Low-rank approximation of integral operators by using the Green formula and quadrature

Authors:
Steffen Börm;Jessica Gördes
Affiliations:
Christian-Albrechts-Universität zu Kiel, Kiel, Germany;Christian-Albrechts-Universität zu Kiel, Kiel, Germany
Venue:
Numerical Algorithms
Year:
2013

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Abstract

Approximating integral operators by a standard Galerkin discretisation typically leads to dense matrices. To avoid the quadratic complexity it takes to compute and store a dense matrix, several approaches have been introduced including $\mathcal {H}$-matrices. The kernel function is approximated by a separable function, this leads to a low rank matrix. Interpolation is a robust and popular scheme, but requires us to interpolate in each spatial dimension, which leads to a complexity of $m^d$ for $m$-th order. Instead of interpolation we propose using quadrature on the kernel function represented with Green's formula. Due to the fact that we are integrating only over the boundary, we save one spatial dimension compared to the interpolation method and get a complexity of $m^{d-1}$.