Krylov Subspace Methods for Radial Basis Function Interpolation 1
نویسندگان
چکیده
Radial basis function methods for interpolation to values of a function of several variables are particularly useful when the data points are in general positions, but hardly any sparsity occurs in the matrix of the linear system of interpolation equations. Therefore an iterative procedure for solving the system is studied. The k-th iteration calculates the element in a k-dimensional linear sub-space of radial functions that is closest to the required interpolant, the subspaces being generated by a Krylov construction that employs a self-adjoint operator A. Distances between functions are measured by the semi-norm that is induced by the well-known conditional positive or negative deenite properties of the matrix of the interpolation problem, and conditions on A are found that guarantee successful termination of the iterations in exact arithmetic. A particular choice of A is recommended and is tried in numerical experiments. Fortunately, the number of iterations to achieve high accuracy is much less than the number that provides termination in theory, at most ten iterations being usual in practice. Thus the given procedure solves the interpolation equations very eeciently when the number of data is large.
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