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

Markus Bachmayr; Huqing Yang

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arXiv · 2506.19584

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

Markus Bachmayr·Huqing Yang

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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.

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Markus Bachmayr, Huqing Yang. 2025-06-24. Sparse and low-rank approximations of parametric elliptic PDEs: the best of both worlds. https://arxiv.org/abs/2506.19584

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