Sparse and low-rank approximations of parametric elliptic PDEs: the best of both worlds
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| Format: | Preprint |
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2025
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| _version_ | 1866916809690578944 |
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| author | Bachmayr, Markus Yang, Huqing |
| author_facet | Bachmayr, Markus Yang, Huqing |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_19584 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sparse and low-rank approximations of parametric elliptic PDEs: the best of both worlds Bachmayr, Markus Yang, Huqing Numerical Analysis 41A46, 41A63, 42C10, 65D99, 65J10, 65N12, 65N15 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. |
| title | Sparse and low-rank approximations of parametric elliptic PDEs: the best of both worlds |
| topic | Numerical Analysis 41A46, 41A63, 42C10, 65D99, 65J10, 65N12, 65N15 |
| url | https://arxiv.org/abs/2506.19584 |