless DiracCoulomb Equation (Abstract)
We describe a work in progress regarding the longtime asymptotics of the massless DiracCoulomb equation. We obtain a complete joint asymptotic expansion for solutions near future null infinity, where the exponents of the expansion appear as resonances of a related hyperbolic operator. Key techniques include microlocal propagation estimates which reduce the proof to a Fredholm problem on variable coefficient Sobolev spaces. This work builds on prior related work by BaskinVasyWunsch. This project is joint with Dean Baskin and Jesse GellRedman.

Zoom Meeting ID: 916 3502 2832


12:30pm  2:30pm 
M04: Theoretical and computational studies of PDEs driven by random processes
12:30pm  1:00pm 
12:30pm  1:00pm 
Balci, Anna: Elliptic Equations With Degenerate Weights (Abstract)
In recent years Calder'onZygmund type regularity estimates were established for solutions of different classes of linear weighed degenerate elliptic problems with matrix coefficients. For nonlinear setting
$$$$
\textrm{div} \big( \mathbb{M} \nabla u^{p2} \mathbb M^2 \nabla u\big)
= \textrm{div} \big( \mathbb M G^{p2} \mathbb M^2 G\big),
$$$$
where $$1 0$$ and get the local higher integrability of
weak solutions.
The talk is based on joint work with Lars
Diening, Raffaella Giova and Antonia Passarelli di Napoli.

Zoom Meeting ID: 962 5244 6504

1:00pm  1:30pm 
1:00pm  1:30pm 
Padgett, Joshua: Beating The Curse Of Dimensionality In HighDimensional Stochastic FixedPoint Equations (Abstract)
In recent years, highdimensional partial differential equations (PDEs) have become a topic of extreme interest due to their occurrence in numerous scientific fields. Examples of such equations include the Schrodinger equation in quantum manybody problems, the nonlinear BlackScholes equation for pricing financial derivatives, and the HamiltonJacobiBellman equation in dynamic programming. In each of these cases, standard numerical techniques suffer from the socalled curse of dimensionality, which refers to the computational complexity of an employed approximation method growing exponentially as a function of the dimension of the underlying problem. This phenomenon is what prevents traditional numerical algorithms, such as finite differences and finite element methods, from being efficiently employed in problems with more than, say, ten dimensions. The purpose of this talk is to introduce a novel approximation algorithm known as the multilevel Picard (MLP) approximation method for beating the curse of dimensionality in the case of semilinear PDEs. We accomplish this task by considering the equivalent stochastic fixedpoint equations associated to such PDEs.
The primary focus of this talk will be motivating the development of this novel algorithm and then providing rigorous $$L^p$$error and computational complexity analysis with optimal constants.
Numerical examples will be provided in order to provide experimental verification of the obtained results.

Zoom Meeting ID: 962 5244 6504

1:30pm  2:00pm 
1:30pm  2:00pm 
Wang, Chunmei: Structure Probing Neural Network Deflation (Abstract)
Deep learning is a powerful tool for solving nonlinear differential equations, but usually, only the solution corresponding to the flattest local minimizer can be found due to the implicit regularization of stochastic gradient descent. This paper proposes Structure Probing Neural Network Deflation (SPNND) to make deep learning capable of identifying multiple solutions that are ubiquitous and important in nonlinear physical models. First, we introduce deflation operators built with known solutions to make known solutions no longer local minimizers of the optimization energy landscape. Second, to facilitate the convergence to the desired local minimizer, a structure probing technique is proposed to obtain an initial guess close to the desired local minimizer. Together with neural network structures carefully designed in this paper, the new regularized optimization can converge to new solutions efficiently. Due to the meshfree nature of deep learning, SPNND is capable of solving highdimensional problems on complicated domains with multiple solutions, while existing methods focus on merely one or twodimensional regular domains and are more expensive than SPNND in operation counts. Numerical experiments also demonstrate that SPNND could find more solutions than exiting methods.

Zoom Meeting ID: 962 5244 6504

2:00pm  2:30pm 
2:00pm  2:30pm 
Kara, Erdi: Diffusion Tensor Imaging (Dti) Based Drug Diffusion Model In A Solid Tumor (Abstract)
In this work, we study the effect of drug distribution on tumor cell death when the drug is internally injected in the
tumorous tissue. We derive a full 3dimensional inhomogeneous – anisotropic diffusion model. To capture the
anisotropic nature of the diffusion process in the model, we use an MRI data of a 35year old patient diagnosed
with Glioblastoma multiform(GBM) which is the most common and most aggressive primary brain tumor. After
preprocessing the data with a medical image processing software, we employ finite element method in MPIbased
parallel setting to numerically simulate the full model and produce doseresponse curves. We then illustrate the
apoptosis (cell death) fractions in the tumor region over the course of simulation and proposed several ways to
improve the drug efficacy. Our model also allows us to visually examine the toxicity. Since the model is built
directly on the top of a patientspecific data, we hope that this study will contribute to the individualized cancer
treatment efforts from a computational biomechanics viewpoint.
[ref]

Zoom Meeting ID: 962 5244 6504


12:30pm  2:30pm 
M05: Dynamical Systems and Mathematical Biology
12:30pm  1:00pm 
12:30pm  1:00pm 
Cantrell, Robert Stephen: Perspective On The Connection Between Ideal Free Dispersal And The Evolution Of Dispersal (Abstract)
In this talk I will discuss our perspective on the study of the evolution of dispersal and its connections to the ecological notion of the ideal free distribution. This will include some insight into its development first in the context of spatially heterogeneous but temporally constant habitats and more recently into the case where temporal change is periodic. This work is in collaboration with Chris Cosner and Adrian Lam.

Zoom Meeting ID: 981 2270 2181

1:00pm  1:30pm 
1:00pm  1:30pm 
Shi, Junping: Spatial Modeling And Dynamics Of Organic Matter Biodegradation In The Absence Or Presence Of Bacterivorous Grazing (Abstract)
Biodegradation is a pivotal natural process for elemental recycling and preservation of an ecosystem. Mechanistic modeling of biodegradation has to keep track of chemical elements via stoichiometric theory, under which we propose and analyze a spatial movement model in the absence or presence of bacterivorous grazing. Sensitivity analysis shows that the organic matter degradation rate is most sensitive to the grazer's death rate when the grazer is present and most sensitive to the bacterial death rate when the grazer is absent. Therefore, these two death rates are chosen as the primary parameters in the conditions of most mathematical theorems. The existence, stability and persistence of solutions are proven by applying linear stability analysis, local and global bifurcation theory, and the abstract persistence theory. Through numerical simulations, we obtain the transient and asymptotic dynamics and explore the effects of all parameters on the organic matter decomposition. Grazers either facilitate biodegradation or has no impact on biodegradation, which resolves the ``decompositionfacilitation paradox" in the spatial context.

Zoom Meeting ID: 981 2270 2181

1:30pm  2:00pm 
1:30pm  2:00pm 
Lenhart, Suzanne: Optimal Control Of Directed Flow In A Pde Model Of An Invasive Species In A River (Abstract)
Invasive species in rivers may be controlled by adjustment of flow rates. Using a parabolic PDE model representing an invasive population in a river, we investigate optimal control of the water discharge rate to keep an invasive population downstream. We show some numerical simulations to illustrate how far upstream the invasive population reaches.

Zoom Meeting ID: 981 2270 2181

2:00pm  2:30pm 
2:00pm  2:30pm 
Jin, Yu: The Dynamics Of A ZooplanktonFish System In Aquatic Habitats (Abstract)
Diel vertical migration is a common movement pattern of zooplankton in marine and freshwater habitats. In this paper, we use a temporally periodic reactiondiffusionadvection system to describe the dynamics of zooplankton and fish in aquatic habitats. Zooplankton live in both the surface water and the deep water, while fish only live in the surface water. Zooplankton undertake diel vertical migration to avoid predation by fish during the day and to consume sufficient food in the surface water during the night. We establish the persistence theory for both species as well as the existence of a timeperiodic positive solution to investigate how zooplankton manage to maintain a balance with their predators via vertical migration. Numerical simulations discover the effects of migration strategy, advection rates, domain boundary conditions, as well as spatially varying growth rates, on persistence of the system.

Zoom Meeting ID: 981 2270 2181


12:30pm  2:30pm 
M06: Structure preserving techniques for nonlinear conservation equations
12:30pm  1:00pm 
12:30pm  1:00pm 
Appelo, Daniel: An EnergyBased Discontinuous Galerkin Method For Semilinear Wave Equations (Abstract)
We generalize the energybased discontinuous Galerkin method proposed in [ref] to secondorder semilinear wave equations. A stability and convergence analysis is presented along with numerical experiments demonstrating optimal convergence for certain choices of the interelement fluxes. Applications to the sineGordon equation include simulations of breathers, kink, and antikink solitons.

Zoom Meeting ID: 963 2421 1162

1:00pm  1:30pm 
1:00pm  1:30pm 
Valseth, Eirik: An Unconditionally Stable SpaceTime Fe Method For The Shallow Water Equations (Abstract)

Zoom Meeting ID: 963 2421 1162

1:30pm  2:00pm 
1:30pm  2:00pm 
Maier, Matthias: Massively Parallel 3D Computation Of The Compressible Euler
Equations With An InvariantDomain Preserving SecondOrder FiniteElement
Scheme (Abstract)
We discuss the efficient implementation of a highperformance
secondorder collocationtype finiteelement scheme for solving the
compressible Euler equations of gas dynamics on unstructured meshes. The
solver is based on the convex limiting technique introduced by
Guermond et al. (SIAM J. Sci. Comput. 40, A3211A3239, 2018). As such it
is invariantdomain preserving, i.\,e., the solver maintains
important physical invariants and is guaranteed to be stable without the
use of adhoc tuning parameters. This stability comes at the expense of a
significantly more involved algorithmic structure that renders
conventional highperformance discretizations challenging.
We demonstrate that it is nevertheless possible to achieve an appreciably
high throughput of the computing kernels of such a scheme. We discuss the
algorithmic design that allows a SIMD vectorization of the compute
kernel, analyze the nodelevel performance and report excellent weak and
strong scaling of a hybrid thread/MPI parallelization.

