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Title: Adaptive Multilevel Krylov Methods

Abstract

Inexact (variable) preconditioning of Multilevel Krylov methods (MK methods) for the solution of linear systems of equations is considered. MK methods approximate the solution of the local systems on a subspace using a few, but fixed, number of iteration steps of a preconditioned flexible Krylov method. In this paper, using the philosophy of inexact Krylov subspace methods, we use a theoretically-derived criterion to choose the number of iterations needed on each level to achieve a desired tolerance. We use this criterion on one level and obtain an improved MK method. Inspired by these results, a second ad hoc method is also explored. Numerical experiments for the Poisson, Helmholtz, and the convection-diffusion equations illustrate the efficiency and robustness of this adaptive Multilevel Krylov method.

Authors:
 [1];  [2];  [3]
  1. Weierstraß-Inst., Berlin (Germany)
  2. Technische Univ. Berlin (Germany). Inst. für Mathematik,
  3. Temple Univ., Philadelphia, PA (United States)
Publication Date:
Research Org.:
Temple Univ., Philadelphia, PA (United States)
Sponsoring Org.:
USDOE Office of Science (SC); National Science Foundation (NSF)
OSTI Identifier:
1612659
Grant/Contract Number:  
SC0016578; DMS-1115520; DMS-1418882
Resource Type:
Accepted Manuscript
Journal Name:
Electronic Transactions on Numerical Analysis
Additional Journal Information:
Journal Volume: 51; Journal ID: ISSN 1068-9613
Publisher:
Kent State University - Johann Radon Institute for Computational and Applied Mathematics (RICAM)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; mathematics; multilevel Krylov methods; flexible GMRES; inexact Krylov subspace methods; inexact preconditioning

Citation Formats

Kehl, René, Nabben, Reinhard, and Szyld, Daniel B. Adaptive Multilevel Krylov Methods. United States: N. p., 2019. Web. https://doi.org/10.1553/etna_vol51s512.
Kehl, René, Nabben, Reinhard, & Szyld, Daniel B. Adaptive Multilevel Krylov Methods. United States. https://doi.org/10.1553/etna_vol51s512
Kehl, René, Nabben, Reinhard, and Szyld, Daniel B. Mon . "Adaptive Multilevel Krylov Methods". United States. https://doi.org/10.1553/etna_vol51s512. https://www.osti.gov/servlets/purl/1612659.
@article{osti_1612659,
title = {Adaptive Multilevel Krylov Methods},
author = {Kehl, René and Nabben, Reinhard and Szyld, Daniel B.},
abstractNote = {Inexact (variable) preconditioning of Multilevel Krylov methods (MK methods) for the solution of linear systems of equations is considered. MK methods approximate the solution of the local systems on a subspace using a few, but fixed, number of iteration steps of a preconditioned flexible Krylov method. In this paper, using the philosophy of inexact Krylov subspace methods, we use a theoretically-derived criterion to choose the number of iterations needed on each level to achieve a desired tolerance. We use this criterion on one level and obtain an improved MK method. Inspired by these results, a second ad hoc method is also explored. Numerical experiments for the Poisson, Helmholtz, and the convection-diffusion equations illustrate the efficiency and robustness of this adaptive Multilevel Krylov method.},
doi = {10.1553/etna_vol51s512},
journal = {Electronic Transactions on Numerical Analysis},
number = ,
volume = 51,
place = {United States},
year = {2019},
month = {12}
}

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