Proof of convergence of an iterative technique for thin plate spline interpolation in two dimensions (Q1966318)

The paper studies the thin plate spline solution of the interpolation problem \(s(x_i) = f_i\), \(i=1,\dots,n\), where \(x_i \in \mathbb{R}^2\), \(f_i \in \mathbb{R}\) are data and the points \(x_i\) are required to be all different and not collinear. A generalization of the iterative algorithm described by \textit{M. J. D. Powell} [Ann. Numer. Math. 4, No. 1-4, 519-527 (1997; Zbl 0885.65012)] is presented, and a convergence proof of this method is given. The interpolation method and the analysis also apply to radial basis functions and to other functions that are conditionally positive definite. The speed of convergence of the method is shown by some numerical results.