Multiscale algorithms for structured matrices : fast solvers and deep learning
Abstract/Contents
- Abstract
- This dissertation introduces fast factorizations for sparse and dense matrices, allowing to represent or invert pseudo-differential and Fourier integral operators. This thesis is organized in the following three main parts. In the first part we develop fast and robust hierarchical solvers for sparse matrices based on the hierarchical interpolative factorization (HIF). HIF is a fast algorithm for approximate sparse matrix inversion whose accuracy can degrade, however, when applied to strongly ill-conditioned problems. In this work we introduce the recursively preconditioned interpolative factorization which can significantly improve the accuracy at no additional asymptotic cost. This dramatically limits the impact of the underlying system conditioning and enables the construction of robust and highly efficient preconditioners. Most state-of-the-art fast solvers for sparse matrices focus on the SPD case. In this work we also develop a new strategy for factorizing unsymmetric matrices by providing optimal compression of fill-ins. In the second part, we introduce a machine learning approach to parameterize pseudo-differential operators with deep neural networks. With the help of the nonstandard wavelet form, the pseudo-differential operators can be approximated in a compressed form with a collection of vectors. The nonlinear map from the parameter to this collection of vectors and the wavelet transform are learned together from a small number of matrix-vector multiplications of the pseudo-differential operator. Numerical results for Green's functions of elliptic partial differential equations and the radiative transfer equations demonstrate the efficiency and accuracy of the proposed approach. Lastly, the third part introduces a factorization for the inverse of discrete Fourier integral operators that can be applied in quasi-linear time. The resulting approximate inverse factorization can be used as a direct solver or as a preconditioner. Numerical examples on 1D and 2D Fourier integral operators, including a generalized Radon transform, demonstrate the efficiency of the factorization.
Description
| Type of resource | text |
|---|---|
| Form | electronic resource; remote; computer; online resource |
| Extent | 1 online resource. |
| Place | California |
| Place | [Stanford, California] |
| Publisher | [Stanford University] |
| Copyright date | 2021; ©2021 |
| Publication date | 2021; 2021 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Author | Feliu Fabà, Jordi |
|---|---|
| Degree supervisor | Ying, Lexing |
| Thesis advisor | Ying, Lexing |
| Thesis advisor | Darve, Eric |
| Thesis advisor | Iaccarino, Gianluca |
| Degree committee member | Darve, Eric |
| Degree committee member | Iaccarino, Gianluca |
| Associated with | Stanford University, Institute for Computational and Mathematical Engineering |
Subjects
| Genre | Theses |
|---|---|
| Genre | Text |
Bibliographic information
| Statement of responsibility | Jordi Feliu Fabà. |
|---|---|
| Note | Submitted to the Institute for Computational and Mathematical Engineering. |
| Thesis | Thesis Ph.D. Stanford University 2021. |
| Location | https://purl.stanford.edu/xv565rm8453 |
Access conditions
- Copyright
- © 2021 by Jordi Feliu Faba
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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