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Spectral analysis and preconditioning techniques for radial basis function collocation matrices

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SUMMARY

Meshless collocation methods based on radial basis functions lead to structured linear systems, which, for equispaced grid points, have almost a multilevel Toeplitz structure. In particular, if we consider partial differential equations (PDEs) in two dimensions, then we find almost (up to a ‘low-rank’ correction given by the boundary conditions) two-level Toeplitz matrices, i.e. block Toeplitz with Toeplitz blocks structures, where both the number of blocks and the block-size grow with the number of collocation points. In Bini et al. (Linear Algebra Appl. 2008; 428:508–519), upper bounds for the condition number of the Toeplitz matrices approximating a one-dimensional model problem were proved. Here, we refine the one-dimensional results, by explaining some numerics reported in the previous paper, and we show a preliminary analysis concerning conditioning, extremal spectral behavior, and global spectral results in the two-dimensional case for the structured part. By exploiting the recent tools in the literature, a global distribution theorem in the sense of Weyl is proved also for the complete matrix-sequence, where the low-rank correction due to the boundary conditions is taken into consideration. The provided spectral analysis is then applied to design effective preconditioning techniques in order to overcome the ill-conditioning of the matrices. A wide numerical experimentation, both in the one- and two-dimensional cases, confirms our analysis and the robustness of the proposed preconditioners. Copyright © 2011 John Wiley & Sons, Ltd.

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