Toward more geometrically adaptive compression of moment matrices

In recent years, fast direct integral equation solvers have been developed, as an alternative to iterative solver, for problems that suffer from ill-conditioning or when a solution is sought for many right-hand-sides. These solvers rely on the fast computation of a compressed representation of the impedance matrix. Then, the compressed representation's factorized (effectively 'solved') form is computed and applied to each right-hand-side. If the factorized form inherits the original matrix's compressibility, the savings in memory are maintained and the solution for each right-hand-side is, indeed, fast. The compression of the impedance matrix is often performed in a hierarchical manner. The geometry is first partitioned into clusters of basis and testing functions. Then, a hierarchical block structure for compression is defined, in accordance with the choice of hierarchical algebraic procedure for computing the compressed factorized form, e.g., [1], [2], or [3]. Matrix blocks, corresponding to interactions between sources and observers, that are assumed compressible, in some sense, are identified and compressed.