Zoom Meeting ID: 963 2421 1162

2:00pm  2:30pm 
2:00pm  2:30pm 
Guermond, JeanLuc: SecondOrder Invariant Domain Preserving Approximation Of The
Compressible NavierStokes Equations (Abstract)
The objective of this talk is to present a fullydiscrete
approximation technique for the compressible NavierStokes equations.
The method is implicitexplicit, secondorder accurate in time and space,
and guaranteed to be invariant domain preserving. The restriction on
the timestep size is the standard hyperbolic CFL condition. To the best of our knowledge, this
method is the first one that is guaranteed to be invariant domain
preserving under the standard hyperbolic CFL condition and be
secondorder accurate in time and space.
Of course there are countless papers in the literature describing techniques
to approximate the timedependent compressible NavierStokes equations, but
there are very few papers establishing invariant domain properties. Among the
latest results in this direction we refer the reader to
[ref] where a firstorder method using
upwinding and staggered grid is developed (see Eq. (3.1) therein). The authors
prove positivity of the density and the internal energy (Lem. 4.4 therein).
Unconditional stability is obtained by solving a nonlinear system involving the
mass conservation equation and the internal energy equation. One important
aspect of this method is that it is robust in the low Mach regime. A similar
technique is developed in [ref] for the
compressible barotropic NavierStokes equations (see \S3.6 therein). We also
refer to [ref] where a fully explicit dG scheme is proposed with
positivity on the internal energy enforced by limiting. The invariant domain
properties are proved there under the parabolic time step restriction $$\tau
\le \mathcal{O}(h^2)/\mu$$, where $$\mu$$ is some reference viscosity scale.
The key idea of the present talk is to build on
[ref] and use an operator splitting technique to
treat separately the hyperbolic part and the parabolic part of the problem.
The hyperbolic substep is treated explicitly and the parabolic substep is
treated implicitly. This idea is not new and we refer for instance to
[ref] for an early attempt in this direction. The novelty
of our approach is that each substep is guaranteed to be invariant domain
preserving. In addition, the scheme is conservative and fullycomputable (e.g.
the method is fullydiscrete and there are no openended questions regarding
the solvability of the subproblems). One key ingredient of our method is that
the parabolic substep is reformulated in terms of the velocity and the
internal energy in a way that makes the method conservative, invariant domain
preserving, and secondorder accurate.

Zoom Meeting ID: 963 2421 1162


12:30pm  2:30pm 
M07: Topics in qualitative and quantitative properties of partial differential equations
12:30pm  1:00pm 
12:30pm  1:00pm 
Malinnikova, Eugenia: On The Landis Conjecture On The Plane (Abstract)
We consider solutions to the twodimensional Schrodinger equation $$\Delta u+Vu=0$$ with a bounded real potential and show that they cannot decay too fast by confirming the conjecture of Landis from the 1960s, which states that a solution $$u$$ satisfying $$u(x)\le C\exp(x^{1+\varepsilon)}$$ on the plane $$\mathbb{R}^2$$ is trivial. The vanishing order of solutions to the Schrodinger equations will be also discussed in the talk if time permits. The talk is based on a recent joint work with A. Logunov, N. Nadirashvili, and F. Nazarov [ref]

Zoom Meeting ID: 974 0661 3400

1:00pm  1:30pm 
1:00pm  1:30pm 
Shen, Zhongwei: Weighted $$L^2$$ Estimates For Elliptic Homogenization (Abstract)
We develop a new realvariable method for weighted $$L^p$$ estimates.
The method is applied to the study of weighted $$W^{1, 2}$$ estimates in Lipschitz domains for
weak solutions of secondorder elliptic systems in divergence form with bounded measurable coefficients.
It produces a necessary and sufficient condition, which depends on the weight function,
for the weighted $$W^{1,2}$$ estimate to hold in a fixed Lipschitz domain with a given weight.
Using this condition, for elliptic systems in Lipschitz domains with rapidly oscillating, periodic and VMO coefficients,
we reduce the problem of weighted estimates to the case of constant coefficients.

Zoom Meeting ID: 974 0661 3400

1:30pm  2:00pm 
1:30pm  2:00pm 
Patrizi, Stefania: From The PeierlsNabarro Model To The Equation Of Motion Of The Dislocation Continuum (Abstract)
We consider a semilinear integrodifferential equation in dimension one associated to the half Laplacian
whose solution represents the atom dislocation in a crystal.
The equation comprises the evolutive version of the classical
PeierlsNabarro model.
We show that for a large number of dislocations, the solution, properly rescaled, converges to the solution of a
well known equation called by Head [ref]
``the equation of motion of the dislocation continuum". The limit equation is a model for the macroscopic
crystal plasticity with density of dislocations.
In particular, we recover the so called Orowan's law which states that dislocations move at a velocity proportional to the effective stress.

Zoom Meeting ID: 974 0661 3400

2:00pm  2:30pm 
2:00pm  2:30pm 
Zhang, Yuming: Long Time Dynamics For Combustion In Random Media (Abstract)
We study long time dynamics of combustive processes in random media, modeled by reactiondiffusion equations with random ignition reactions. One expects that under reasonable hypotheses on the randomness, large spacetime scale dynamics of solutions to these equations is almost surely governed by a homogeneous HamiltonJacobi equation. While this was previously shown in one dimension as well as for radially symmetric reactions in several dimensions, we prove this phenomenon in the general nonisotropic multidimensional setting. Our results hold for reactions that are close to reactions with finite ranges of dependence (i.e., their values are independent at sufficiently distant points in space), and are based on proving existence of deterministic front (propagation) speeds in all directions for these reactions.

Zoom Meeting ID: 974 0661 3400


12:30pm  2:30pm 
M11: Recent advances in the numerical approximation of geometric partial differential equations
12:30pm  1:00pm 
12:30pm  1:00pm 
Plucinsky, Paul: Forward And Inverse Design Of Deployable Origami Structures (Abstract)
Shapemorphing finds widespread utility, from the deployment of small stents and large solar sails to actuation and propulsion in soft robotics. Origami structures provide a template for shapemorphing, but rules for designing and folding the structures are challenging to integrate into a broad and versatile design tool. Here, we address this challenge in the context of rigidly and flatfoldable quadrilateral mesh origami (RFFQM). First, we explicitly characterize the designs and deformations of all possible RFFQM [ref]. Our key idea is a rigidity theorem that characterizes compatible crease patterns surrounding a single panel and enables us to march from panel to panel to compute the pattern and its corresponding deformations explicitly. The marching procedure is computationally efficient. So we also employ it in an inverse design framework to approximate a general surface by this family of origami [ref]. The structures produced by our framework are ``deployable": they can be easily manufactured on a flat reference sheet, deployed to their target state by a controlled folding motion, then to a compact folded state in applications involving storage and portability. We demonstrate the accuracy, versatility and efficiency of our framework through a rich series of examples.

Zoom Meeting ID: 967 5576 1385

1:00pm  1:30pm 
1:00pm  1:30pm 
Yang, Shuo: Ldg Approximation Of Large Deformations Of Prestrained Plates (Abstract)
A reduced model for large deformations of prestrained plates consists of minimizing a second order bending energy subject to a nonconvex metric constraint. We discuss a formal derivation of this reduced model along with an equivalent formulation that makes it amenable computationally. We propose a local discontinuous Galerkin (LDG) finite element approach that hinges on the notion of reconstructed Hessian. We design discrete gradient flows to minimize the ensuing nonconvex problem.

Zoom Meeting ID: 967 5576 1385

1:30pm  2:00pm 
1:30pm  2:00pm 
Walker, Shawn: Optimal Control Of VolumePreserving Mean Curvature Flow (Abstract)
We develop a framework and numerical method for controlling the full spacetime tube of a geometrically driven flow. We consider an optimal control problem for the mean curvature flow of a curve or surface with a volume constraint, where the control parameter acts as a forcing term in the motion law. The control of the trajectory of the flow is achieved by minimizing an appropriate trackingtype cost functional. The gradient of the cost functional is obtained via a formal sensitivity analysis of the spacetime tube generated by the mean curvature flow. We show that the perturbation of the tube may be described by a transverse field satisfying a parabolic equation on the tube. We propose a numerical algorithm to approximate the optimal control and demonstrate it with several results in two and three dimensions.

Zoom Meeting ID: 967 5576 1385

2:00pm  2:30pm 
2:00pm  2:30pm 
Zhang, Wujun: Rates Of Convergence For Optimal Transport Problem With Quadratic Cost In Two Or Three Dimensions (Abstract)
We consider the optimal transport problem minimizing the quadratic transport cost between two probability measures.
It is well known that the transport mapping is related to the gradient map of the solution $$\gradv u$$ of a MongeAmp\`{e}retype of partial differential equations with second boundary condition.
Based on the stability estimate established for the MongeAmp\`{e}retype of problem [ref], we establish a measure weighted $$W^1_1$$norm for the numerical optimal transport mapping$$\gradv u_h$$ and show that if $$u \in C^{2, \alpha}$$ where $$0< \alpha \le 2$$, then the measure weighted $$W^1_1$$error of $$u_h$$ converges in order $$\ln \left ( \frac 1 h \right ) h^{\alpha}$$.
We will also present several results when measures contains vacuum region, in which case solution $$u$$ is of low regularity.

Zoom Meeting ID: 967 5576 1385


12:30pm  2:30pm 
M12: Scientific Machine Learning
12:30pm  1:00pm 
12:30pm  1:00pm 
Mang, Andreas: Classification Of 3D Shapes And Shape Deformations (Abstract)
We present a mathematical framework and computational methods for classification and clustering of shapes and shape deformations in an infinitedimensional shape space $$\mathcal{S}$$. Our goal is to discriminate between clinically distinct patient groups through the lens of anatomical shape variability. In a Riemannian setting, we can express the similarity between two $$k$$dimensional ($$k\in\{1,2,3\}$$) shapes $$s_i\in\mathcal{S}, i = 0,1$$, in terms of an energy minimizing $$\mathbb{R}^3$$diffeomorphism $$y\in\mathcal{Y}$$ such that $$y(s_0) = s_1$$. We use an optimal control formulation, in which the diffeomorphism $$y$$ is parameterized by a smooth, timedependent velocity field $$v \in L^2([0,1],\mathcal{V})$$, with associated Hilbert space $$\mathcal{V}$$ of $$\mathbb{R}^3$$ vector fields. After computing an optimal $$v^\star$$, we derive the strain distribution of $$y^\star$$ as well as a Hilbert norm of $$v^\star$$ to characterize the dissimilarities between $$s_0$$ and $$s_1$$. Using these features, we implement machine learning techniques to achieve a classification of shapes extracted from cardiac imaging.

Zoom Meeting ID: 976 1776 9801

1:00pm  1:30pm 
1:00pm  1:30pm 
Actor, Jonas: Upwind Schemes And Neural Networks For Image Segmentation (Abstract)
Two common techniques for image segmentation  level set methods and convolutional neural
networks (CNN)  rely on alternating convolutions with nonlinearities to describe image features:
neural networks with meanzero convolution kernels can be viewed as upwind finite difference
discretizations of differential equations. Such a comparison provides a wellestablished framework for
proving properties of CNNs, such as stability and approximation accuracy. We test this relationship by
constructing a level set network, a CNN whose structure is determined by an upwind discretization of
the level set equation, so that by construction, each layer of the network becomes a timestep in our
discretization. In this sense, forward propagation through the CNN is equivalent to solving the level
set equation. We train our network on abdomen CT data from the MICCAI LiTS 2017 Challenge, with
the goal of performing image segmentation of the liver and hepatocellular carcinoma tumors. The level
set network achieves comparable segmentation accuracy to solving the level set equation, while requiring
substantially fewer parameters than conventional CNN architectures.
% linear convolution features commonly used by radiologists for manual image segmentation.

