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Learning Elliptic Partial Differential Equations with Randomized Linear Algebra

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Bibliographic Details
Main Authors: Nicolas, Boullé, Alex, Townsend
Format: Book
Language:English
Published: Springer 2023
Subjects:
PDE
error of O(Γ−1/2ϵlog3(1/ϵ)ϵ) using
Online Access:https://link.springer.com/article/10.1007/s10208-022-09556-w
https://dlib.phenikaa-uni.edu.vn/handle/PNK/7458
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spelling oai:localhost:PNK-74582023-04-04T01:17:41Z Learning Elliptic Partial Differential Equations with Randomized Linear Algebra Nicolas, Boullé Alex, Townsend PDE error of O(Γ−1/2ϵlog3(1/ϵ)ϵ) using CC BY 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 a relative error of O(Γ−1/2ϵlog3(1/ϵ)ϵ) using at most O(ϵ−6log4(1/ϵ)) input–output training pairs with high probability, for any 0<ϵ<1. The quantity 0<Γϵ≤1 characterizes the quality of the training dataset. Along the way, we extend the randomized singular value decomposition algorithm for learning matrices to Hilbert–Schmidt operators and characterize the quality of covariance kernels for PDE learning. 2023-04-04T01:17:41Z 2023-04-04T01:17:41Z 2022 Book https://link.springer.com/article/10.1007/s10208-022-09556-w https://dlib.phenikaa-uni.edu.vn/handle/PNK/7458 en application/pdf Springer
institution Digital Phenikaa
collection Digital Phenikaa
language English
topic PDE
error of O(Γ−1/2ϵlog3(1/ϵ)ϵ) using
spellingShingle PDE
error of O(Γ−1/2ϵlog3(1/ϵ)ϵ) using
Nicolas, Boullé
Alex, Townsend
Learning Elliptic Partial Differential Equations with Randomized Linear Algebra
description CC BY
format Book
author Nicolas, Boullé
Alex, Townsend
author_facet Nicolas, Boullé
Alex, Townsend
author_sort Nicolas, Boullé
title Learning Elliptic Partial Differential Equations with Randomized Linear Algebra
title_short Learning Elliptic Partial Differential Equations with Randomized Linear Algebra
title_full Learning Elliptic Partial Differential Equations with Randomized Linear Algebra
title_fullStr Learning Elliptic Partial Differential Equations with Randomized Linear Algebra
title_full_unstemmed Learning Elliptic Partial Differential Equations with Randomized Linear Algebra
title_sort learning elliptic partial differential equations with randomized linear algebra
publisher Springer
publishDate 2023
url https://link.springer.com/article/10.1007/s10208-022-09556-w
https://dlib.phenikaa-uni.edu.vn/handle/PNK/7458
_version_ 1762274896423944192
score 8.881002

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