MiS Preprint Repository

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 $\mathcal{O}(d{k}^3+dk(m+1))$ 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.