Zoom Meeting ID: 976 1776 9801

1:30pm  2:00pm 
1:30pm  2:00pm 
Hanin, Boris: Data Augmentation As Stochastic Optimization (Abstract)
Recent advancements in data augmentation have led to stateoftheart performance on diverse machine learning tasks. In these cases, augmentation improves, sometimes dramatically, averagecase performance and robustness. However, it remains unclear how to choose, compare, and schedule augmentations in a principled way. The present work provides a theoretical framework for data augmentation in which such issues can be studied directly, in contrast to adhoc, computationally intensive search typical in practice. Our framework is general enough to unify augmentations such as synthetic noise (additive noise, CutOut and labelpreserving transformations (color jitter, geometric transformations) together with more traditional stochastic optimization methods (SGD, Mixup). The essence of our approach is that any augmentation corresponds to noisy gradient descent on a timevarying sequence of proxy losses.
Specializing our framework to overparameterized linear models, we obtain a MunroRobbins type result, which provides conditions for jointly scheduling learning rate and augmentation strength. Although it holds only in this limited context, we emphasize that our framework as whole covers a broad family of models including kernels and neural networks. Our results in the linear case give a rigorous baseline to compare to more complex settings and uncover nontrivial scheduling phenomena even for linear models.

Zoom Meeting ID: 976 1776 9801

2:00pm  2:30pm 
2:00pm  2:30pm 
Liu, Lizuo: A Phase Shift Deep Neural Network For High Frequency Approximation And Wave Problems (Abstract)
In this talk[ref], we propose a phase shift deep neural network, i.e., (PhaseDNN), which provides a uniform wideband convergence in approximating high frequency functions and solutions of wave equations. The PhaseDNN makes use of the fact that common DNNs often achieve convergence in the low frequency range first[ref], and constructs a series of moderatelysized DNNs trained for selected high frequency ranges. With the help of phase shifts in the frequency domain, each of the DNNs will be trained to approximate the function's specific high frequency range at the speed of learning for low frequency. As a result, the proposed PhaseDNN is able to convert high frequency learning to low frequency one, allowing a uniform learning to wideband functions. The PhaseDNN is then applied to learn solutions of high frequency wave problems in inhomogeneous media through least square residual of either differential or integral equations. Numerical results have demonstrated the capability of the PhaseDNN as a meshless method in general domains in learning high frequency functions and oscillatory solutions of interior and exterior Helmholtz problems.

Zoom Meeting ID: 976 1776 9801


12:30pm  2:30pm 
M14: Spectral Theory and Mathematical Physics
12:30pm  1:00pm 
12:30pm  1:00pm 
Ntekoume, Maria: Integrable Dispersive Pde At Low Regularity (Abstract)
In this talk, we will discuss recent results in the intersection of dispersive PDE and completely integrable systems at low regularity, exploiting the Lax pair formulation.

Zoom Meeting ID: 942 7878 2474

1:00pm  1:30pm 
1:00pm  1:30pm 
Sottile, Frank: Critical Points Of Discrete Periodic Operators (Abstract)
It is believed that the dispersion relation of a Schrodinger operator
with a periodic potential has nondegenerate critical points. In work
with Kuchment and Do [ref], we considered this for discrete operators on a
periodic graph $$\Gamma$$, for then the dispersion relation is an algebraic
hypersurface. A consequence is a dichotomy; either almost all
parameters have all critical points nondegenerate or almost all
parameters give degenerate critical points, and we showed how tools
from computational algebraic geometry may be used to study the
dispersion relation.
With Matthew Faust, we use ideas from combinatorial algebraic
geometry to give an upper bound for the number of critical points at
generic parameters, and also a criterion for when that bound is
obtained. The dispersion relation has a natural compactification in a
toric variety, and the criterion concerns the smoothness of the
dispersion relation at toric infinity.

Zoom Meeting ID: 942 7878 2474

1:30pm  2:00pm 
1:30pm  2:00pm 
Sukhtaiev, Selim: Anderson Localization For Kirchhoff And Discrete Laplacians On Random Trees (Abstract)
In this talk, we will discuss a mathematical treatment of a disordered system modeling localization of quantum waves on metric and discrete trees. We will show that the transport properties of several natural Hamiltonians on metric and discrete trees with random branching numbers are suppressed by disorder. This is a joint work with D. Damanik (Rice University) and J. Fillman (Texas State University)

Zoom Meeting ID: 942 7878 2474

2:00pm  2:30pm 
2:00pm  2:30pm 
Matos, Rodrigo: Finite Volume Criterion For Localization On Correlated Environments (Abstract)
In the context of the Anderson model with correlated potentials, we shall present an abstract finite volume criterion which yields subexponential dynamical localization as long as a finite volume condition is verified and suitable decorrelation assumptions are met. As a corollary, in the above setting, polynomial decay of the Green's function fractional moments at a finite scale implies subexponential decay of them at any scale. As an application, the Hubbard model within HartreeFock theory will be discussed.

Zoom Meeting ID: 942 7878 2474


12:30pm  2:30pm 
M15: Graph Theory
12:30pm  1:00pm 
12:30pm  1:00pm 
Liu, ChunHung: Asymptotic Dimension Of MinorClosed Families And Beyond (Abstract)
The asymptotic dimension of metric spaces is an important notion in geometric group theory. The metric spaces considered in this talk are the ones whose underlying spaces are the vertexsets of (edge)weighted graphs and whose metrics are the distance functions in weighted graphs. A standard compactness argument shows that it suffices to consider the asymptotic dimension of classes of finite weighted graphs. We prove that the asymptotic dimension of any minorclosed family of weighted graphs, any class of weighted graphs of bounded treewidth, and any class of graphs of bounded layered treewidth are at most 2, 1, and 2, respectively. The first result solves a question of Fujiwara and Papasoglu; the second and third results solve a number of questions of Bonamy, Bousquet, Esperet, Groenland, Pirot and Scott. These bounds for asymptotic dimension are optimal and generalize and improve some results in the literature, including results for Riemannian surfaces and Cayley graphs of groups with a forbidden minor.

Zoom Meeting ID: 940 6262 2447

1:00pm  1:30pm 
1:00pm  1:30pm 
Postle, Luke: Edge Colouring With Local List Sizes (Abstract)
The wellknown List Colouring Conjecture from the 1970s states that for every graph $$G$$ the chromatic index of $$G$$ is equal to its list chromatic index. In a seminal paper in 1996, Kahn proved that the List Colouring Conjecture holds asymptotically. Our main result is a local generalization of Kahn's theorem. More precisely, we show that, for a graph $$G$$ with sufficiently large maximum degree $$\Delta$$ and minimum degree $$\delta \geq \ln^{25} \Delta$$, the following holds: For every assignment of lists of colours to the edges of $$G$$, such that $$L(e) \geq (1+o(1)) \cdot \max\left\{\deg(u),\deg(v)\right\}$$ for each edge $$e=uv$$, there is an $$L$$edgecolouring of $$G$$. Furthermore, Kahn showed that the List Colouring Conjecture holds asymptotically for linear, $$k$$uniform hypergraphs, and recently Molloy generalized this to correspondence colouring. We also prove a local version of Molloy's result. In fact, we prove a weighted version that simultaneously implies all of our results. Joint work with Marthe Bonamy, Michelle Delcourt, and Richard Lang.

Zoom Meeting ID: 940 6262 2447

1:30pm  2:00pm 
1:30pm  2:00pm 
Lidicky, Bernard: Flexibility In Planar Graphs (Abstract)
Recently, Dvořák, Norin, and Postle introduced flexibility as an extension of list coloring on graphs [ref]. In this new setting, each vertex $$v$$ in some subset of $$V(G)$$ has a request for a certain color $$r(v)$$ in its list of colors $$L(v)$$. The goal is to find an $$L$$ coloring satisfying many, but not necessarily all, of the requests.
The main studied question is whether there exists a universal constant $$\epsilon >0$$ such that any graph $$G$$ in some graph class $$\mathcal{C}$$ satisfies at least $$\epsilon$$ proportion of the requests.
More formally, for $$k > 0$$ the goal is to prove that for any graph $$G \in \mathcal{C}$$ on vertex set $$V$$, with any list assignment $$L$$ of size $$k$$ for each vertex, and for every $$R \subseteq V$$ and a request vector $$(r(v): v\in R, r(v) \in L(v))$$, there exists an $$L$$coloring of $$G$$ satisfying at least $$\epsilonR$$ requests.
If this is true, then $$\mathcal{C}$$ is called {\em $$\epsilon$$flexible for lists of size $$k$$}.
In this talk, we explain the notion, describe methods for obtaining results and survey the known results.

Zoom Meeting ID: 940 6262 2447

2:00pm  2:30pm 
2:00pm  2:30pm 
A graph $$H$$ is $$k$$common if the number of monochromatic copies of $$H$$ in a $$k$$edgecoloring of $$K_n$$ is
asymptotically minimized by a random coloring. A consequence of the famous Sidorenko’s conjecture is that every graph that is Sidorenko is also kcommon. In fact, we showed that a graph is kcommon for every k if and only if the graph is Sidorenko (which implies the graph is bipartite). However, it is not known whether there is a kcommon nonbipartite graph for some fixed k.
In this talk, I will talk about a recent result which shows that for every $$k$$, there is a connected nonbipartite $$k$$common graph.
This resolves a problem raised by Jagger, \v{S}\v{t}ov'{\i}\v{c}ek and Thomason.
This is a joint work with Daniel Kr'al', Jon Noel, Sergey Norin, and Jan Volec.

Zoom Meeting ID: 940 6262 2447


12:30pm  2:30pm 
M19: Nonlocal models in mathematics and computation
12:30pm  1:00pm 
12:30pm  1:00pm 
Martowicz, Adam: Peridynamic Model For Shape Memory Alloys (Abstract)
Peridynamics offers unique capabilities for the computational tools used to simulate behavior of solid components [ref]. The above mentioned modeling approach provides means for convenient handling various types of model nonlinearities, including material and geometric properties as well as boundary conditions. An integral based formulation of the governing equation assumes an extended region for the force interactions, i.e., the socalled longrange interactions, considered within the body of the nonlocally modeled solid component. The authors of the present work make use of the advantages of peridynamics and propose a nonlocal formulation for the model of shape memory alloys (SMA) [ref]. In particular, numerical aspects of the elaborated model are discussed in details to confirm its usability. The peridynamic model is applied to simulate phase transformations in SMA originating from the phenomenon of superelasticity. The phenomenological model is created using the concept of Gibbs free energy and thermoelasticity [ref]. \\The authors acknowledge the project \textit{Mechanisms of stability loss in highspeed foil bearings – modeling and experimental validation of thermomechanical couplings}, no. OPUS 2017/27/B/ST8/01822 financed by the National Science Center, Poland.

