Preconditioning in implicit initial value problem methods on parallel computers:
Abstract: "Implicit step-by-step methods for numerically solving the initial-value problem [y' = f(y), y(0) = y b0 s] usually lead to implicit relations of which the Jacobian can be approximated by a matrix of the special form K = I - hM[symbol]J, where M is a matrix characterizing the ste...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Amsterdam
1992
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| Schriftenreihe: | Centrum voor Wiskunde en Informatica <Amsterdam> / Afdeling Numerieke Wiskunde: Report NM
1992,16 |
| Schlagworte: | |
| Zusammenfassung: | Abstract: "Implicit step-by-step methods for numerically solving the initial-value problem [y' = f(y), y(0) = y b0 s] usually lead to implicit relations of which the Jacobian can be approximated by a matrix of the special form K = I - hM[symbol]J, where M is a matrix characterizing the step-by-step method and J is the Jacobian of f. Similar implicit relations are encountered in descretizing initial-value problems for other types of functional equations such as VIEs, VIDs and DDEs. Application of (modified) Newton iteration for solving these implicit relations requires the LU-decomposition of K. If s and d are the dimensions of M and J, respectively, then this LU-decomposition is an O(s p3 sd p3 s)-process, which is extremely costly for large values of sd We shall discuss parallel iteration methods for solving the implicit relations that exploit the special form of Jacobian matrix K. Their main characteristic is that each processor is required to compute LU- decompositions of matrices of dimension d, so that this part of the computational work is reduced by a factor s p3 s. On the other hand, the number of iterations in these parallel iteration methods is usually much larger than in Newton iteration. In this contribution, we will try to reduce the number of iterations by improving the convergence of such parallel iteration methods by means of preconditioning. |
| Beschreibung: | 14 S. |
Internformat
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| 100 | 1 | |a Houwen, Pieter J. van der |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Preconditioning in implicit initial value problem methods on parallel computers |c P. J. van der Houwen |
| 264 | 1 | |a Amsterdam |c 1992 | |
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| 490 | 1 | |a Centrum voor Wiskunde en Informatica <Amsterdam> / Afdeling Numerieke Wiskunde: Report NM |v 1992,16 | |
| 520 | 3 | |a Abstract: "Implicit step-by-step methods for numerically solving the initial-value problem [y' = f(y), y(0) = y b0 s] usually lead to implicit relations of which the Jacobian can be approximated by a matrix of the special form K = I - hM[symbol]J, where M is a matrix characterizing the step-by-step method and J is the Jacobian of f. Similar implicit relations are encountered in descretizing initial-value problems for other types of functional equations such as VIEs, VIDs and DDEs. Application of (modified) Newton iteration for solving these implicit relations requires the LU-decomposition of K. If s and d are the dimensions of M and J, respectively, then this LU-decomposition is an O(s p3 sd p3 s)-process, which is extremely costly for large values of sd | |
| 520 | 3 | |a We shall discuss parallel iteration methods for solving the implicit relations that exploit the special form of Jacobian matrix K. Their main characteristic is that each processor is required to compute LU- decompositions of matrices of dimension d, so that this part of the computational work is reduced by a factor s p3 s. On the other hand, the number of iterations in these parallel iteration methods is usually much larger than in Newton iteration. In this contribution, we will try to reduce the number of iterations by improving the convergence of such parallel iteration methods by means of preconditioning. | |
| 650 | 4 | |a Numerical analysis | |
| 810 | 2 | |a Afdeling Numerieke Wiskunde: Report NM |t Centrum voor Wiskunde en Informatica <Amsterdam> |v 1992,16 |w (DE-604)BV010177152 |9 1992,16 | |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-006766869 | ||
Datensatz im Suchindex
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| author | Houwen, Pieter J. van der |
| author_facet | Houwen, Pieter J. van der |
| author_role | aut |
| author_sort | Houwen, Pieter J. van der |
| author_variant | p j v d h pjvd pjvdh |
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| bvnumber | BV010186056 |
| ctrlnum | (OCoLC)29451541 (DE-599)BVBBV010186056 |
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| id | DE-604.BV010186056 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:48:01Z |
| institution | BVB |
| language | English |
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| physical | 14 S. |
| publishDate | 1992 |
| publishDateSearch | 1992 |
| publishDateSort | 1992 |
| record_format | marc |
| series2 | Centrum voor Wiskunde en Informatica <Amsterdam> / Afdeling Numerieke Wiskunde: Report NM |
| spelling | Houwen, Pieter J. van der Verfasser aut Preconditioning in implicit initial value problem methods on parallel computers P. J. van der Houwen Amsterdam 1992 14 S. txt rdacontent n rdamedia nc rdacarrier Centrum voor Wiskunde en Informatica <Amsterdam> / Afdeling Numerieke Wiskunde: Report NM 1992,16 Abstract: "Implicit step-by-step methods for numerically solving the initial-value problem [y' = f(y), y(0) = y b0 s] usually lead to implicit relations of which the Jacobian can be approximated by a matrix of the special form K = I - hM[symbol]J, where M is a matrix characterizing the step-by-step method and J is the Jacobian of f. Similar implicit relations are encountered in descretizing initial-value problems for other types of functional equations such as VIEs, VIDs and DDEs. Application of (modified) Newton iteration for solving these implicit relations requires the LU-decomposition of K. If s and d are the dimensions of M and J, respectively, then this LU-decomposition is an O(s p3 sd p3 s)-process, which is extremely costly for large values of sd We shall discuss parallel iteration methods for solving the implicit relations that exploit the special form of Jacobian matrix K. Their main characteristic is that each processor is required to compute LU- decompositions of matrices of dimension d, so that this part of the computational work is reduced by a factor s p3 s. On the other hand, the number of iterations in these parallel iteration methods is usually much larger than in Newton iteration. In this contribution, we will try to reduce the number of iterations by improving the convergence of such parallel iteration methods by means of preconditioning. Numerical analysis Afdeling Numerieke Wiskunde: Report NM Centrum voor Wiskunde en Informatica <Amsterdam> 1992,16 (DE-604)BV010177152 1992,16 |
| spellingShingle | Houwen, Pieter J. van der Preconditioning in implicit initial value problem methods on parallel computers Numerical analysis |
| title | Preconditioning in implicit initial value problem methods on parallel computers |
| title_auth | Preconditioning in implicit initial value problem methods on parallel computers |
| title_exact_search | Preconditioning in implicit initial value problem methods on parallel computers |
| title_full | Preconditioning in implicit initial value problem methods on parallel computers P. J. van der Houwen |
| title_fullStr | Preconditioning in implicit initial value problem methods on parallel computers P. J. van der Houwen |
| title_full_unstemmed | Preconditioning in implicit initial value problem methods on parallel computers P. J. van der Houwen |
| title_short | Preconditioning in implicit initial value problem methods on parallel computers |
| title_sort | preconditioning in implicit initial value problem methods on parallel computers |
| topic | Numerical analysis |
| topic_facet | Numerical analysis |
| volume_link | (DE-604)BV010177152 |
| work_keys_str_mv | AT houwenpieterjvander preconditioninginimplicitinitialvalueproblemmethodsonparallelcomputers |