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IMA Journal of Numerical Analysis Advance Access originally published online on September 26, 2008
IMA Journal of Numerical Analysis 2009 29(3):773-789; doi:10.1093/imanum/drm046
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A symmetric collocation method with fast evaluation

Michael J. Johnson{dagger}

Department of Mathematics and Computer Science, Kuwait University, PO Box 5969, Safat 13060, Kuwait

{dagger} Email: johnson{at}sci.kuniv.edu.kw

Received on 4 July 2005. Revised on 27 March 2007.


  Abstract

Symmetric collocation, which can be used to numerically solve linear partial differential equations, is a natural generalization of the well-established scattered data interpolation method known as radial basis function interpolation. As with radial basis function interpolation, a major shortcoming of symmetric collocation is the high cost, in terms of floating-point operations, of evaluating the obtained function. When solving a linear partial differential equation, one usually has some freedom in choosing the collocation points. We explain how this freedom can be exploited to allow the fast evaluation of the obtained function provided the basic function is chosen as a tensor product of compactly supported piecewise polynomials. Our proposed fast evaluation method, which is exact in exact arithmetic, is initially designed and analysed in the univariate case. The multivariate case is then reduced, recursively, to multiple univariate evaluations. Along with the theoretical development of the method, we report the results of selected numerical experiments which help to clarify expectations.

Key Words: spline; multivariate; interpolation; positive-definite function; scattered data

In memory of Georg Heinig


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