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Block preconditioners for linear systems arising from multiscale collocation with compactly supported RBFs
Author(s) -
Farrell Patricio,
Pestana Jennifer
Publication year - 2015
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.1984
Subject(s) - mathematics , collocation (remote sensing) , block (permutation group theory) , block matrix , positive definite matrix , grid , collocation method , algorithm , diagonal , mathematical optimization , mathematical analysis , computer science , differential equation , combinatorics , geometry , ordinary differential equation , eigenvalues and eigenvectors , physics , quantum mechanics , machine learning
Summary 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. Copyright © 2015 John Wiley & Sons, Ltd.