Parallelization of Hierarchical Matrix Algorithms for Electromagnetic Scattering Problems

Numerical solution methods for electromagnetic scattering problems lead to large systems of equations with millions or even billions of unknown variables. The coefficient matrices are dense, leading to large computational costs and storage requirements if direct methods are used. A commonly used technique is to instead form a hierarchical representation for the parts of the matrix that corresponds to far-field interactions. The overall computational cost and storage requirements can then be reduced to \(\mathcal {O}(N\log N)\). This still corresponds to a large-scale simulation that requires parallel implementation. The hierarchical algorithms are rather complex, both regarding data dependencies and communication patterns, making parallelization non-trivial. In this chapter, we describe two classes of algorithms in some detail, we provide a survey of existing solutions, we show results for a proof-of-concept implementation, and we provide various perspectives on different aspects of the problem.