
Date Time 
Location  Speaker 
Title – click for abstract 

09/20 3:00pm 
BLOC 628 
Martin Licht 
Smooth commuting projections in rough settings: Weakly Lipschitz domains and mixed boundary conditions 

09/28 3:00pm 
BLOC 506A 
Rob Stevenson University of Amsterdam 
An optimal adaptive Fictitious Domain Method
We consider a Fictitious Domain formulation of an elliptic PDE, and solve the arising saddlepoint problem by an inexact preconditioned Uzawa iteration. Solving the arising `inner' elliptic problems with an adaptive finite element method, we prove that the overall method converges with the best possible rate. So far our results apply to twodimensional domains and lowest order finite elements (continuous piecewise linears on the fictitious domain, and piecewise constants on the boundary of the physical domain).
Joint work with S. Berrone (Torino), A. Bonito (Texas A&M), and M. Verani (Milano). 

10/18 3:00pm 
BLOC 628 
Peter Jantsch TAMU 
The Lebesgue Constant for Leja Points on Unbounded Domains
The standard Leja points are a nested sequence of points defined on a compact subset of the real line, and can be extended to unbounded domains with the introduction of a weight function $w : R \rightarrow [0, 1]$. Due to a simple recursive formulation, such abcissas show promise as a foundation for highdimensional approximation methods such as sparse grid collocation, deterministic least squares, and compressed sensing. Just as in the unweighted case of interpolation on a compact domain, we use results from potential theory to prove that the Lebesgue constant for the Leja points grows subexponentially with the number of interpolation nodes. 

10/25 3:00pm 
BLOC 628 
Giorgio Bornia Texas Tech 
Some questions arising in PDEconstrained optimal control and numerical linear algebra
The talk will be divided in two parts. In the first one, we address some issues arising in a certain class of boundary optimization problems. Common boundary optimal control problems yield a mismatch between the regularity of solutions on the domain and their boundary data. We discuss a reformulation of these problems with a lifting approach whose goal is to fix this regularity mismatch. Moreover, this approach provides additional benefits when applied to constraints characterized by compatibility conditions on the boundary data, such as those arising for the incompressible NavierStokes equations. In the second part of the talk, we discuss an analysis of preconditioning schemes for the numerical solution of RayleighBénard convection problems discretized with infsup stable finite element spaces. The analysis is carried out using a notion of fieldofvalues (FOV) equivalence between the preconditioner and the system matrix. Numerical results are discussed for both topics. 

11/01 3:00pm 
BLOC 628 
Natalia Kopteva University of Limerick, Ireland 
Fully computable a posteriori error estimators on anisotropic meshes
It is well known that anisotropic meshes offer an efficient way of computing reliable numerical approximations of solutions that exhibit sharp boundary and interior layers. Our goal is to give explicitly and fully computable a posteriori error estimates on reasonably general anisotropic meshes in the energy norm. This goal is achieved by a certain combination of explicit flux reconstruction and flux equilibration. Our approach differs from the previous work, mostly done for shaperegular meshes, in a few ways. The fluxes are equilibrated within a local patch using anisotropic weights depending on the local, possibly anisotropic, mesh geometry. Prior to the flux equilibration, divergencefree corrections are introduced for pairs of anisotropic triangles sharing a short edge. We shall also give an upper bound for the constructed estimator, in which the error constant is independent of the diameters and the aspect ratios of mesh elements, and discuss the efficiency of a posteriori error estimators on anisotropic meshes. 

11/08 3:00pm 
BLOC 628 
Diane Guignard TAMU 
A posteriori error estimation for PDEs with random input data
In this talk, we perform a posteriori error analysis for partial differential equations with uncertain input data characterized using random variables. Considering first small uncertainties, we use a perturbation approach expanding the solution of the problem with respect to a parameter ε that controls the amount of uncertainty. We derive residualbased a posteriori error estimates that control the two sources of error: the finite element discretization and the truncation in the expansion. The methodology is presented first on an elliptic equation with random coefficients and then on the steadystate NavierStokes equations on random domains. In the case of large uncertainties, we use instead the stochastic collocation method for the random space approximation. We present a residualbased a posteriori error estimate that provides an upper bound for the total error, which is composed of the finite element and the stochastic collocation errors. The stochastic error estimator is then used to drive an adaptive sparse grid algorithm. Several numerical examples are presented to illustrate the theoretical findings. 

11/15 3:00pm 
BLOC 628 
Paul Hand Rice University 
Deep Compressed Sensing
Combining principles of compressed sensing with deep neural networkbased generative image priors has recently been empirically shown to require 10X fewer measurements than traditional compressed sensing in certain scenarios. As deep generative priors (such as those obtained via generative adversarial training) improve, analogous improvements in the performance of compressed sensing and other inverse problems may be realized across the imaging sciences. In joint work with Vladislav Voroninski, we provide a theoretical framework for studying inverse problems subject to deep generative priors. In particular, we prove that with high probability, the nonconvex empirical risk objective for enforcing random deep generative priors subject to compressive random linear observations of the last layer of the generator has no spurious local minima, and that for a fixed network depth, these guarantees hold at orderoptimal sample complexity. 

11/29 3:00pm 
BLOC 628 
Laura Saavedra Universidad Politécnica de Madrid 
TBA 

12/06 3:00pm 
BLOC 628 
Chris Kees Coastal and Hydraulics Laboratory, US Army ERDC 
TBA 

12/12 3:00pm 
BLOC 506A 
Prof. Kai Diethelm Technische University at Braunschweig, Germany 
On the Principle of ``Fractionalization'' in Mathematical Modeling
Traditional mathematical models for many phenomena in various different
fields of science and engineering are based on the use of classical
differential equations, i.e. on equations containing integer order
derivatives. These models are usually well understood from an analytic
point of view, in particular regarding the qualitative behavior of their
solutions. The availability of such information is important for
evaluating whether the mathematical model really reflects
the actual properties that the process in question has, and thus for
showing that the set of equations is indeed a suitable model
for the concrete process.
In many cases, it has been observed that a generalization of the classical
models obtained by replacing the integer order derivative(s) by
a derivative of fractional (i.e., noninteger) order leads to better
quantitative agreement between the mathematical model and experimental
data, but the knowledge about qualitative properties is frequently
lacking. Thus, the question whether the fractional order model is in
fact able to correctly reproduce the behavior that the underlying
process must exhibit frequently remains unanswered. In this talk,
specific examples from the life sciences are used to demonstrate
potential approaches to handle such issues and point out possible
pitfalls in this ``fractionalization'' procedure for differential
equation based mathematical models.
Parts of the work described in this presentation are based on results
of the project READEX that has received funding
from the European Union's Horizon 2020 research and innovation
program under Grant Agreement No.\ 671657. 