Zoom Meeting ID: 964 9811 8679

1:00pm  1:30pm 
1:00pm  1:30pm 
Bhattacharya, Debdeep: Modeling Particle Beds Using Peridynamics (Abstract)
We model the interaction of particle beds with a plate due to gravity, where the particle shapes are allowed to be nonconvex. The intraparticle force is modeled using peridynamics and interparticle repulsive, friction, and damping forces are incorporated when the particles are close by. The collision between nonconvex grains is detected dynamically to allow large displacements and breakage. The plate acts as a peridynamic material that settles under its own weight.
This work is a part of a MURI project for predicting and controling the response of particulate systems through grainscale engineering.
%Our abstract with possibly some references [ref]

Zoom Meeting ID: 964 9811 8679

1:30pm  2:00pm 
1:30pm  2:00pm 
Aksoylu, Burak: The Choice Of Kernel Function In Nonlocal Wave Propagation With Local Boundary Conditions (Abstract)
It is a challenge to choose the appropriate kernel function in
nonlocal problems. We tackle this challenge from the aspect of
nonlocal wave propagation and study the dispersion relation at the
analytical level. The kernel function enters the formulation as an
input. Any effort to narrow down this function family is valuable.
Dispersion relations of the nonlocal governing operators are
identified. Using a Taylor expansion, a selection criterion is
devised to determine the kernel function that provides the best
approximation to the classical (linear) dispersion relation. The
criterion is based on selecting the smallest coefficient in magnitude
of the dominant term in the Taylor expansion after the constant term.
The governing operators are constructed using functional calculus
[ref]
which provides the explicit expression of the eigenvalues of the
operators. The ability to express eigenvalues explicitly allows us to
obtain dispersion relation at the analytical level, thereby isolating
the effect of discretization on the dispersion relation. With the
presence of expressions of eigenvalues of the governing operator, the
analysis is clear and accessible. The choices made to obtain the best
approximation to the classical dispersion relation become completely
transparent. We find that the truncated Gaussian family is the most
effective compared to power and rational function families [ref].

Zoom Meeting ID: 964 9811 8679

2:00pm  2:30pm 
2:00pm  2:30pm 
Diehl, Patrick: On The Treatment Of Boundary Conditions For BondBased Peridynamic Models (Abstract)
In this talk, we propose two approaches to apply boundary conditions for bondbased peridynamic models. There has been in recent years a renewed interest in the class of socalled nonlocal models, which include peridynamic models, for the simulation of structural mechanics problems as an alternative approach to classical local continuum models. However, a major issue, which is often disregarded when dealing with this class of models, is concerned with the manner by which boundary conditions should be prescribed. Our point of view here is that classical boundary conditions, since applied on surfaces of solid bodies, are naturally associated with local models. The paper describes two methods to incorporate classical Dirichlet and Neumann boundary conditions into bondbased peridynamics. The first method consists in artificially extending the domain with a thin boundary layer over which the displacement field is required to behave as an odd function with respect to the boundary points. The second method resorts to the idea that peridynamic models and local models should be compatible in the limit that the socalled horizon vanishes. The approach consists then in decreasing the horizon from a constant value in the interior of the domain to zero at the boundary so that one can directly apply the classical boundary conditions. We present the continuous and discrete formulations of the two methods and assess their performance on several numerical experiments dealing with the simulation of a onedimensional bar.

Zoom Meeting ID: 964 9811 8679


12:30pm  2:30pm 
M20: Dynamics of Nonlinear PDE and Applications
12:30pm  1:00pm 
12:30pm  1:00pm 
Lopera, Emer: Existence Of Positive Solutions For A Semipositone $$\Phi$$ Laplacian Problem. (Abstract)
In this talk we present an overview of the $$\Phi$$Laplacian operator as well as some related boundary value problems. In particular we are interested in the study of positive solutions of semipositone problems. %[ref]

Zoom Meeting ID: 929 4166 8231

1:00pm  1:30pm 
1:00pm  1:30pm 
Suazo, Erwin: On Explicit And Numerical Solutions For Stochastic Partial Differential Equations: Fisher And Burger Type Equations (Abstract)
We will introduce exact and numerical solutions to some stochastic
Fisher and Burgers equations with variable coefficients. The solutions are found using a
coupled system of deterministic Burgers equations and stochastic differential
equations.

Zoom Meeting ID: 929 4166 8231

1:30pm  2:00pm 
1:30pm  2:00pm 
Ibdah, Hussain: Strong Solutions To A Modified MichelsonSivashinsky Equation (Abstract)
I will explain how to obtain a global wellposedness and regularity result of strong solutions to a slight modification of the so called MichelsonSivashinsky equation [ref]. Regularity is shown to persist by studying the propagation of moduli of continuity, as introduced by Kiselev, Nazarov, Volberg and Shterenberg [ref] to handle the critically dissipative SQG and Burgers equation. Namely, the Lipschitz constant of the solution is shown to be under control by constructing a modulus of continuity that must be obeyed by the solution. If time permits, I will briefly explain how one can extend such ideas to driftdiffusion systems with nonlocal source terms, such as the incompressible NSE and viscous BurgersHilbert equations.

Zoom Meeting ID: 929 4166 8231

2:00pm  2:30pm 
2:00pm  2:30pm 
Leiva, Hugo: Solvability Of Semilinear Equations In Hilbert Spaces And Applications To Control System Governed By Pdes (Abstract)
In this work we study the existence of solutions for a broad class of abstract semilinear equations in Hilbert spaces. This is done by applying Rothe's Fixed Point Theorem and a characterization of dense range linear operators in Hilbert spaces. As an applications we study the approximate controllability of a semilinear control system governed by a semilinear evolution equations, and a particular case of this is a control system governed by a semilinear heat equation with interior control.

Zoom Meeting ID: 929 4166 8231


12:30pm  2:30pm 
M25: Applications of Algebraic Geometry
12:30pm  1:00pm 
12:30pm  1:00pm 
Brysiewicz, Taylor: Nodes On Quintic Spectrahedra (Abstract)
A quintic spectrahedron in $$\mathbb{R}^3$$ is the intersection of a $$3$$dimensional affine linear subspace of $$5 \times 5$$ real symmetric matrices with the cone of positivesemidefinite matrices. The algebraic boundary of a quintic spectrahedra is an algebraic surface of degree $$5$$ in $$\mathbb{C}^3$$ called a symmetroid. Quintic symmetroids generically have $$20$$ singular points and the real singularities are partitioned into those which lie on the spectrahedra and those which do not. We determine which such partitions are possible and compute explicit spectrahedra witnessing each. We do this using numerical algebraic geometry and an augmented hillclimbing algorithm.

Zoom Meeting ID: 942 1371 3534

1:00pm  1:30pm 
1:00pm  1:30pm 
Tran, Ngoc M: A Simple And Effective Method For LowRank Completion For The Shared Response Model In Fmri Experiment Design (Abstract)
Lowrank matrix completion is a typical instance of the general problem of learning an algebraic variety from data. This talk presents a new lowrank matrix completion challenge stemming from the shared response model in the analysis fMRI data. In this application, the matrix has a special block structure dictated by a bipartite graph, which we exploit to give a simple, fast and provably optimal algorithm to learn the parameters of the shared response model under a sparse experiment design, where not all subjects are exposed to all stimuli. Our algorithm outperforms existing methods for lowrank matrix completion in simulations such as nuclear norm minimization. Our work opens up opportunities to run larger fMRI experiments on more subjects and stimuli on the same query budget, and raises interesting interesting theory questions on lowrank matrix completion, random graphs and experiment design. Joint work with Daniel Bernstein.

Zoom Meeting ID: 942 1371 3534

1:30pm  2:00pm 
1:30pm  2:00pm 
Coons, Jane: QuasiIndependence Models With Rational Maximum Likelihood Estimates (Abstract)
Let X and Y be random variables. Quasiindependence models are loglinear models that describe a situation in which some states of X and Y cannot occur together, but X and Y are otherwise independent. In [ref], we characterize which quasiindependence models have rational maximum likelihood estimate based on combinatorial features of the bipartite graph associated to the model. In this case, we give an explicit formula for the maximum likelihood estimate.

Zoom Meeting ID: 942 1371 3534

2:00pm  2:30pm 
2:00pm  2:30pm 
Kileel, Joe: Fast Symmetric Tensor Decomposition (Abstract)
We present a new method for lowrank symmetric tensor decomposition, building on Sylvester's catalecticant method from classical algebraic geometry and the power method from numerical linear algebra. The approach achieves a speedup over the stateoftheart by roughly one order of magnitude. We also sketch an ``implicit" variant of the algorithm for the case of moment tensors, which avoids the explicit formation of higherorder moments. This talk is based on joint works with Jo{\ a}o Pereira and Tammy Kolda.

Zoom Meeting ID: 942 1371 3534


2:30pm  2:45pm 
Break / discussions 
3:00pm  4:00pm 
XiaoHui Wu (ExxonMobil Upstream Integrated Solutions)
Decision Support under Subsurface Uncertainty.
Almost all decisions made in the upstream oil and gas industry, from exploration to development and production, must account for subsurface uncertainty. Despite advances in computational and data sciences, effective decision support under subsurface uncertainty remains extremely challenging. In this talk, a holistic view of the key components, both technical and cognitive, involved in the decision support process are presented. Both decision making and inference are approached from a Bayesian point of view. We review the computational challenges and some recent progresses. The need to manage computational complexity through goaloriented inference (GOI) is highlighted. In addition, we examine the challenges associated with specification of priors and validation of the efficacy of the decision process, which are of fundamental importance in practice but have received relatively little attention in research.

4:00pm  4:30pm 
Break, discussion with the plenary speaker

4:30pm  6:30pm 
M02: New Developments in PDE Constrained Optimization
4:30pm  5:00pm 
4:30pm  5:00pm 
Luo, Dingcheng: Optimal Control Of Block Copolymer Systems Governed By Nonlocal Cahn Hilliard Equations (Abstract)
Directed selfassembly (DSA) of blockcopolymers (BCPs) is one of the most promising strategies for the costeffective production of nanoscale devices [ref].
The process makes use of the natural tendency for BCP mixtures to form nanoscale structures upon phase separation.
Furthermore, this can be directed through the placement of chemically patterned substrates to promote the formation of morphologies of interest.
Adopting a nonlocal CahnHilliard equation to model the phaseseparation of BCPs, the DSA process can be cast as a PDE constrained optimization problem
in which one seeks an optimal design for the substrate pattern while respecting manufacturing constraints.
In this talk, we present a formulation of this optimization problem and discuss its computational solutions as well as the associated challenges.

