Some comments on preconditioning for normal equations and least squares
The solution of systems of linear(ized) equations lies at the heart of many problems in scientific computing. In particular, for large systems, iterative methods are a primary approach. For many symmetric (or self-adjoint) systems, there are effective solution methods based on the conjugate gradient...
| Main Author: | |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
Society for Industrial and Applied Mathematics
2022
|
| _version_ | 1826308241316904960 |
|---|---|
| author | Wathen, A |
| author_facet | Wathen, A |
| author_sort | Wathen, A |
| collection | OXFORD |
| description | The solution of systems of linear(ized) equations lies at the heart of many problems in scientific computing. In particular, for large systems, iterative methods are a primary approach. For many symmetric (or self-adjoint) systems, there are effective solution methods based on the conjugate gradient method (for definite problems) or MINRES (for indefinite problems) in combination with an appropriate preconditioner, which is required in almost all cases. For nonsymmetric systems there are two principal lines of attack: the use of a nonsymmetric iterative method such as GMRES or transformation into a symmetric problem via the normal equations and application of LSQR. In either case, an appropriate preconditioner is generally required. We consider the possibilities here, particularly the idea of preconditioning the normal equations via approximations to the original nonsymmetric matrix. We highlight dangers that readily arise in this approach. Our comments also apply in the context of linear least squares problems.
|
| first_indexed | 2024-03-07T07:15:06Z |
| format | Journal article |
| id | oxford-uuid:4f421c4c-16f1-49d4-ba6a-8b48167719bc |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:15:06Z |
| publishDate | 2022 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:4f421c4c-16f1-49d4-ba6a-8b48167719bc2022-08-15T08:51:03ZSome comments on preconditioning for normal equations and least squaresJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:4f421c4c-16f1-49d4-ba6a-8b48167719bcEnglishSymplectic ElementsSociety for Industrial and Applied Mathematics2022Wathen, AThe solution of systems of linear(ized) equations lies at the heart of many problems in scientific computing. In particular, for large systems, iterative methods are a primary approach. For many symmetric (or self-adjoint) systems, there are effective solution methods based on the conjugate gradient method (for definite problems) or MINRES (for indefinite problems) in combination with an appropriate preconditioner, which is required in almost all cases. For nonsymmetric systems there are two principal lines of attack: the use of a nonsymmetric iterative method such as GMRES or transformation into a symmetric problem via the normal equations and application of LSQR. In either case, an appropriate preconditioner is generally required. We consider the possibilities here, particularly the idea of preconditioning the normal equations via approximations to the original nonsymmetric matrix. We highlight dangers that readily arise in this approach. Our comments also apply in the context of linear least squares problems. |
| spellingShingle | Wathen, A Some comments on preconditioning for normal equations and least squares |
| title | Some comments on preconditioning for normal equations and least squares |
| title_full | Some comments on preconditioning for normal equations and least squares |
| title_fullStr | Some comments on preconditioning for normal equations and least squares |
| title_full_unstemmed | Some comments on preconditioning for normal equations and least squares |
| title_short | Some comments on preconditioning for normal equations and least squares |
| title_sort | some comments on preconditioning for normal equations and least squares |
| work_keys_str_mv | AT wathena somecommentsonpreconditioningfornormalequationsandleastsquares |