### Abstract

Language | English |
---|---|

Pages | 731-747 |

Number of pages | 17 |

Journal | Numerical Linear Algebra with Applications |

Volume | 22 |

Issue number | 4 |

Early online date | 30 Apr 2015 |

DOIs | |

Publication status | Published - Aug 2015 |

### Fingerprint

### Keywords

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

### Cite this

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*Numerical Linear Algebra with Applications*, vol. 22, no. 4, pp. 731-747. https://doi.org/10.1002/nla.1984

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - Block preconditioners for linear systems arising from multiscale collocation with compactly supported RBFs

AU - Farrell, Patricio

AU - Pestana, Jennifer

PY - 2015/8

Y1 - 2015/8

N2 - 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.

AB - 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.

KW - partial differential equation

KW - multiscale collocation

KW - compactly supported RBFs

KW - Krylov subspace methods

KW - preconditioning

KW - additive Schwarz methods

U2 - 10.1002/nla.1984

DO - 10.1002/nla.1984

M3 - Article

VL - 22

SP - 731

EP - 747

JO - Numerical Linear Algebra with Applications

T2 - Numerical Linear Algebra with Applications

JF - Numerical Linear Algebra with Applications

SN - 1070-5325

IS - 4

ER -