Zoom Meeting ID: 937 8161 9980

5:00pm  5:30pm 
5:00pm  5:30pm 
Bornia, Giorgio: On The Treatment Of Full Dirichlet Optimal Control Problems With DivergenceFree Constraints (Abstract)
Full Dirichlet boundary optimal control problems
pose challenges when the state equations contain divergencefree constraints.
Such constraints impose integral compatibility conditions on the Dirichlet controls.
We compare two different approaches for the treatment of these compatibility conditions:
one that is based on the use of a scalar Lagrange multiplier,
the other one that uses lifting functions to treat boundary controls as distributed controls.
The differences between the two formulations are described.
Numerical results are presented for systems with divergencefree conditions,
such as the incompressible NavierStokes equations or the incompressible elasticity equations.
These results are obtained by solving the finite element approximation
of the fully coupled optimality systems
arising from the firstorder necessary conditions.
Preliminary comparisons with implementations involving fractional norms are also discussed.

Zoom Meeting ID: 937 8161 9980

5:30pm  6:00pm 
5:30pm  6:00pm 
Ridzal, Denis: Alesqp: An Augmented Lagrangian EqualityConstrained Sqp Method For FunctionSpace Optimization With General Constraints (Abstract)
We present a new algorithm for infinitedimensional optimization with general constraints, called ALESQP.
In a nutshell, ALESQP is an augmented Lagrangian method that penalizes inequality constraints and solves equalityconstrained nonlinear optimization subproblems at every iteration.
The subproblems are solved using a matrixfree trustregion sequential quadratic programming (SQP) method [ref] that takes advantage of iterative, i.e., inexact linear solvers and is suitable for PDEconstrained optimization and other largescale applications.
We analyze convergence of ALESQP under different assumptions. We show that strong accumulation points are stationary, i.e., in finite dimensions ALESQP converges to a stationary point. In infinite dimensions we establish that weak accumulation points are feasible in many practical situations. Under additional assumptions we show that weak accumulation points are stationary.
In the context of optimal control problems, e.g., in PDEconstrained optimization, ALESQP provides a unified framework to efficiently handle general constraints on both the state variables and the control variables. A key algorithmic feature is a constraint decomposition strategy that allows ALESQP to exploit problemspecific variable scalings and inner products. We present several examples with state and control inequality constraints where ALESQP shows remarkable \emph{meshindependent} performance, requiring only a handful of outer (AL) iterations to meet constraint tolerances at the level of machine precision. At the same time, ALESQP uses the inner (SQP) loop economically, requiring only a few dozen SQP iterations in total.

Zoom Meeting ID: 937 8161 9980

6:00pm  6:30pm 
6:00pm  6:30pm 
Lin, Shengchao: MultigridInTime Algorithm For Optimal Control Problems (Abstract)
This talk introduces and analyzes an efficient multigridintime parallel algorithm that reduces time and storage requirements for solving optimal control problems. The optimality system for such problems involves the forwardintime state equation coupled with a backwardintime adjoint equation. Therefore, solution of such problems is computing time and memory intensive. To introduce parallelism, I first introduce a twogrid approach based on a timedomain decomposition. This approach eliminates the variables in time subdomains, which can be done on parallel, and then replaces the resulting Schur complement by a less expensive coarse grid discretization. To further reduce the cost of the coarse grid approximation, I extended it to a multigrid algorithm using an algebraic multigrid approach. This reduces sequential computation therefore increases parallel efficiency. Finally, I will present convergence results for the twogrid algorithm and numerical result for the multigrid extension.

Zoom Meeting ID: 937 8161 9980


4:30pm  6:30pm 
M03: Nonlinear Waves and Applications
4:30pm  5:00pm 
4:30pm  5:00pm 
Ntekoume, Maria: Symplectic NonSqueezing For The Kdv Flow On The Line (Abstract)
We prove that the KdV flow on the line cannot squeeze a ball in $$\dot H^{\frac 1 2}(\mathbb R)$$ into a cylinder of lesser radius. This is a PDE analogue of Gromov’s famous symplectic nonsqueezing theorem for an infinite dimensional PDE in infinite volume.

Zoom Meeting ID: 930 7699 3802

5:00pm  5:30pm 
5:00pm  5:30pm 
Young, Giorgio: Uniqueness Of Solutions Of The KdvHierarchy Via DubrovinType Flows (Abstract)
We consider the Cauchy problem for the KdV hierarchy  a family of integrable PDEs with a Lax pair representation involving onedimensional Schrodinger operators  under a local in time boundedness assumption on the solution.
For reflectionless initial data, we prove that the solution stays reflectionless. For almost periodic initial data with absolutely continuous spectrum, we prove that under Craigtype conditions on the spectrum, Dirichlet data evolve according to a Lipschitz Dubrovintype flow, so the solution is uniquely recovered by a trace formula. This applies to algebrogeometric (finite gap) solutions; more notably, we prove that it applies to small quasiperiodic initial data with analytic sampling functions and Diophantine frequency.
This also gives a uniqueness result for the Cauchy problem on the line for periodic initial data, even in the absence of Craigtype conditions.
This is joint work with M. Lukic.

Zoom Meeting ID: 930 7699 3802

5:30pm  6:00pm 
5:30pm  6:00pm 
Parker, Ross: Instability Bubbles For MultiPulse Solutions To Hamiltonian Systems On A Periodic Domain (Abstract)
In this talk, I will look at multipulse solitary wave solutions to Hamiltonian systems which are translation invariant and reversible. The fifth order Kortewegde Vries equation is a prototypical example. In particular, I will look at the spectral stability of these solutions on a periodic domain. Using Lin's method, an implementation of the LyapunovSchmidt reduction, the spectral problem can be reduced to computing the determinant of a block matrix which encodes both eigenvalues resulting from interactions between neighboring pulses (interaction eigenvalues) and eigenvalues associated with the background state. Using this matrix, we can compute the spectrum associated with single and double pulses on a periodic domain. In addition, for periodic double pulses, we prove that brief instability bubbles form when interaction eigenvalues and background state eigenvalues collide on the imaginary axis as the periodic domain size is altered. These analytical results are in good agreement with numerical computations.

Zoom Meeting ID: 930 7699 3802

6:00pm  6:30pm 
6:00pm  6:30pm 
Zhang, Lu: An EnergyBased Discontinuous Galerkin Method For A Nonlinear Variational Wave Equation Modelling Nematic Liquid Crystal (Abstract)
We generalize the energybased discontinuous Galerkin method proposed in [ref] to compute solutions to the Cauchy problem for a nonlinear variational wave equation proposed as a model for the dynamics of nematic liquid crystals. The solution is known to form singularities in finite time. Beyond the singularity time, both conservative and dissipative Hölder continuous weak solutions exist. We present results with both energyconserving schems and energydissipating schemes.
Numerical experiments demonstrating optimal convergence in energy norm for upwind fluxes.

Zoom Meeting ID: 930 7699 3802


4:30pm  6:30pm 
M05: Dynamical Systems and Mathematical Biology
4:30pm  5:00pm 
4:30pm  5:00pm 
Thieme, Horst: DiscreteTime Population Dynamics Of Spatially
Distributed Semelparous TwoSex Populations (Abstract)
Spatially distributed populations with two sexes may face the problem that males
and females concentrate in different parts of the habitat and
mating and reproduction does not happen sufficiently often
for the population to persist.
For simplicity, to explore the impact of sexdependent dispersal
on population survival, we consider a discretetime model for a semelparous population
where individuals reproduce only once in their
lifetime, during a very short reproduction season. The dispersal
of females and males is modeled by Feller kernels [ref] and the
mating
by a homogeneous pair formation function [ref]. The spectral radius
of a homogeneous operator is established as basic reproduction
number of the population, $$\cR_0$$. If $$\cR_0 <1$$,
the extinction state is locally stable, and if $$\cR_0 >1$$
the population shows various degrees of persistence that
depend on the irreducibility properties of the dispersal
kernels. Special cases exhibit how sexbiased dispersal
affects the persistence of the population.

Zoom Meeting ID: 929 3748 4155

5:00pm  5:30pm 
5:00pm  5:30pm 
Cushing, Jim: Does Evolution Select Against Chaos? (Abstract)
Despite the ubiquity of chaotic attractors in many theoretical equations of
population dynamics, unequivocal evidence of its occurrence in biological
populations is sparse and is, for the most part, limited to populations
manipulated in laboratory settings. One of the numerous hypotheses offered
to explain this is that evolution selects against complex dynamics in favor
of equilibrium dynamics. We investigate this hypothesis by means of a
Darwinian dynamics version of the iconic Ricker difference equation. We
investigate how the threshold $$e^{2}$$ for the onset of complexity (i.e. the
destabilization of an equilibrium and a perioddoubling bifurcation cascade
to chaos) is affected by allowing the model coefficients to evolve according
to Darwinian principles. We find that when evolution is slow, the Darwinian
Ricker equation has an onset of complexity threshold larger than $$e^{2}$$ and
that, in this sense evolution, selects against complexity. On the other
hand, when evolution is fast the threshold can be less than $$e^{2}$$ and, in
this sense, evolution selects for complexity. In the latter case, the onset
of complexity is by means of a NaimarkSacker bifurcation, not a
perioddoubling bifurcation.

Zoom Meeting ID: 929 3748 4155

5:30pm  6:00pm 
5:30pm  6:00pm 
De Leenheer, Patrick: The Basic Reproduction Number For Linear Maps That Preserve A Cone (Abstract)
We review some mathematical results that are part of the folklore about the basic reproduction number, a concept that is prevalent in epidemiology and population biology. The reason why the basic reproduction number is
commonly used in applications, is that it is often easier to calculate than the spectral radius of the nonnegative matrix to which it is associated. Moreover, its value helps to establish the stability or instability of the linear recursion defined by the matrix, because, as the saying goes, `` the spectral radius of a nonnegative matrix, and its associated basic reproduction number, lie on the same side of $$1$$".
Perhaps not as wellknown in the community of mathematical biology, these results had already been obtained by Vandergraft in 1968 in [ref], and are applicable to the more general class of linear maps that preserve a cone in $$R^n$$, and not just to linear maps described by a nonnegative matrix. Note that Vandergraft's work was done well before the notion of the basic reproduction number became popular in mathematical biology, yet interestingly, Vandergraft attributes the ideas to even earlier work in optimization. We strengthen one of Vandergraft's results, albeit very slightly, using an idea in [ref] that was proposed for
linear maps which preserve the nonnegative orthant cone. Looming in the background, and grounding all the proofs of these results, is the celebrated PerronFrobenius Theorem for linear maps that preserve a cone, which is presented in a very nice, yet comprehensive way in [ref].

Zoom Meeting ID: 929 3748 4155

6:00pm  6:30pm 
6:00pm  6:30pm 
Vaidya, Naveen: Modeling Transmission Dynamics Of Covid19 In Nepal (Abstract)
While ongoing COVID19 pandemic remains disastrous all over the world, the situation of COVID19 transmission in Nepal can be considered unique because of openboarder provision with India and control programs implemented by Nepal Government. In this talk, I will present a dynamical system model to describe transmission dynamics of COVID19 in Nepal. Using our model and case data from Nepal, we compute the basic reproduction number and the effective reproduction number. Furthermore, we analyze our model to evaluate the impact of openboarder and control policies on the burden of COVID19 cases in Nepal.

