Hierarchical approximate factorization methods for the solution of sparse linear systems with application in geoscience
Abstract/Contents
- Abstract
- Hierarchical approximate factorization methods are a recent class of approaches to solving sparse linear systems such as those arising from discretized partial differential equations (PDEs). These methods sparsify the fill-in matrix blocks that arise in block Gaussian elimination (assuming that these blocks are low-rank) to obtain an approximate factorization of the given matrix. In the first part of the dissertation, we describe an approach to improving the accuracy of hierarchical approximate factorization methods on the eigenvectors corresponding to the smallest eigenvalues, which is critical for achieving fast convergence of the Krylov subspace methods. We exploit the fact that for a large class of problems, including many elliptic equations, the eigenvectors corresponding to the smallest eigenvalues are smooth functions of the PDE grid. In the second part, we describe an approach to obtaining an additional order of accuracy in the sparsification of the low-rank matrix blocks. To achieve fast convergence, such sparsification must introduce only a small amount of error. In our new approach, the 2-norm of the incurred error in the sparsification of a matrix block is squared, compared to previous approaches, with a small amount of additional computation. The third part is concerned with developing efficient hierarchical factorization methods for symmetric quasi-definite systems such as those arising from coupled poromechanics, as well as saddle-point systems that arise from fractured mechanics. While for symmetric positive definite matrices the existing hierarchical factorization methods are quite efficient, highly indefinite matrices pose a bigger challenge. We develop novel methods that exploit the global (quasi-definite or saddle-point) structure of the matrix for speed, accuracy, and stability.
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 | Klockiewicz, Bazyli Jan |
|---|---|
| Degree supervisor | Darve, Eric |
| Thesis advisor | Darve, Eric |
| Thesis advisor | Saunders, Michael A |
| Thesis advisor | Tchelepi, Hamdi |
| Degree committee member | Saunders, Michael A |
| Degree committee member | Tchelepi, Hamdi |
| Associated with | Stanford University, Institute for Computational and Mathematical Engineering |
Subjects
| Genre | Theses |
|---|---|
| Genre | Text |
Bibliographic information
| Statement of responsibility | Bazyli Klockiewicz. |
|---|---|
| Note | Submitted to the Institute for Computational and Mathematical Engineering. |
| Thesis | Thesis Ph.D. Stanford University 2021. |
| Location | https://purl.stanford.edu/jb111ff2523 |
Access conditions
- Copyright
- © 2021 by Bazyli Jan Klockiewicz
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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