A multi-level solution of scalar and vectorial interpolation problems based on iterated elliptic operators

• Published in: Proceedings in applied mathematics and mechanics Vol. 3; no. 1; pp. 535 - 536
• Main Author: Gáspár, Csaba
• Format: Journal Article
• Language: English
• Published: Berlin WILEY-VCH Verlag 01.12.2003; WILEY‐VCH Verlag
• ISSN: 1617-7061; 1617-7061
• DOI: 10.1002/pamm.200310537

One of the most popular methods to solve scattered data interpolation problems is the method of radial basis functions. However, it leads to a linear system with large, dense and ill‐conditioned matrix, which causes severe numerical difficulties. Here a relatively new approach is presented which completely avoids the large and dense matrices and significantly reduces the computational cost. The interpolation problem is converted to a higher order (typically to an iterated elliptic) partial differential equation supplied with the interpolation condition as special boundary conditions. This new problem is well‐posed in a suitable Sobolev space and can be solved by using robust, multi‐level methods. The approach is generalized also to vectorial interpolation problems, where the interpolation vector field is assumed to satisfy some additional conditions e.g. it is irrotational or divergence‐free.