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A review of algorithms for thin plate spline interpolation in two dimensions. (English) Zbl 1273.65019

Fontanella, F. (ed.) et al., Proceedings of the international workshop on Advanced topics in multivariate approximation, Montecatini Terme, Italy, September 27–October 3, 1995. Singapore: World Scientific (ISBN 981-02-2852-X/hbk). Ser. Approx. Decompos. 8, 303-322 (1996).
Summary: The thin plate spline interpolation problem can be expressed as the solution of an \((n+3) \times (n+3)\) system of linear equations, where \(n\) is the number of interpolations points. If the system is solved by a direct method, then \(\mathcal {O}(n^3)\) computer operations are required. A particular direct method is recommended, that gives good results when this amount of work is tolerable. Conjugate gradient methods for positive definite or semi-definite symmetric systems are also applicable, and it is shown that the well-known variational formulation of the thin plate spline method assists the choice of pre-conditioners. Further, the work of each iteration can be reduced greatly by the use of Laurent expansion techniques when \(n\) is large. We also consider some iterative methods that are developed from the localization properties of the Lagrange functions of interpolation. Thus we identify a new algorithm that typically requires fewer than 15 iterations to achieve 10 decimals accuracy for large \(n\).
For the entire collection see [Zbl 0901.00042].

MSC:

65D07 Numerical computation using splines
41A15 Spline approximation
41A63 Multidimensional problems
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