Zoom Meeting ID: 929 3748 4155


4:30pm  6:30pm 
M09: Numerical Methods and Deep Learning for PDEs
4:30pm  5:00pm 
4:30pm  5:00pm 
Xiu, Dongbin: Data Driven Governing Equations Recovery With Deep Neural Networks (Abstract)
We present effective numerical algorithms for recovering unknown
governing equations from measurement data. Several recovery
strategies using deep neural networks (DNNs) are presented. We
demonstrate that residual network (ResNet) is particularly suitable
for equation discovery, as it can produce exact time integrator for
numerical prediction. We also discuss extensions to learning systems
with missing variables and learning partial differential equations.

Zoom Meeting ID: 928 3781 8963

5:00pm  5:30pm 
5:00pm  5:30pm 
Yang, Haizhao: A Few Thoughts On Deep LearningBased Scientific Computing (Abstract)
The remarkable success of deep learning in computer science has evinced potentially great applications of deep learning in computational and applied mathematics. Understanding the mathematical principles of deep learning is crucial to validating and advancing deep learningbased scientific computing. We present a few thoughts on the theoretical foundation of this topic and our methodology for designing efficient solutions of highdimensional and highly nonlinear partial differential equations, mainly focusing on the approximation and optimization of deep neural networks.

Zoom Meeting ID: 928 3781 8963

5:30pm  6:00pm 
5:30pm  6:00pm 
Hao, Wenrui: A Randomized Newton's Method For Solving Differential Equations Based On The Neural Network Discretization (Abstract)
In this talk, I will present a randomized Newton's method for solving differential equations, based on a fully connected neural network discretization. In particular, the randomized Newton's method randomly chooses equations from the overdetermined nonlinear system resulting from the neural network discretization and solves the nonlinear system adaptively. We prove theoretically that the randomized Newton's method has a quadratic convergence locally. I will also show various numerical examples, from one to highdimensional differential equations, in order to verify its feasibility and efficiency. Moreover, the randomized Newton's method can allow the neural network to "learn" multiple solutions for nonlinear systems of differential equations, such as pattern formation problems, and provides an alternative way to study the solution structure of nonlinear differential equations overall.

Zoom Meeting ID: 928 3781 8963

6:00pm  6:30pm 
6:00pm  6:30pm 
Guo, Wei: Low Rank Tensor Methods For Vlasov Simulations (Abstract)
In this talk, we present a lowrank tensor approach for solving the Vlasov equation. Among many existing challenges for Vlasov simulations (e.g. multiscale features, nonlinearity, formation of filamentation structures), the curse of dimensionality and the associated huge computational cost have been a longstanding key obstacle for realistic highdimensional simulations. In this work we propose to overcome the curse of dimensionality by dynamically
and adaptively exploring a lowrank tensor representation of Vlasov solutions in a general highdimensional setting. In particular, we develop two different approaches: one is to directly solve the unknown function, and the other is to solve the underlying flow map, aiming to obtain a lowrank approximation with optimal complexity. The performance of both proposed algorithms are benchmarked for standard VlasovPoisson/Maxwell test problems.

Zoom Meeting ID: 928 3781 8963


4:30pm  6:30pm 
M10: Recent advances in numerical methods for shallow water flows
4:30pm  5:00pm 
4:30pm  5:00pm 
Kees, Chris: A Simple Cutfem Implementation With Applications To Shallow Water Waves And FluidStructure Interaction (Abstract)
Simulating wave attenuation by natural and engineered
coastal structures with complex and deforming geometry, such as
vegetation [ref], is an increasingly important
application of coastal hydrodynamic models. Geometry for structures
of interest can come in the form of CAD, LIDAR, and photogrammetry
data, which are problematic input sources for automated,
boundaryconforming mesh generation. While a range of immersed and
embedded interface methods avoiding boundaryconforming mesh
generation have been advanced in recent decades, attainment of
optimal accuracy or practical implementation of the methods have
proven difficult. In this work, we present an alternative
implementation of the CutFEM approach [ref] for embedded
solid boundaries, which was analyzed and verified to achieve
$$O(h^2)$$ accuracy for smooth solutions of the NavierStokes equation
using linear polynomials on tetrahedra in [ref]. Our
implementation does not explicitly cut mesh cells or use special
quadrature rules. Instead, we employ polynomial approximations of
the Dirac and Heaviside functions appearing in the CutFEM
formulation following the equivalent polynomials construction from
[ref], which have the suprising property that FEM
integrals on cut cells and cut cell boundaries is exact to machine
precision. This approach yields a practical implementation for
embedding solid obstacles in unstructured FEM for a range of 2D and
3D nonlinear wave models.

Zoom Meeting ID: 997 4185 8846

5:00pm  5:30pm 
5:00pm  5:30pm 
Kazhyken, Kazbek: Discontinuous Galerkin Methods For A Dispersive Wave HydroSedimentMorphodynamic Model (Abstract)
Sediment transport and bed morphodynamic processes attract a growing engineering interest as these processes, driven by currents and waves, expose coastal infrastructure and environment to potential hazards, e.g. elements of coastal infrastructure, such as bridges, levees and piers, can become structurally compromised due to excessive sediment bed scour. Capturing these processes in a mathematical model involves combining hydrodynamic, sediment transport, and bed morphodynamic models and their twoway interactions into a single set of equations. Although a threedimensional hydrosedimentmorphodynamic model can capture the physics of the processes with a greater accuracy, their computational costs limit them to applications with shorter time and length scales. A more computationally efficient alternative is a twodimensional depthaveraged model such as the shallow water hydrosedimentmorphodynamic (SHSM) equations, which couple the nonlinear shallow water equations (NSWE) with sediment transport and bed morphodynamic models. Although the NSWE provide an accurate approximation to shallow water flow dynamics, they do not possess a capacity to capture dispersive wave effects; and, thus, the model cannot be applied in coastal regions where the dispersive effects are prevalent. To overcome this limitation of the model, the NSWE part of the SHSM equations is augmented with dispersive terms from the GreenNaghdi equations, which have the capacity to resolve the dispersive effects. The resulting set of equations forms a dispersive wave hydrosedimentmorphodynamic model. A numerical solution algorithm is proposed for the developed model based on the secondorder Strang operator splitting technique and discontinuous Galerkin methods. The solution algorithm is validated against a set of numerical experiments that model one and twodimensional dam breaks over mobile beds, and solitary wave runs over an erodible sloping beach.

Zoom Meeting ID: 997 4185 8846

5:30pm  6:00pm 
5:30pm  6:00pm 
Wu, Tong: Equilibrium Preserving Schemes For Shallow Water Models (Abstract)
Shallow water models are widely used to describe and study freesurface water flow. They are hyperbolic systems of balance laws and are usually solved by finite volume methods, which are appropriate numerical tools for computing nonsmooth solutions. One requirement when designing numerical schemes for shallow water models is to preserve a delicate balance between the flux and source terms since many physical related solutions are small perturbations of some steadystate solutions. I will present a general approach of designing wellbalanced centralupwind schemes for shallow water models, from the "lake at rest" steady states to the moving steadystates with bottom frictions and illustrate their performance on a number of numerical examples.

Zoom Meeting ID: 997 4185 8846

6:00pm  6:30pm 
6:00pm  6:30pm 
Tovar, Eric: Hyperbolic Relaxation Technique For Solving The Dispersive SerreSaintVenant Equations With Topography (Abstract)
We propose a relaxation technique for solving the dispersive SerreSaintVenant equations (also known as the Serre equations or
fully nonlinear Boussinesq equations) that
accounts for the full topography effects introduced in [ref].
This is done by revisiting the techniques introduced in
[ref] and its drystate compliant version
from [ref]. We then give a space/time approximation of the relaxed
model using continuous finite elements and explicit time stepping.
Finally, we illustrate the performance of the proposed method.

Zoom Meeting ID: 997 4185 8846


4:30pm  6:30pm 
M12: Scientific Machine Learning
4:30pm  5:00pm 
4:30pm  5:00pm 
Gildin, Eduardo: PhysicsAware DeepLearningBased Proxy Reservoir Simulation Model Equipped With State And Well Output Prediction (Abstract)
Sustainable hydrocarbon production demands complex decisionsupport strategies involving optimal production scheduling. At the core of these decisions is the prediction of reservoir performance, usually done by running computationally demanding complex reservoir simulators. As a substitute, physicsaware machine learning (ML) techniques have been used to endow datadriven proxy models with features closely related to the ones encountered in nature, especially conservation laws. They can lead to fast, reliable, and interpretable simulations used in many reservoir management workflows. In this talk, we build upon the recently developed deeplearningbased reducedorder modeling framework [ref] for fast and reliable proxy for reservoir simulation by adding a new step related to information of the inputoutput behavior (e.g., well rates) of the reservoir and not only the states (e.g., pressure and saturation). I will use here a combination of datadriven model reduction strategies and machine learning (deepneural networks  DNN) to achieve simultaneously state and inputoutput matching. Such a nonintrusive method does not need to have access to reservoir simulation internal structure, so it can be easily applied in tandem with reservoir simulations. I will show preliminary results based on an oilwater model with heterogeneous permeability, 4 injectors, and 5 producers wells. Comparisons will be made regarding training, accuracy and speedups.

Zoom Meeting ID: 950 7630 4599

5:00pm  5:30pm 
5:00pm  5:30pm 
Henriksen, Amelia: Components And Principles Of Streaming Principal Components (Abstract)
Big data is big.
A consequence of this fact, of course, is that principal component analysis is itself one of the principal components of many algorithms that process large data sets.
Unfortunately, it is much harder to choose \emph{how} to implement principal component analysis than whether or not you \emph{should} use itespecially when your data so large it has to be processed in a stream.
In this presentation, we examine key approaches to streaming principal component analysis: methods based on Oja's subspace method [ref] and methods based on random projections.
We illustrate the advantages and disadvantages of both approaches, and address specific ways to overcome some of the traditional disadvantages of Ojabased streaming PCA.
In particular, we discuss the advantages of AdaOjaa new adaptive form of Oja's subspace method that solves one of the biggest disadvantages of the algorithm: the need to choose a stepsize scheme.

Zoom Meeting ID: 950 7630 4599

5:30pm  6:00pm 
5:30pm  6:00pm 
Zhang, Zecheng: Multiagent Reinforcement Learning Accelerated Mcmc On Multiscale Inversion Problem (Abstract)
In this work, we proposed a multiagent actorcritic reinforcement learning (RL) algorithm to accelerate the multilevel Monte Carlo Markov Chain (MCMC) sampling. The policies (actors) of the agents are used to generate the proposal in the MCMC steps; and the critic which is centralized is in charge of estimating the reward to go. We apply our algorithm to solve an inverse problem with multiscales. We use generalized multiscale finite element methods (GMsFEM) as the forward solvers in evaluating posterior distribution in the multilevel rejection procedure. Our experiments show that the proposed method is a good alternative to the classical sampling process.

