Learning Elliptic Partial Differential Equations with Randomized Linear Algebra
CC BY
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| Ngôn ngữ: | English |
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Springer
2023
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| Truy cập trực tuyến: | https://link.springer.com/article/10.1007/s10208-022-09556-w https://dlib.phenikaa-uni.edu.vn/handle/PNK/7458 |
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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 |