Algorithms for the equilibration of matrices and their application to limited-memory quasi-newton methods
Abstract/Contents
- Abstract
- Diagonally scaling a matrix can reduce its condition number. Equilibration scales a matrix so that the row and column norms are equal. We review the existence and uniqueness theory for exact equilibration. Then we introduce a formalization of approximate equilibration and develop its existence and uniqueness theory. Next we develop approximate equilibration algorithms that access a matrix only by matrix-vector products. We address both the nonsymmetric and symmetric cases. Limited-memory quasi-Newton methods may be thought of as changing the metric so that the steepest-descent method works effectively on the problem. Quasi-Newton methods construct a matrix using vectors of two types involving the iterates and gradients. The vectors are related by an approximate matrix-vector product. Using our approximate matrix-free symmetric equilibration method, we develop a limited-memory quasi-Newton method in which one part of the quasi-Newton matrix approximately equilibrates the Hessian. Often a differential equation is solved by discretizing it on a sequence of increasingly fine meshes. This technique can be used when solving differential-equation-constrained optimization problems. We describe a method to interpolate our limited-memory quasi-Newton matrix from a coarse to a fine mesh.
Description
| Type of resource | text |
|---|---|
| Form | electronic; electronic resource; remote |
| Extent | 1 online resource. |
| Publication date | 2010 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Associated with | Bradley, Andrew Michael |
|---|---|
| Associated with | Stanford University, Department of Computational and Mathematical Engineering. |
| Primary advisor | Murray, Walter |
| Thesis advisor | Murray, Walter |
| Thesis advisor | Saunders, Michael A |
| Thesis advisor | Ye, Yinyu |
| Advisor | Saunders, Michael A |
| Advisor | Ye, Yinyu |
Subjects
| Genre | Theses |
|---|
Bibliographic information
| Statement of responsibility | Andrew Michael Bradley. |
|---|---|
| Note | Submitted to the Department of Computational and Mathematical Engineering. |
| Thesis | Ph. D. Stanford University 2010 |
| Location | electronic resource |
Access conditions
- Copyright
- © 2010 by Andrew Michael Bradley
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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