A Multigrid Preconditioner for Jacobian-free Newton–Krylov Methods

Hardik Kothari; Alena Kopaničáková; Rolf Krause

doi:10.1007/978-3-030-95025-5_38

A Multigrid Preconditioner for Jacobian-free Newton–Krylov Methods

Hardik Kothari^*, Alena Kopaničáková, Rolf Krause

^*Corresponding author for this work

Applied Mathematics and Computational Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

1 Scopus citations

Abstract

The numerical solution of partial differential equations (PDEs) is often carried out using discretization techniques, such as the finite element method (FEM), and typically requires the solution of a nonlinear system of equations. These nonlinear systems are often solved using some variant of the Newton method, which utilizes a sequence of iterates generated by solving a linear system of equations. However, for problems such as inverse problems, optimal control problems, or higher-order coupled PDEs, it can be computationally expensive, or even impossible to assemble a Jacobian matrix.

Original language	English (US)
Title of host publication	Domain Decomposition Methods in Science and Engineering XXVI
Editors	Susanne C. Brenner, Axel Klawonn, Jinchao Xu, Eric Chung, Jun Zou, Felix Kwok
Publisher	Springer Science and Business Media Deutschland GmbH
Pages	365-372
Number of pages	8
ISBN (Print)	9783030950248
DOIs	https://doi.org/10.1007/978-3-030-95025-5_38
State	Published - 2022
Event	26th International Conference on Domain Decomposition Methods, 2020 - Virtual, Online Duration: Dec 7 2020 → Dec 12 2020

Publication series

Name	Lecture Notes in Computational Science and Engineering
Volume	145
ISSN (Print)	1439-7358
ISSN (Electronic)	2197-7100

Conference

Conference	26th International Conference on Domain Decomposition Methods, 2020
City	Virtual, Online
Period	12/7/20 → 12/12/20

Bibliographical note

Publisher Copyright:
© 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.

ASJC Scopus subject areas

Modeling and Simulation
General Engineering
Discrete Mathematics and Combinatorics
Control and Optimization
Computational Mathematics

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10.1007/978-3-030-95025-5_38

Cite this

Kothari, H., Kopaničáková, A., & Krause, R. (2022). A Multigrid Preconditioner for Jacobian-free Newton–Krylov Methods. In S. C. Brenner, A. Klawonn, J. Xu, E. Chung, J. Zou, & F. Kwok (Eds.), Domain Decomposition Methods in Science and Engineering XXVI (pp. 365-372). (Lecture Notes in Computational Science and Engineering; Vol. 145). Springer Science and Business Media Deutschland GmbH. https://doi.org/10.1007/978-3-030-95025-5_38

Kothari, Hardik ; Kopaničáková, Alena ; Krause, Rolf. / A Multigrid Preconditioner for Jacobian-free Newton–Krylov Methods. Domain Decomposition Methods in Science and Engineering XXVI. editor / Susanne C. Brenner ; Axel Klawonn ; Jinchao Xu ; Eric Chung ; Jun Zou ; Felix Kwok. Springer Science and Business Media Deutschland GmbH, 2022. pp. 365-372 (Lecture Notes in Computational Science and Engineering).

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abstract = "The numerical solution of partial differential equations (PDEs) is often carried out using discretization techniques, such as the finite element method (FEM), and typically requires the solution of a nonlinear system of equations. These nonlinear systems are often solved using some variant of the Newton method, which utilizes a sequence of iterates generated by solving a linear system of equations. However, for problems such as inverse problems, optimal control problems, or higher-order coupled PDEs, it can be computationally expensive, or even impossible to assemble a Jacobian matrix.",

author = "Hardik Kothari and Alena Kopani{\v c}{\'a}kov{\'a} and Rolf Krause",

note = "Publisher Copyright: {\textcopyright} 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.; 26th International Conference on Domain Decomposition Methods, 2020 ; Conference date: 07-12-2020 Through 12-12-2020",

year = "2022",

doi = "10.1007/978-3-030-95025-5_38",

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Kothari, H, Kopaničáková, A & Krause, R 2022, A Multigrid Preconditioner for Jacobian-free Newton–Krylov Methods. in SC Brenner, A Klawonn, J Xu, E Chung, J Zou & F Kwok (eds), Domain Decomposition Methods in Science and Engineering XXVI. Lecture Notes in Computational Science and Engineering, vol. 145, Springer Science and Business Media Deutschland GmbH, pp. 365-372, 26th International Conference on Domain Decomposition Methods, 2020, Virtual, Online, 12/7/20. https://doi.org/10.1007/978-3-030-95025-5_38

A Multigrid Preconditioner for Jacobian-free Newton–Krylov Methods. / Kothari, Hardik; Kopaničáková, Alena; Krause, Rolf.
Domain Decomposition Methods in Science and Engineering XXVI. ed. / Susanne C. Brenner; Axel Klawonn; Jinchao Xu; Eric Chung; Jun Zou; Felix Kwok. Springer Science and Business Media Deutschland GmbH, 2022. p. 365-372 (Lecture Notes in Computational Science and Engineering; Vol. 145).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N2 - The numerical solution of partial differential equations (PDEs) is often carried out using discretization techniques, such as the finite element method (FEM), and typically requires the solution of a nonlinear system of equations. These nonlinear systems are often solved using some variant of the Newton method, which utilizes a sequence of iterates generated by solving a linear system of equations. However, for problems such as inverse problems, optimal control problems, or higher-order coupled PDEs, it can be computationally expensive, or even impossible to assemble a Jacobian matrix.

AB - The numerical solution of partial differential equations (PDEs) is often carried out using discretization techniques, such as the finite element method (FEM), and typically requires the solution of a nonlinear system of equations. These nonlinear systems are often solved using some variant of the Newton method, which utilizes a sequence of iterates generated by solving a linear system of equations. However, for problems such as inverse problems, optimal control problems, or higher-order coupled PDEs, it can be computationally expensive, or even impossible to assemble a Jacobian matrix.

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A2 - Brenner, Susanne C.

A2 - Klawonn, Axel

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A2 - Kwok, Felix

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Kothari H, Kopaničáková A, Krause R. A Multigrid Preconditioner for Jacobian-free Newton–Krylov Methods. In Brenner SC, Klawonn A, Xu J, Chung E, Zou J, Kwok F, editors, Domain Decomposition Methods in Science and Engineering XXVI. Springer Science and Business Media Deutschland GmbH. 2022. p. 365-372. (Lecture Notes in Computational Science and Engineering). doi: 10.1007/978-3-030-95025-5_38

A Multigrid Preconditioner for Jacobian-free Newton–Krylov Methods

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