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Learning elliptic partial differential equations with randomized linear algebra

Abstract:

Given input-output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically-rigorous scheme for learning the associated Green's function $G$. By exploiting the hierarchical low-rank structure of $G$, we show that one can construct an approximant to $G$ that converges almost surely and achieves an expected relative error of $\epsilon$ using at most $\mathcal{O}(\epsilon^{-6}\log^4(1/\epsilon)/\Gamma_\epsilon)$ input-output training pairs, ...

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Publication status:
Published
Peer review status:
Peer reviewed

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Publisher copy:
10.1007/s10208-022-09556-w

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Institution:
University of Oxford
Division:
MPLS
Department:
Mathematical Institute
Role:
Author
Publisher:
Springer
Journal:
Foundations of Computational Mathematics More from this journal
Volume:
23
Issue:
2
Pages:
709-739
Publication date:
2022-01-18
Acceptance date:
2021-11-20
DOI:
EISSN:
1615-3383
ISSN:
1615-3375
Language:
English
Keywords:
Pubs id:
1161835
Local pid:
pubs:1161835
Deposit date:
2021-11-21

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