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Radial basis function methods for interpolation to functions of many variables. (English) Zbl 1026.65009

Summary: A review of interpolation to values of a function \(f(x)\), \(x\in{\mathcal R}^d\), by radial basis function methods is given. It addresses the nonsingularity of the interpolation equations, the inclusion of polynomial terms, and the accuracy of the approximation \(s\approx f\), where \(s\) is the interpolant. Then some numerical experiments investigate the situation when the data points are on a low-dimensional nonlinear manifold in \({\mathcal R}^d\). They suggest that the number of data points that are necessary for good accuracy on the manifold is independent of \(d\), even if \(d\) is very large. The experiments employ linear and multiquadric radial functions, because an iterative algorithm for these cases was developed at Cambridge recently [cf. A. C. Faul, G. Goodsell and M. J. D. Powell, A Krylov subspace algorithm for multiquadric interpolation in many dimensions, Report No. DAMTP 2002/NA5, University of Cambridge (2002)]. The algorithm is described briefly. Fortunately, the number of iterations is small when the data points are on a low-dimensional manifold. We expect these findings to be useful, because the manifold setting for large \(d\) is similar to typical distributions of interpolation points in data mining applications.

MSC:

65D05 Numerical interpolation
41A05 Interpolation in approximation theory
41A63 Multidimensional problems
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