IAS Seminar "Metric-Based Optimization of Approximation Spaces for PDE Problems"

Speaker:

Prof. Georg May, Aachen Institute for Advanced Study in Computational Engineering Science AICES, RWTH Aachen University

Abstract:

One of the most fundamental goals in the numerical solution of PDEs is to achieve the best possible approximation, given the available resources. Here we investigate this problem from an approximation theory perspective. The goal is to optimize piecewise polynomial approximation spaces, defined on simplex meshes, and constraint by the number of degrees-of-freedom. The cost function can either be the error, measured in some norm, or the error in more general solution-dependent functionals.

As a discrete optimization problem, this task is hardly tractable by analytic means. We use a continuous-mesh framework to convert the problem into a formulation which may be tackled using relatively simple calculus of variations. A continuous mesh is essentially a metric tensor field, defined for each point in the computational domain. Using suitable error models, the metric field may be optimized using analytical methods. Subsequently, a discrete simplex mesh can be generated by requiring that mesh elements be (approximately) equilateral under the Riemannian metric induced by the tensor field.

This approach leads to tailor-suited anisotropic meshes for a given problem. It is sufficiently general to be used for a variety of problems and with many solvers. We show examples in the context of convection-diffusion problems, including highly anisotropic convection-dominated cases.

Prof. Georg May was invited by Prof. Dr. Dr. Thomas Lippert (JSC)