Block preconditioners for linear systems arising from multiscale collocation with compactly supported RBFs
Authors
- Farrell, Patricio
ORCID: 0000-0001-9969-6615 - Pestana, Jennifer
2010 Mathematics Subject Classification
- 65F08 65N35 65N55
Keywords
- partial differential equation, multiscale collocation, compactly supported radial basis functions, Krylov subspace methods, preconditioning, additive Schwarz method
DOI
Abstract
Symmetric collocation methods with radial basis functions 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 radial basis functions 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 (ARASM). Numerical results verify the effectiveness of the preconditioners.
Appeared in
- Numer. Linear Algebra Appl., 22 (2015) pp. 731--747.
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