How descriptive are GMRES convergence bounds?
Eigenvalues with the eigenvector condition number, the field of values, and pseudospectra have all been suggested as the basis for convergence bounds for minimum residual Krylov subspace methods applied to non-normal coefficient matrices. This paper analyzes and compares these bounds, illustrating w...
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| Format: | Report |
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Unspecified
1999
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| _version_ | 1797081112744296448 |
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| author | Embree, M |
| author_facet | Embree, M |
| author_sort | Embree, M |
| collection | OXFORD |
| description | Eigenvalues with the eigenvector condition number, the field of values, and pseudospectra have all been suggested as the basis for convergence bounds for minimum residual Krylov subspace methods applied to non-normal coefficient matrices. This paper analyzes and compares these bounds, illustrating with six examples the success and failure of each one. Refined bounds based on eigenvalues and the field of values are suggested to handle low-dimensional non-normality. It is observed that pseudospectral bounds can capture multiple convergence stages. Unfortunately, computation of pseudospectra can be rather expensive. This motivates an adaptive technique for estimating GMRES convergence based on approximate pseudospectra taken from the Arnoldi process that is the basis for GMRES. |
| first_indexed | 2024-03-07T01:09:53Z |
| format | Report |
| id | oxford-uuid:8ca2d383-4d7d-4e21-805c-98e16537d3d3 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T01:09:53Z |
| publishDate | 1999 |
| publisher | Unspecified |
| record_format | dspace |
| spelling | oxford-uuid:8ca2d383-4d7d-4e21-805c-98e16537d3d32022-03-26T22:45:50ZHow descriptive are GMRES convergence bounds?Reporthttp://purl.org/coar/resource_type/c_93fcuuid:8ca2d383-4d7d-4e21-805c-98e16537d3d3Mathematical Institute - ePrintsUnspecified1999Embree, MEigenvalues with the eigenvector condition number, the field of values, and pseudospectra have all been suggested as the basis for convergence bounds for minimum residual Krylov subspace methods applied to non-normal coefficient matrices. This paper analyzes and compares these bounds, illustrating with six examples the success and failure of each one. Refined bounds based on eigenvalues and the field of values are suggested to handle low-dimensional non-normality. It is observed that pseudospectral bounds can capture multiple convergence stages. Unfortunately, computation of pseudospectra can be rather expensive. This motivates an adaptive technique for estimating GMRES convergence based on approximate pseudospectra taken from the Arnoldi process that is the basis for GMRES. |
| spellingShingle | Embree, M How descriptive are GMRES convergence bounds? |
| title | How descriptive are GMRES convergence bounds? |
| title_full | How descriptive are GMRES convergence bounds? |
| title_fullStr | How descriptive are GMRES convergence bounds? |
| title_full_unstemmed | How descriptive are GMRES convergence bounds? |
| title_short | How descriptive are GMRES convergence bounds? |
| title_sort | how descriptive are gmres convergence bounds |
| work_keys_str_mv | AT embreem howdescriptivearegmresconvergencebounds |