Block preconditioners for linear systems arising from multiscale collocation with compactly supported RBFs

Patricio Farrell, Jennifer Pestana

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

Symmetric collocation methods with RBFs allow approximation of the solution of a partial differential equation, even if the right-hand side is only known at scattered data points, without needing to generate a grid. However, the benefit of a guaranteed symmetric positive definite block system comes at a high computational cost. This cost can be alleviated somewhat by considering compactly supported RBFs and a multiscale technique. But the condition number and sparsity will still deteriorate with the number of data points. Therefore, we study certain block diagonal and triangular preconditioners. We investigate ideal preconditioners and determine the spectra of the preconditioned matrices before proposing more practical preconditioners based on a restricted additive Schwarz method with coarse grid correction. Numerical results verify the effectiveness of the preconditioners.
LanguageEnglish
Pages731-747
Number of pages17
JournalNumerical Linear Algebra with Applications
Volume22
Issue number4
Early online date30 Apr 2015
DOIs
Publication statusPublished - Aug 2015

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Collocation
Preconditioner
Linear systems
Linear Systems
Partial differential equations
Costs
Additive Schwarz Method
Grid
Scattered Data
Condition number
Collocation Method
Sparsity
Positive definite
Computational Cost
Triangular
Partial differential equation
Verify
Numerical Results
Approximation

Keywords

  • partial differential equation
  • multiscale collocation
  • compactly supported RBFs
  • Krylov subspace methods
  • preconditioning
  • additive Schwarz methods

Cite this

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Block preconditioners for linear systems arising from multiscale collocation with compactly supported RBFs. / Farrell, Patricio; Pestana, Jennifer.

In: Numerical Linear Algebra with Applications, Vol. 22, No. 4, 08.2015, p. 731-747.

Research output: Contribution to journalArticle

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KW - partial differential equation

KW - multiscale collocation

KW - compactly supported RBFs

KW - Krylov subspace methods

KW - preconditioning

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