Zoom Meeting ID: 950 7630 4599

6:00pm  6:30pm 
6:00pm  6:30pm 
Goh, Hwan: Solving Forward And Inverse Problems With ModelAware Autoencoders (Abstract)
This work develops modelaware autoencoder networks as a new method for solving
scientific forward and inverse problems. Autoencoders are unsupervised neural
networks that are able to learn new representations of data through
appropriately selected architecture and regularization. The resulting mappings
to and from the latent representation can be used to encode and decode the data.
In our work, we set the input and output space to be space of observations of
physicallygoverned phenomena. Further, we enforce the latent space of the
autoencoder to be the parameter space of a parameter we wish to invert for. In doing so,
the decoder acts as a regularizer for learning the inverse map from
the observations to the parameter of interest. The results suggest that
regularization using a learned forward map or a numerical model of a forward map improves
the learning of the inverse map.

Zoom Meeting ID: 950 7630 4599


4:30pm  6:30pm 
M14: Spectral Theory and Mathematical Physics
4:30pm  5:00pm 
4:30pm  5:00pm 
Berkolaiko, Gregory: Locating Conical Degeneracies In The Spectra Of Parametric
SelfAdjoint Matrices (Abstract)
A generic 2parameter family of real symmetric matrices has
isolated points in the parameter space where a pair of eigenvalues
coincides. A simple iterative scheme is proposed for locating these
parameter values. The convergence is proved to be
quadratic. An extension of the scheme to complex Hermitian matrices
(with 3 parameters) and to location of triple eigenvalues (5
parameters for real symmetric matrices) is also described. Algorithm
convergence is illustrated in several examples: a real symmetric
family, a complex Hermitian family, a family of matrices with an
``avoided crossing'' (no convergence) and a 5parameter family of real
symmetric matrices with a triple eigenvalue. Joint work with Advait
Parulekar, {\tt arXiv:2001.02753}.

Zoom Meeting ID: 914 3647 5021

5:00pm  5:30pm 
5:00pm  5:30pm 
Gesztesy, Fritz: The Limiting Absorption Principle And Continuity Properties Of The Spectral Shift
Function For Massless DiracType Operators (Abstract)
We report on recent results regarding the limiting absorption principle for multidimensional,
massless Diractype operators and continuity properties of the associated spectral shift function.
This is based on various joint work with A. Carey, J. Kaad, G. Levitina, R. Nichols, D. Potapov, F. Sukochev, and
D. Zanin.

Zoom Meeting ID: 914 3647 5021

5:30pm  6:00pm 
5:30pm  6:00pm 
Kuchment, Peter: Spectral Shift Via Lateral Perturbation (Abstract)
Let $$\mathcal{H}$$ be a separable Hilbert space and $$A:\mathcal{H}\to\mathcal{H}$$ be a selfadjoint operator bounded from below. Assume that below its essential spectrum, $$A$$ has an eigenvalue $$\lambda_0$$ with the eigenfunction $$f$$. Consider further a nonnegative selfadjoint perturbation operator $$K^{*}_{0}K_0$$, where $$K_0$$ is a compact operator from $$\mathcal{H}$$ to an auxiliary Hilbert space $$\mathcal{K}$$ and $$K_0 f=0$$ (signindefinite perturbations can also be included). Thus, $$\lambda$$ is also an isolated eigenvalue of the perturbed operator $$H=A+K^{*}_{0}K_0$$, say $$\lambda=\lambda_n (A +K^{*}_{0}K_0)$$. We now allow the operator $$K_0$$ to vary, and consider the continuation of the eigenvalue $$\lambda_0$$ as a function of $$K:\Lambda(K):=\lambda(A+K^* K)$$ such that $$\Lambda(K_0 )\lambda$$. Due to the standard perturbation theory, this function is (real)analytic with respect to $$K$$. We establish that $$\Lambda(K)$$ has a critical point at $$K=K_0$$ and, if the family of variations $$K$$ is ``rich enough,'' the Morse index of this critical point is equal to the spectral shift $$\sigma$$, where $$\lambda=\lambda_n +\sigma(A)$$.

Zoom Meeting ID: 914 3647 5021

6:00pm  6:30pm 
6:00pm  6:30pm 
Ettehad, Mahmood: Three Dimensional Elastic Frames: Rigid Joint Conditions In Variational And Differential Formulation. (Abstract)
We consider elastic frames constructed out of EulerBernoulli beams. Correct vertex conditions corresponding to rigid joints have been a subject of active interest in both mathematical and structural engineering literature, with consideration usually limited to planar frames. In this talk we will describe a simple process of generating joint conditions out of the geometric description of an arbitrary threedimensional frame. The corresponding differential operator is
shown to be selfadjoint. Furthermore, in the presence of symmetry, one can restrict the operator onto reducing subspaces corresponding to irreducible representations of the symmetry group. This decomposition is demonstrated in general planar frames and in a three dimensional example with rotational symmetry.

Zoom Meeting ID: 914 3647 5021


4:30pm  6:30pm 
M15: Graph Theory
4:30pm  5:00pm 
4:30pm  5:00pm 
Milans, Kevin: Longest Path Transversals And Gallai Families (Abstract)
A \emph{longest path transveral} in a graph $$G$$ is a set of vertices $$S$$ such that every longest path in $$G$$ contains a vertex in $$S$$. Let $$\mathrm{lpt}(G)$$ be the minimum size of a longest path transversal in $$G$$. Gallai asked whether $$\mathrm{lpt}(G)=1$$ when $$G$$ is connected. The answer is no; the best known construction is due to Gr\"unbaum (1973), giving a connected graph $$G$$ with $$\mathrm{lpt}(G)=3$$. In 2014, Rautenbach and Sereni showed that $$\mathrm{lpt}(G)\le \left\lceil \frac{n}{4}  \frac{n^{2/3}}{90} \right\rceil$$ when $$G$$ is an $$n$$vertex connected graph. We show that $$\mathrm{lpt}(G)\le O(n^{3/4})$$ when $$G$$ is an $$n$$vertex connected graph. Our results also provide sublinear sets in $$G$$ which intersect all maximum subdivisions of any fixed graph $$F$$.
A family of graphs is \emph{Gallai} if every connected graph $$G$$ in the family satisfies $$\mathrm{lpt}(G)=1$$. We present progress toward a characterization of the graphs $$H$$ such that the $$H$$free graphs form a Gallai family. We also show that $$\mathrm{lpt}(G)=1$$ when $$G$$ is a sufficiently large $$k$$connected graph with independence number at most $$k + 2$$.

Zoom Meeting ID: 944 2654 9797

5:00pm  5:30pm 
5:00pm  5:30pm 
Delcourt, Michelle: Progress Towards NashWilliams' Conjecture On Triangle Decompositions (Abstract)
Partitioning the edges of a graph into edge disjoint triangles forms a triangle decomposition of the graph. A famous conjecture by NashWilliams [ref] from 1970 asserts that any sufficiently large, triangle divisible graph on $$n$$ vertices with minimum degree at least $$0.75 n$$ admits a triangle decomposition. In the light of recent results, the fractional version of this problem is of central importance. A fractional triangle decomposition is an assignment of nonnegative weights to each triangle in a graph such that the sum of the weights along each edge is precisely 1.
We show that for any graph on $$n$$ vertices with minimum degree at least $$0.827327 n$$ admits a fractional triangle decomposition. Combined with results of Barber, K\"{u}hn, Lo, and Osthus [ref], this implies that for every sufficiently large triangle divisible graph on n vertices with minimum degree at least $$0.82733 n$$ admits a triangle decomposition. This is a significant improvement over the previous asymptotic result of Dross [ref] showing the existence of fractional triangle decompositions of sufficiently large graphs with minimum degree more than $$0.9 n$$. This is joint work with Luke Postle.

Zoom Meeting ID: 944 2654 9797

5:30pm  6:00pm 
5:30pm  6:00pm 
Park, Jinyoung: Tuza's Conjecture For Random Graphs (Abstract)
A celebrated conjecture of Zs. Tuza says that in any (finite) graph, the minimum size of a cover of triangles by edges is at most twice the maximum size of a set of edgedisjoint triangles. Resolving a recent question of Bennett, Dudek, and Zerbib, we show that this is true for random graphs; more precisely: \[ \mbox{for any $$p=p(n)$$, $$\mathbb P(\mbox{$$G_{n,p}$$ satisfies Tuza's Conjecture})\rightarrow 1 $$ (as $$n\rightarrow\infty$$).} \]

Zoom Meeting ID: 944 2654 9797

6:00pm  6:30pm 
6:00pm  6:30pm 
Luo, Ruth: Forbidden Traces In Hypergraphs (Abstract)
Fix a graph $$F$$ and a hypergraph $$H$$ with $$V(F) \subseteq V(H)$$. We say that $$H$$ is an $$F$$trace if there exists a bijection $$\phi$$ between the edges of $$F$$ and the edges of $$H$$ such that for every $$xy \in E(F)$$, $$\phi(xy) \cap V(F) = \{x,y\}$$. In this talk, we show asymptotics for the maximum number of edges in an $$r$$uniform hypergraph with no copy of an $$F$$trace in terms of the generalized Tur'an number $$ex(n, K_r, F)$$. We also give better bounds for the case $$F = K_{2,t}$$. This is joint work with Zolt'an F\"uredi and Sam Spiro.

Zoom Meeting ID: 944 2654 9797


4:30pm  6:30pm 
M16: Analytic and computational approaches for metamaterial and nanoscale optics
4:30pm  5:00pm 
4:30pm  5:00pm 
Shipman, Stephen: Fano Resonance In In A Periodic Array Of Narrow Slits In Metal (Abstract)
We investigate resonant scattering by a perfectly conducting slab with periodically arranged subwavelength slits, with two slits per period. There are two classes of resonances, corresponding to poles of a scattering problem. A sequence of resonances has an imaginary part that is nonzero and on the order of the width $$\varepsilon$$ of the slits; these are associated with FabryPerot resonance. The focus of this study is another class of resonances induced by symmetry; they become real valued at normal incidence, when the Bloch wavenumber $$\kappa$$ is zero.
These are spectrally embedded eigenvalues corresponding to surface waves of the slab that lie within the radiation continuum. When $$0<\kappa\ll 1$$, the real embedded eigenvalues are perturbed into complex resonances.
We prove that Fanotype anomalies occur for the transmission of energy through the slab, and we show that the field enhancement is of order $$1/(\kappa\varepsilon)$$, which is stronger than FabryPerot resonance, which is merely order $$1/\varepsilon$$.
This work is published in [ref].

