Approximation and interpolation employing divergence-free radial basis functions with applications

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Title: Approximation and interpolation employing divergence-free radial basis functions with applications
Author: Lowitzsch, Svenja
Abstract: Approximation and interpolation employing radial basis functions has found important applications since the early 1980's in areas such as signal processing , medical imaging , as well as neural networks . Several applications demand that certain physical properties be fulfilled , such as a function being divergence free . No such class of radial basis functions that reflects these physical properties was known until 1994 , when Narcowich and Ward introduced a family of matrix -valued radial basis functions that are divergence free . They also obtained error bounds and stability estimates for interpolation by means of these functions . These divergence -free functions are very smooth , and have unbounded support . In this thesis we introduce a new class of matrix -valued radial basis functions that are divergence free as well as compactly supported . This leads to the possibility of applying fast solvers for inverting interpolation matrices , as these matrices are not only symmetric and positive definite , but also sparse because of this compact support . We develop error bounds and stability estimates which hold for a broad class of functions . We conclude with applications to the numerical solution of the Navier -Stokes equation for certain incompressible fluid flows .
URI: http : / /hdl .handle .net /1969 .1 /21
Date: 2004-09-30

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Approximation and interpolation employing divergence-free radial basis functions with applications. Available electronically from http : / /hdl .handle .net /1969 .1 /21 .

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