A universal preconditioner for linear systems

Tom Vettenburg; Ivo M. Vellekoop

doi:10.48550/arXiv.2207.14222

A universal preconditioner for linear systems

Tom Vettenburg, Ivo M. Vellekoop

Physics

Research output: Working paper/Preprint › Preprint

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Abstract

We present a universal preconditioner Γ that is applicable to all invertible linear problems Ax=y for which an approximate inverse is available. After preconditioning, the condition number depends on the norm of the discrepancy of this approximation instead of that of the original, potentially unbounded, system.

We prove that our construct is the only universal approach that ensures that ∥1−Γ−1A∥
We demonstrate and evaluate our approach for wave problems, diffusion problems, and the pantograph delay differential equation.

Original language	English
Place of Publication	Cornell University
Publisher	arXiv
Number of pages	9
DOIs	https://doi.org/10.48550/arXiv.2207.14222
Publication status	Published - 28 Jul 2022

Keywords

math.NA
cs.NA
physics.comp-ph
65F08
accretive linear systems
preconditioning
iterative methods
partial differential equations
generalized Born series
Richardson iteration

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10.48550/arXiv.2207.14222

Preprint
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Submitted manuscript, 2.15 MB

Correlative Refractive Index Light-Sheet Microscopy
Vettenburg, T. (Investigator)
Medical Research Council
1/01/20 → 1/07/27
Project: Research

Cite this

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title = "A universal preconditioner for linear systems",

abstract = "We present a universal preconditioner Γ that is applicable to all invertible linear problems Ax=y for which an approximate inverse is available. After preconditioning, the condition number depends on the norm of the discrepancy of this approximation instead of that of the original, potentially unbounded, system.We prove that our construct is the only universal approach that ensures that ∥1−Γ−1A∥We demonstrate and evaluate our approach for wave problems, diffusion problems, and the pantograph delay differential equation.",

keywords = "math.NA, cs.NA, physics.comp-ph, 65F08, accretive linear systems, preconditioning, iterative methods, partial differential equations, generalized Born series, Richardson iteration",

author = "Tom Vettenburg and Vellekoop, {Ivo M.}",

note = "IMV is supported by the Dutch Research Council (NWO, Vidi grant n◦ 14879) and the European Research Council under the European Union{\textquoteright}s Horizon 2020 Programme / ERC Grant Agreement n◦ 678919. TV is a UKRI Future Leaders Fellow supported by grant MR/S034900/1.",

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AU - Vettenburg, Tom

AU - Vellekoop, Ivo M.

N1 - IMV is supported by the Dutch Research Council (NWO, Vidi grant n◦ 14879) and the European Research Council under the European Union’s Horizon 2020 Programme / ERC Grant Agreement n◦ 678919. TV is a UKRI Future Leaders Fellow supported by grant MR/S034900/1.

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N2 - We present a universal preconditioner Γ that is applicable to all invertible linear problems Ax=y for which an approximate inverse is available. After preconditioning, the condition number depends on the norm of the discrepancy of this approximation instead of that of the original, potentially unbounded, system.We prove that our construct is the only universal approach that ensures that ∥1−Γ−1A∥We demonstrate and evaluate our approach for wave problems, diffusion problems, and the pantograph delay differential equation.

AB - We present a universal preconditioner Γ that is applicable to all invertible linear problems Ax=y for which an approximate inverse is available. After preconditioning, the condition number depends on the norm of the discrepancy of this approximation instead of that of the original, potentially unbounded, system.We prove that our construct is the only universal approach that ensures that ∥1−Γ−1A∥We demonstrate and evaluate our approach for wave problems, diffusion problems, and the pantograph delay differential equation.

KW - math.NA

KW - cs.NA

KW - physics.comp-ph

KW - 65F08

KW - accretive linear systems

KW - preconditioning

KW - iterative methods

KW - partial differential equations

KW - generalized Born series

KW - Richardson iteration

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DO - 10.48550/arXiv.2207.14222

M3 - Preprint

BT - A universal preconditioner for linear systems

PB - arXiv

CY - Cornell University

ER -

A universal preconditioner for linear systems

Abstract

Keywords

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