MiS Preprint Repository

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

Fast Evaluation of Singular BEM Integrals Based on Tensor Approximations

Jonas Ballani


In this paper we propose a method for the fast evaluation of integrals stemming from boundary element methods. Our method is based on the parametrisation of boundary elements in terms of a $d$-dimensional parameter tuple. We interpret the integral as a real-valued function $f$ depending on $d$ parameters and show that $f$ is smooth in a $d$-dimensional box. A standard interpolation of $f$ by polynomials leads to a $d$-dimensional tensor which is given by the values of $f$ at the interpolation points. This tensor may be approximated in a low rank tensor format like the (CP) format or the $\mathcal{H}$-Tucker format. The tensor approximation has to be done only once and allows us to evaluate interpolants in $\mathcal{O}(dr(m+1))$ operations in the (CP) format, or $\mathcal{O}(dk^3+dk(m+1))$ operations in the $\mathcal{H}$-Tucker format, where $m$ denotes the interpolation order and the ranks $r$, $k$ are small integers. We demonstrate that highly accurate integral values can be obtained at very moderate costs.

MSC Codes:
15A69, 65499, 65N38

Related publications

2012 Repository Open Access
Jonas Ballani

Fast evaluation of singular BEM integrals based on tensor approximations

In: Numerische Mathematik, 121 (2012) 3, pp. 433-460