| Date: | January 23, 2019 |
| Time: | 1:00pm |
| Location: | Mitchell Institu |
| Speaker: | A. Cohen, E. Tadmor and W. Dahmen, mini-symposium |
| Title: | ``Challenges in Computational Mathematics’' |
| Abstract: | Albert Cohen, Optimal non-intrusive methods in high dimension;
Wolfgang Dahmen, Error controlled Computation - the merit of residuals;
Eitan Tadmor, Emergent behavior in collective dynamics.
More information can be found at
http://gain.math.tamu.edu/Compmath2019.html
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| Date: | March 6, 2019 |
| Time: | 4:00pm |
| Location: | BLOC 628 |
| Speaker: | Uwe Kocher, Helmut-Schmidt-University, Hamburg |
| Title: | Numerical simulation of coupled flow and deformation in porous media with space-time methods |
| Abstract: | The efficient and accurate simulation of coupled flow and deformation in porous media in space and time is of fundamental importance in many engineering fields. The quasi-static and dynamic poroelastic models appear for instance in simulation studies to support the development of next-generation batteries. Such must support fast-charging and fast-draining with currents of a factor of at least 100 or more compared to nowadays cutting-edge technologies. Future generation numerical simulation tools must incorporate multiphysics phenomena in which sharp concentration and pressure gradients, high mechanical stresses, elastic wave propagation, memory-effects on the media parameters, multi-phase behavior, crack propagation as well as electro-chemical reactions occur. In this contribution high-order space-time discretisations, including mixed finite elements (MFEM) for the flow variables and interior-penalty discontinuous Galerkin finite elements (IPDG) for the displacement and velocity variables, are presented. The arising linear block systems are solved with a sophisticated monolithic solver technology with flexible multi-step fixed-stress physical preconditioning. Inside the preconditioner highly optimized system solvers for low order approximations can be used. Additionally, our solver technology allows for parallel-in-time application. The performance properties of the solver and for further applications are illustrated by numerical experiments. |
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| Date: | March 8, 2019 |
| Time: | 12:45pm |
| Location: | BLOC 628 |
| Speaker: | Sara Pollock, University of Florida |
| Title: | Anderson acceleration improves the convergence rate in linearly converging fixed point methods |
| Abstract: | The extrapolation method known as Anderson acceleration has been used for decades to speed up nonlinear solvers in many applications, however a mathematical justification of the improved convergence rate has remained elusive. Here, we provide theory to establish the improved convergence rate. The key ideas of the analysis are relating the difference of consecutive iterates to residuals based on performing the inner-optimization in a Hilbert space setting, and explicitly defining the gain in the optimization stage to be the ratio of improvement over a step of the unaccelerated fixed point iteration. The main result we prove is that this method of acceleration improves the convergence rate of a fixed point iteration to first order by a factor of the gain at each step. |
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| Date: | March 20, 2019 |
| Time: | 4:00pm |
| Location: | BLOC 220 |
| Speaker: | Antoine Mellet, UMD |
| Title: | Anomalous Diffusion Phenomena: A Kinetic Approach |
| Abstract: | The derivation of diffusion or drift-diffusion equations from transport equations (such as Vlasov-Fokker-Planck or Boltzmann equations) is a classical problem. In this talk, we will discuss situations in which the usual derivation fails because the mean squared displacement of the particles does not grow linearly with
time. We will show that such "anomalous diffusion" regimes typically lead to fractional diffusion equations. We will present results in both bounded and unbounded domain and we will discuss some applications to the description of anomalous energy transport in chains of non-harmonic oscillators (FPU-alpha and FPU-beta chains).
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| Date: | March 27, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Serge Prudhomme, Polytechnique Montreal |
| Title: | A goal-oriented formulation for reduced-order modeling |
| Abstract: | The subject of the talk will be concerned with a mathematical formulation for constructing reduced-order models optimized with respect to quantities of interest. The main idea is to formulate a minimization problem that includes an equality, or inequality constraint on the error in the goal functional so that the resulting model is capable of delivering predictions of the quantity of interest within some prescribed tolerance. The formulation will be tested on the so-called Proper Generalized Decomposition (PGD) method. Such a paradigm represents a departure from classical goal-oriented approaches in which a reduced model is first derived by minimization of the energy, or of residual functionals, and then adapted via a greedy approach by controlling the error with respect to quantities of interest using dual-based error estimates. Numerical examples will be presented in order to demonstrate the efficiency of the proposed approach. In particular, we will consider the simulation of a delaminated composite material by the Proper Generalized Decomposition approach. |
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| Date: | April 24, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Soeren Bartels, University of Freiburg |
| Title: | Approximating gradient flow evolutions of self-avoiding inextensible curves and elastic knots |
| Abstract: | We discuss a semi-implicit numerical scheme that allows for minimizing the bending energy of curves within certain isotopy classes. To this end we consider a weighted sum of a bending energy and a so-called tangent-point functional. We define evolutions via the gradient flow for the total energy within a class of arclength parametrized curves, i.e., given an initial curve we look for a family of inextensible curves that solves the nonlinear evolution equation. Our numerical approximation scheme for the evolution problem is specified via a semi-implicit discretization of the equation with an explicit treatment of the tangent-point functional and a linearization of the arclength condition. The scheme leads to sparse systems of linear equations in the time steps for cubic splines and a nodal treatment of the constraints. The explicit treatment of the nonlocal and nonlinear tangent-point functional avoids working with fully populated matrices and furthermore allows for a straightforward parallelization of its computation. Based on estimates for the second derivative of the tangent-point functional and a uniform bi-Lipschitz radius, we prove a stability result implying energy decay during the evolution as well as maintenance of arc length parametrization. We present some numerical experiments exploring the energy landscape, targeted to the question how to obtain global minimizers of the bending energy in certain knot classes, so-called elastic knots. This is joint work with Philipp Reiter (University of Georgia). |
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