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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

${\cal H^2}$-Matrix approximation of Integral operators by interpolation

Wolfgang Hackbusch and Steffen Börm


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 pd) 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.


Related publications

2002 Repository Open Access
Wolfgang Hackbusch and Steffen Börm

\(\mathscr {H}^2\)-matrix approximation of integral operators by interpolation

In: Applied numerical mathematics, 43 (2002) 1-2, pp. 129-143