Kernel-based Greedy Approximation of Parametric Elliptic Boundary Value Problems

TL;DR

This paper introduces an improved kernel-based greedy method for solving elliptic boundary value problems, achieving exponential convergence with smooth kernels and extending to parametric PDEs, demonstrating efficiency in complex scenarios.

Contribution

It extends previous work by establishing exponential convergence rates for smooth kernels and solutions, and adapts the scheme for parametric PDEs using state-parameter product kernels.

Findings

Achieves exponential convergence rates for smooth kernels and solutions.

Extends the approximation scheme to parametric PDEs with no precomputed solutions.

Demonstrates efficiency in complex, high-dimensional, and non-affine geometry problems.

Abstract

We recently introduced a scale of kernel-based greedy schemes for approximating the solutions of elliptic boundary value problems. The procedure is based on a generalized interpolation framework in reproducing kernel Hilbert spaces and was coined PDE-$\beta$-greedy procedure, where the parameter $\beta \geq 0$ is used in a greedy selection criterion and steers the degree of function adaptivity. Algebraic convergence rates have been obtained for Sobolev-space kernels and solutions of finite smoothness. We now report a result of exponential convergence rates for the case of infinitely smooth kernels and solutions. We furthermore extend the approximation scheme to the case of parametric PDEs by the use of state-parameter product kernels. In the surrogate modelling context, the resulting approach can be interpreted as an a priori model reduction approach, as no solution snapshots need to be precomputed. Numerical results show the efficiency of the approximation procedure for problems which occur as challenges for other parametric MOR procedures: non-affine geometry parametrizations, moving sources or high-dimensional domains.