Structured low-rank representations for frequency-dependent wave scattering simulations using boundary element methods

The numerical simulation of wave phenomena is increasingly
relevant in several disciplines of science and engineering, including
acoustics, electromagnetics and medical imaging. Boundary element
methods (BEM) are a popular simulation tool for wave problems,
often applied to integral equations in the frequency domain. In the
last three decades, enormous research efforts have went into
efficient BEM solvers at a specific, application-dependent frequency.
These solvers are based, either implicitly or explicitly, on structured
linear algebraic representation of the system matrix involved: though
the latter is dense, it is well approximated block-wise by low rank
matrices, and this is a natural consequence of the smoothness of
Green's function. However, in many applications similar systems
have to be solved for a range of frequencies. In this project we
introduce compact structured representations of frequencydependent
BEM matrices, giving to structured low-rank tensor
approximations that generalize state-of-the-art hierarchical matrices
and H2-matrices for a single frequency. We identify and aim to solve
several mathematical and algorithmic challenges, including in
particular algebraic computations involving a Hadamard product of
matrices that very compactly captures the oscillatory nature of the
representation regardless of the geometrical complexity of the
scattering problem involved. This representation lies at the heart of
significant computational benefits.