-Matrix approximation of Integral operators by interpolation

Wolfgang Hackbusch and Steffen Börm

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Submission date: 17. Dec. 2001
Pages: 15
published in: Applied numerical mathematics, 43 (2002) 1-2, p. 129-143 
DOI number (of the published article): 10.1016/S0168-9274(02)00121-6
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Abstract:
Typical panel clustering methods for the fast evaluation of integral operators are based on the Taylor expansion of the kernel function and therefore usually require the user to implement the evaluation of the derivatives of this function up to an arbitrary degree.

We propose an alternative approach that replaces the Taylor expansion by simple polynomial interpolation. By applying the interpolation idea to the approximating polynomials on different levels of the cluster tree, the matrix vector multiplication can be performed in only O(n p^d) operations for a polynomial order of p and an n-dimensional trial space.

The main advantage of our method, compared to other methods, is its simplicity: Only pointwise evaluations of the kernel and of simple polynomials have to be implemented.