Convergence of Iterations for Linear Equations:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Basel
Birkhäuser Basel
1993
|
| Schriftenreihe: | Lectures in Mathematics ETH Zürich, Department of Mathematics Research Institute of Mathematics Department of Mathematics Research Institute of Mathematics
|
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | Assume that after preconditioning we are given a fixed point problem x = Lx + f (*) where L is a bounded linear operator which is not assumed to be symmetric and f is a given vector. The book discusses the convergence of Krylov subspace methods for solving fixed point problems (*), and focuses on the dynamical aspects of the iteration processes. For example, there are many similarities between the evolution of a Krylov subspace process and that of linear operator semigroups, in particular in the beginning of the iteration. A lifespan of an iteration might typically start with a fast but slowing phase. Such a behavior is sublinear in nature, and is essentially independent of whether the problem is singular or not. Then, for nonsingular problems, the iteration might run with a linear speed before a possible superlinear phase. All these phases are based on different mathematical mechanisms which the book outlines. The goal is to know how to precondition effectively, both in the case of "numerical linear algebra" (where one usually thinks of first fixing a finite dimensional problem to be solved) and in function spaces where the "preconditioning" corresponds to software which approximately solves the original problem |
| Beschreibung: | 1 Online-Ressource |
| ISBN: | 9783034885478 9783764328658 |
| DOI: | 10.1007/978-3-0348-8547-8 |
Internformat
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| 500 | |a Assume that after preconditioning we are given a fixed point problem x = Lx + f (*) where L is a bounded linear operator which is not assumed to be symmetric and f is a given vector. The book discusses the convergence of Krylov subspace methods for solving fixed point problems (*), and focuses on the dynamical aspects of the iteration processes. For example, there are many similarities between the evolution of a Krylov subspace process and that of linear operator semigroups, in particular in the beginning of the iteration. A lifespan of an iteration might typically start with a fast but slowing phase. Such a behavior is sublinear in nature, and is essentially independent of whether the problem is singular or not. Then, for nonsingular problems, the iteration might run with a linear speed before a possible superlinear phase. All these phases are based on different mathematical mechanisms which the book outlines. The goal is to know how to precondition effectively, both in the case of "numerical linear algebra" (where one usually thinks of first fixing a finite dimensional problem to be solved) and in function spaces where the "preconditioning" corresponds to software which approximately solves the original problem | ||
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Datensatz im Suchindex
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| author | Nevanlinna, Olavi |
| author_facet | Nevanlinna, Olavi |
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| author_sort | Nevanlinna, Olavi |
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| dewey-hundreds | 000 - Computer science, information, general works |
| dewey-ones | 050 - General serial publications |
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| dewey-tens | 050 - General serial publications |
| discipline | Allgemeine Naturwissenschaft Mathematik |
| doi_str_mv | 10.1007/978-3-0348-8547-8 |
| format | Electronic eBook |
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| indexdate | 2024-07-10T01:21:10Z |
| institution | BVB |
| isbn | 9783034885478 9783764328658 |
| language | English |
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| series2 | Lectures in Mathematics ETH Zürich, Department of Mathematics Research Institute of Mathematics Department of Mathematics Research Institute of Mathematics |
| spelling | Nevanlinna, Olavi Verfasser aut Convergence of Iterations for Linear Equations by Olavi Nevanlinna Basel Birkhäuser Basel 1993 1 Online-Ressource txt rdacontent c rdamedia cr rdacarrier Lectures in Mathematics ETH Zürich, Department of Mathematics Research Institute of Mathematics Department of Mathematics Research Institute of Mathematics Assume that after preconditioning we are given a fixed point problem x = Lx + f (*) where L is a bounded linear operator which is not assumed to be symmetric and f is a given vector. The book discusses the convergence of Krylov subspace methods for solving fixed point problems (*), and focuses on the dynamical aspects of the iteration processes. For example, there are many similarities between the evolution of a Krylov subspace process and that of linear operator semigroups, in particular in the beginning of the iteration. A lifespan of an iteration might typically start with a fast but slowing phase. Such a behavior is sublinear in nature, and is essentially independent of whether the problem is singular or not. Then, for nonsingular problems, the iteration might run with a linear speed before a possible superlinear phase. All these phases are based on different mathematical mechanisms which the book outlines. The goal is to know how to precondition effectively, both in the case of "numerical linear algebra" (where one usually thinks of first fixing a finite dimensional problem to be solved) and in function spaces where the "preconditioning" corresponds to software which approximately solves the original problem Science (General) Science, general Naturwissenschaft Konvergenz (DE-588)4032326-2 gnd rswk-swf Lineare Gleichung (DE-588)4234490-6 gnd rswk-swf Iteration (DE-588)4123457-1 gnd rswk-swf Lineare Gleichung (DE-588)4234490-6 s Iteration (DE-588)4123457-1 s Konvergenz (DE-588)4032326-2 s 1\p DE-604 https://doi.org/10.1007/978-3-0348-8547-8 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Nevanlinna, Olavi Convergence of Iterations for Linear Equations Science (General) Science, general Naturwissenschaft Konvergenz (DE-588)4032326-2 gnd Lineare Gleichung (DE-588)4234490-6 gnd Iteration (DE-588)4123457-1 gnd |
| subject_GND | (DE-588)4032326-2 (DE-588)4234490-6 (DE-588)4123457-1 |
| title | Convergence of Iterations for Linear Equations |
| title_auth | Convergence of Iterations for Linear Equations |
| title_exact_search | Convergence of Iterations for Linear Equations |
| title_full | Convergence of Iterations for Linear Equations by Olavi Nevanlinna |
| title_fullStr | Convergence of Iterations for Linear Equations by Olavi Nevanlinna |
| title_full_unstemmed | Convergence of Iterations for Linear Equations by Olavi Nevanlinna |
| title_short | Convergence of Iterations for Linear Equations |
| title_sort | convergence of iterations for linear equations |
| topic | Science (General) Science, general Naturwissenschaft Konvergenz (DE-588)4032326-2 gnd Lineare Gleichung (DE-588)4234490-6 gnd Iteration (DE-588)4123457-1 gnd |
| topic_facet | Science (General) Science, general Naturwissenschaft Konvergenz Lineare Gleichung Iteration |
| url | https://doi.org/10.1007/978-3-0348-8547-8 |
| work_keys_str_mv | AT nevanlinnaolavi convergenceofiterationsforlinearequations |