Zoom Meeting ID: 981 9720 3440

5:00pm  5:30pm 
5:00pm  5:30pm 
Li, Wei: Lorentz Resonance In The Homogenization Of Plasmonic Crystals (Abstract)
We explain the sharp Lorentz resonances in plasmonic crystals that consist of 2D nano dielectric inclusions as the interaction between resonant material properties and geometric resonances of electrostatic nature. One example of such plasmonic crystals are graphene nanosheets that are periodically arranged within a nonmagnetic bulk dielectric. We derive an analytic formula for the Lorentz resonances which decouples the geometric contribution and the frequency dependance. This formula comes rigorously from the corrector equation in the process of homogenization, and it can be used for efficient computation. This is joint work with Matthias Maier and Robert Lipton.

Zoom Meeting ID: 981 9720 3440

5:30pm  6:00pm 
5:30pm  6:00pm 
Perera, Jayasinghage Ruchira Nirmali: Band Structure For Phononic Media (Abstract)
We develop analytic representation formulas and power series to describe the band structure inside periodic phononic crystals made from high contrast inclusions. Our basic approach for this is to identify and utilize the the resonance spectrum for source free modes. By using these modes we represent solution operators associated with elastic waves inside periodic high contrast media. We then recover the convergent power series for the Bloch wave spectrum from representation formulas. The lower bound on the convergence radius is established using derived explicit conditions on the contrast. Finally, the separation of spectral branches of the dispersion relation is achieved using these conditions. [ref]

Zoom Meeting ID: 981 9720 3440

6:00pm  6:30pm 
6:00pm  6:30pm 
Margetis, Dionisios: A Flavor Of Plasmonics In The Time Domain (Abstract)
In this talk, I will discuss aspects of the propagation of electricfield pulses generated by localized sources on homogeneous and isotropic, translation invariant twodimensional (2D) materials such as monolayer graphene via linear response theory. To this end, I will introduce a spatially nonlocal evolution law for the 2D electron charge density in the quasielectrostatic approach. In this context, I will show that the related propagator, or Green's function, in spacetime can exhibit self similarity, and describe related classical solutions for the electric field.

Zoom Meeting ID: 981 9720 3440


4:30pm  6:30pm 
M19: Nonlocal models in mathematics and computation
4:30pm  5:00pm 
4:30pm  5:00pm 
Lipton, Robert: Recovery Of Linear Elastic Fracture Mechanics From Nonlocal Dynamics (Abstract)
We introduce a peridynamic model for calculating dynamic fracture as emergent phenomina. The force interaction is derived from a double well strain energy density function, resulting in a nonmonotonic material model. The material properties change in response to evolving internal forces and fracture emerges from the model.
In the limit of zero nonlocal interaction the model recovers a sharp crack evolution characterized by the classic Griffith free energy of brittle fracture with elastic deformation satisfying the linear elastic wave equation off the crack set, zero traction on crack faces and the kinetic relation between crack tip velocity and crack driving force given in [ref], [ref], [ref], [ref]. These new nonlocal models and results are reported by the authors in [ref], [ref], [ref], [ref], [ref], [ref] . This research is funded through ARO Grant W911NF1610456.

Zoom Meeting ID: 949 9643 8569

5:00pm  5:30pm 
5:00pm  5:30pm 
D'Elia, Marta: A Unified Theoretical And Computational Nonlocal Framework: Generalized Vector Calculus And MachineLearned Nonlocal Models (Abstract)
Nonlocal models provide an improved predictive capability thanks to their ability to capture effects that classical partial differential equations fail to capture. Among these effects we have multiscale behavior (e.g. in fracture mechanics) and anomalous behavior such as super and subdiffusion. These models have become incredibly popular for a broad range of applications, including mechanics, subsurface flow, turbulence, heat conduction and image processing. However, their improved accuracy comes at a price of many modeling and numerical challenges.
In this talk I will first address the problem of connecting nonlocal and fractional calculus by developing a unified theoretical framework that enables the identification of a broad class of nonlocal models [ref]. Then, I will present recently developed machinelearning techniques [ref] for nonlocal and fractional model learning. These physicsinformed, datadriven, tools allow for the reconstruction of model parameters or nonlocal kernels. Numerical tests illustrate our theoretical findings and the robustness and accuracy of our approaches.

Zoom Meeting ID: 949 9643 8569

5:30pm  6:00pm 
5:30pm  6:00pm 
Jha, Prashant: Application Of Peridynamics To Fracture In Solids And Granular Media (Abstract)
In this talk, we will present our recent work on peridynamics and its application. We consider a bondbased peridynamics with a nonlinear constitutive law relating the bondstrain to the pairwise force. For the model considered, we can show wellposedness and existence in the H\"{o}lder and Hilbert H$$^2$$ space under appropriate conditions and obtain apriori bounds on the finitedifference and finiteelement discretization. We will present the application of the model to modeI and mixedmode fracture problems. One particular topic of interest is the kinetic relation for the crack tip velocity in the peridynamics and how it relates to the local kinetic relation (LEFM theory). We will show that in the limit of vanishing nonlocality, we recover the classical kinetic relation from the peridynamics formulation. We will present numerical results that support the theory. Another application of peridynamics recently gaining much attention is in the granular media. DEM based methods can describe the interaction in particulate media very well but lack the capacity to model the intraparticle fracture. We will discuss some advances on the development of hybrid model based on peridynamics and DEM for granular media.

Zoom Meeting ID: 949 9643 8569

6:00pm  6:30pm 
6:00pm  6:30pm 
Tian, Xiaochuan: Reproducing Kernel Collocation Methods For Nonlocal Models: Asymptotic Compatibility And Numerical Stability (Abstract)
Reproducing kernel (RK) approximations are meshfree methods that construct shape unctions from sets of scattered data. We present asymptotically compatible (AC) RK collocation methods for nonlocal models that are robust under the change of the nonlocal horizon parameter. The study of convergent nonvariational AC schemes for nonlocal models was largely hindered by the lack of tools for the study of numerical stability. We show the numerical stability of a special class of RK collocation schemes by establishing connections between the collocation schemes and certain Galerkin schemes. The work applies to both nonlocal diffusion and the statebased peridynamics model [1, 2]. This is a joint work with Yu Leng, Nat Trask, and John Foster.

Zoom Meeting ID: 949 9643 8569


4:30pm  6:30pm 
M23: Clustering Analysis of Novel Corona Virus (COVID19) Cases in U.S. States and Territories
4:30pm  5:00pm 
4:30pm  5:00pm 
Parker, Marqus: Regression Analysis Of Statewide Covid19 Data In The U.S. (Abstract)
Pandemics can cause social, political, and economic turmoil that can interfere with the peoples’ lives and everyday occupations. The COVID19 pandemic is a virus spread from person to person through the release of respiratory substances generated by a cough or sneeze according to the Centers for Disease Control and Prevention (CDC).[ref] The virus’s mode of transmission has inspired the creation of global social distancing laws and transitions to a form of virtual proceedings for many professional and educational settings. Researchers have been studying the COVID19 pandemic in order to create models that predicts the total number of cases and deaths that caused by the virus. In this study, a multiple linear regression and nonlinear regression model was derived to predict the total number of COVID19 deaths since January 2020 in daily increments for each state in the United States. Multiple linear regression and Nonlinear regression models developed in this study in R and Python and the data used to plot daily U.S. state data have been generated from the Johns Hopkins University’s Github Repository. The performance of the linear regression model features a significant pvalue of 2e16 while the nonlinear regression holds a significant pvalue of 0.001. This study will assist doctors and researchers in developing methods of mitigation to the spread of the COVID19 pandemic. Based on the predictions received by the generated models, forecasting of COVID19 deaths could be observed over various period of time.

Zoom Meeting ID: 975 8979 7490

5:00pm  5:30pm 
5:00pm  5:30pm 
Mohideen, Arbaaz: Police Funding And Fatal Police Shootings In The United States (Abstract)
Fatal police shootings have been an interest in the political, social, and academic field, as there has been discrepancy of whether the police used force to protect the people. With the increased awareness of the Black Lives Matter Movement (BLM) and the movement of Defunding the Police, there are speculations of the legitimacy of police useofforce. In this research, we will make use of statistical analysis methods to see whether police funding has a significant relation with fatal police shootings. At the same time, we will try to see which other factors might correlate to rate of police shootings either positively or negatively. We made use of multiple different datasets and using Exploratory Data Analysis we were able to see the univariate and multivariate distributions of all available variables. We will perform analysis on the state and the city levels by making use of Poisson Regression which better fits the data.

Zoom Meeting ID: 975 8979 7490

5:30pm  6:00pm 
5:30pm  6:00pm 
Chhatrala, Bansi: Applications Of Graph Models And Spline Interpolation (Abstract)
In this talk, we describe a graph model for two data sets: one from the
2019 Overwatch League, which is a professional esport, and another concerning breast cancer from the UNC Lineberger Comprehensive Cancer Center. The general framework is the same in both cases. We use this model to predict information using spline interpolation on the graphs. The spline is constructed from a derivative matrix similar to one defined in [ref]. This approach is based on the spline construction in [ref], which uses the Laplacian. A key aspect of this project is the construction of the graph from the data. The distance function between parameter vectors in the data is a weighted vector norm that specifies if an edge exists in the graph as well as the weight of the edges. Our results exhibit a limiting behavior to the predictive accuracy of the model on these data sets, and we discuss potential modifications that could lead to improvements.

Zoom Meeting ID: 975 8979 7490

6:00pm  6:30pm 
6:00pm  6:30pm 
Manivannan, Chandra: Clustering Analysis Of Novel Corona Virus (Covid19) Cases In U.S. States And Territories (Abstract)
The Centers for Disease Control and Prevention (CDC) confirmed the first case of the novel corona virus (COVID19) in the United States on January 21, 2020, in the state of Washington. To date (August 31, 2020) at least 5,998,000 COVID19 cases have been reported and at least 180,000 individuals have died from the novel corona virus in the United States. This minisymposium will discuss applying hierarchical clustering techniques to cluster U.S. states and territories with respect to number of confirmed cases, number of recovered patients, and number of deaths. The resulting clusters of states and territories will prove useful to various government, healthcare, and private sector stakeholders as the clusters can help prioritize different needs for different regions [ref]. We will showcase visualizations that depict confirmed cases, recovered cases, and deaths over time. The visualizations and clusters that resulted from from study can identify resource or policy needs of various clusters (including ventilators, testing kits, masks, and lock down measures) to mitigate the spread and threat of COVID19. We will discuss analytical methods and algorithms implemented in Python as well as R, with a focus on principal component analysis (PCA). Presenting our research in a minisymposium format will allow various stakeholders in the public and private sectors, as well as academia, to view clusters of U.S. regions and make appropriate policy decisions [ref].

Zoom Meeting ID: 975 8979 7490

