Sparse and low-rank approximations of parametric elliptic PDEs: the best of both worlds

TL;DR

This paper introduces a hybrid approximation method combining low-rank tensors and sparse polynomials to efficiently solve parametric elliptic PDEs, especially with complex random coefficients, achieving optimal convergence and computational efficiency.

Contribution

It presents a novel approximation format that combines low-rank tensor and sparse polynomial expansions, along with an adaptive solver for parametric PDEs with complex coefficients.

Findings

Achieves quasi-optimal ranks in approximations

Yields optimal convergence rates for spatial discretizations

Demonstrates effectiveness through numerical tests

Abstract

A new approximation format for solutions of partial differential equations depending on infinitely many parameters is introduced. By combining low-rank tensor approximation in a selected subset of variables with a sparse polynomial expansion in the remaining parametric variables, it addresses in particular classes of elliptic problems where a direct polynomial expansion is inefficient, such as those arising from random diffusion coefficients with short correlation length. A convergent adaptive solver is proposed and analyzed that maintains quasi-optimal ranks of approximations and at the same time yields optimal convergence rates of spatial discretizations without coarsening. The results are illustrated by numerical tests.