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SPARSE HIERARCHICAL PRECONDITIONERS USING PIECEWISE SMOOTH APPROXIMATIONS OF EIGENVECTORS.

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Titel: SPARSE HIERARCHICAL PRECONDITIONERS USING PIECEWISE SMOOTH APPROXIMATIONS OF EIGENVECTORS.
Autoren: KLOCKIEWICZ, BAZYLI, DARVE, ERIC
Quelle: SIAM Journal on Scientific Computing; 2020, Vol. 42 Issue 6, pA3907-A3931, 25p
Schlagwörter: SMOOTHNESS of functions, ELLIPTIC equations, FLOW simulations, LINEAR equations, RIGID bodies
Abstract: When solving linear systems arising from PDE discretizations, iterative methods (such as conjugate gradient (CG), GMRES, or MINRES) are often the only practical choice. To converge in a small number of iterations, however, they have to be coupled with an efficient preconditioner. One approach to preconditioning is provided by the hierarchical approximate factorization methods. However, to guarantee sufficient accuracy on the eigenvectors corresponding to the smallest eigenvalues, these methods typically have to be performed at very stringent accuracies, making the preconditioner expensive to apply. On the other hand, for a large class of problems, including many elliptic equations, the eigenvectors corresponding to small eigenvalues are smooth functions of the PDE grid. In this paper, we describe a hierarchical approximate factorization approach which focuses on improving accuracy on the smooth eigenvectors. The improved accuracy is achieved by preserving the action of the factorized matrix on piecewise polynomial functions of the grid. Based on the factorization, we propose a family of sparse preconditioners with Ϭ(n) or Ϭ (n log n) construction complexities. Our methods exhibit rapid convergence of CG in benchmarks run on large elliptic problems, arising for example in flow or mechanical simulations. In the case of the linear elasticity equation the preconditioners are exact on the near-kernel rigid body modes. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Scientific Computing is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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An: 147971349
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  Data: SPARSE HIERARCHICAL PRECONDITIONERS USING PIECEWISE SMOOTH APPROXIMATIONS OF EIGENVECTORS.
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  Data: <searchLink fieldCode="AR" term="%22KLOCKIEWICZ%2C+BAZYLI%22">KLOCKIEWICZ, BAZYLI</searchLink><br /><searchLink fieldCode="AR" term="%22DARVE%2C+ERIC%22">DARVE, ERIC</searchLink>
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  Data: SIAM Journal on Scientific Computing; 2020, Vol. 42 Issue 6, pA3907-A3931, 25p
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  Data: <searchLink fieldCode="DE" term="%22SMOOTHNESS+of+functions%22">SMOOTHNESS of functions</searchLink><br /><searchLink fieldCode="DE" term="%22ELLIPTIC+equations%22">ELLIPTIC equations</searchLink><br /><searchLink fieldCode="DE" term="%22FLOW+simulations%22">FLOW simulations</searchLink><br /><searchLink fieldCode="DE" term="%22LINEAR+equations%22">LINEAR equations</searchLink><br /><searchLink fieldCode="DE" term="%22RIGID+bodies%22">RIGID bodies</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: When solving linear systems arising from PDE discretizations, iterative methods (such as conjugate gradient (CG), GMRES, or MINRES) are often the only practical choice. To converge in a small number of iterations, however, they have to be coupled with an efficient preconditioner. One approach to preconditioning is provided by the hierarchical approximate factorization methods. However, to guarantee sufficient accuracy on the eigenvectors corresponding to the smallest eigenvalues, these methods typically have to be performed at very stringent accuracies, making the preconditioner expensive to apply. On the other hand, for a large class of problems, including many elliptic equations, the eigenvectors corresponding to small eigenvalues are smooth functions of the PDE grid. In this paper, we describe a hierarchical approximate factorization approach which focuses on improving accuracy on the smooth eigenvectors. The improved accuracy is achieved by preserving the action of the factorized matrix on piecewise polynomial functions of the grid. Based on the factorization, we propose a family of sparse preconditioners with Ϭ(n) or Ϭ (n log n) construction complexities. Our methods exhibit rapid convergence of CG in benchmarks run on large elliptic problems, arising for example in flow or mechanical simulations. In the case of the linear elasticity equation the preconditioners are exact on the near-kernel rigid body modes. [ABSTRACT FROM AUTHOR]
– Name: Abstract
  Label:
  Group: Ab
  Data: <i>Copyright of SIAM Journal on Scientific Computing is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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RecordInfo BibRecord:
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    Identifiers:
      – Type: doi
        Value: 10.1137/20M1315683
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 25
        StartPage: A3907
    Subjects:
      – SubjectFull: SMOOTHNESS of functions
        Type: general
      – SubjectFull: ELLIPTIC equations
        Type: general
      – SubjectFull: FLOW simulations
        Type: general
      – SubjectFull: LINEAR equations
        Type: general
      – SubjectFull: RIGID bodies
        Type: general
    Titles:
      – TitleFull: SPARSE HIERARCHICAL PRECONDITIONERS USING PIECEWISE SMOOTH APPROXIMATIONS OF EIGENVECTORS.
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          Name:
            NameFull: KLOCKIEWICZ, BAZYLI
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          Name:
            NameFull: DARVE, ERIC
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          Dates:
            – D: 01
              M: 11
              Text: 2020
              Type: published
              Y: 2020
          Identifiers:
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              Value: 10648275
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              Value: 42
            – Type: issue
              Value: 6
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            – TitleFull: SIAM Journal on Scientific Computing
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