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MiS Preprint
77/2010

Fast Evaluation of Singular BEM Integrals Based on Tensor Approximations

Jonas Ballani

Abstract

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.

Received:
Dec 21, 2010
Published:
Dec 21, 2010
MSC Codes:
15A69, 65499, 65N38

Related publications

inJournal
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