By Eduard L. Stiefel
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5 0 x iterations (exhibiting a clearly superlinear convergence rate), and then stalls (although actually it keeps on dropping for a few more iterations, albeit at a negligible rate). Figures 6–9 demonstrate how the operator-Newton scheme accurately predicts the correction required to match the current error-notice that the algorithm itself of course ignores the true solution. Concerning R, it increases at first, and then also drops steadily until leveling off at about the 3rd−4th iteration. The fact that the interpolation non-linear residual R does not drop down to zero (or the error ε , for that purpose) is not surprising since the RBF method enforces the sequence of linear PDEs (and therefore the non-linear PDE) on a finite set of collocation nodes only.
V. -S. Chen et al. applied to solving partial differential equations (PDEs) in [20,21] and the theoretical foundation of RBF method for solving PDEs has been well studied . Applications of RBF for solving PDEs include, for example, singularity problems , Hamilton-Jacobi equations , fourth-order elliptic and parabolic problems , approximation in boundary element method for nonlinear elliptic PDEs , hyperbolic conservation laws , and smoothed multilevel approach . When solving boundary value problems, RBF collocation method is shown to be more effective if boundary conditions are properly weighted .
43, 423–438 (2002). 9. G. E. Fasshauer, Solving differential equations with radial basis functions: Multilevel methods and smoothing, Adv. Comput. Math. 11, 139–159 (1999). 10. G. E. Fasshauer, On Smoothing for Multilevel Approximation with Radial Basis Functions, Approximation Theory XI, Vol. II: Computational Aspects, C. K. Chui, and L. L. ), Vanderbilt University Press, Nashville, TN 55–62 (1999). 11. G. E. Fasshauer, E. C. Gartland, and J. W. Jerome, Algorithms defined by Nash iteration: Some implementations via multilevel collocation and smoothing, J.
An Introduction to Numerical Mathematics by Eduard L. Stiefel
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