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On convergent numerical algorithms for unsymmetric collocation. (English) Zbl 1168.65419
Summary: We are interested in some convergent formulations for the unsymmetric collocation method or the so-called Kansa’s method [E. J. Kansa, Comput. Math. Appl. 19, No. 8–9, 127–145 (1990; Zbl 0692.76003); ibid. 147–161 (1990; Zbl 0850.76048)]. We review some newly developed theories on solvability and convergence. The rates of convergence of these variations of Kansa’s method are examined and verified in arbitrary-precision computations. Numerical examples confirm with the theories that the modified Kansa’s method converges faster than the interpolant to the solution; that is, exponential convergence for the multiquadric and Gaussian radial basis functions (RBFs). Some numerical algorithms are proposed for efficiency and accuracy in practical applications of Kansa’s method. In double-precision, even for very large RBF shape parameters, we show that the modified Kansa’s method, through a subspace selection using a greedy algorithm, can produce acceptable approximate solutions. A benchmark algorithm is used to verify the optimality of the selection process.

MSC:
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
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