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A multi‐level solution of scalar and vectorial interpolation problems based on iterated elliptic operators
Author(s) -
Gáspár Csaba
Publication year - 2003
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200310537
Subject(s) - mathematics , interpolation (computer graphics) , sobolev space , mathematical analysis , bilinear interpolation , linear interpolation , boundary value problem , inverse quadratic interpolation , birkhoff interpolation , partial differential equation , nearest neighbor interpolation , computer science , polynomial , animation , statistics , computer graphics (images)
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.