The Inverse Fast Multipole Method

This article introduces a new fast direct solver for linear systems arising out of wide range of applications, integral equations, multivariate statistics, radial basis interpolation, etc., to name a few. The highlight of this new fast direct solver is that the solver scales linearly in the number of unknowns in all dimensions. The solver, termed as Inverse Fast Multipole Method (abbreviated as IFMM), works on the same data-structure as the Fast Multipole Method (abbreviated as FMM). More generally, the solver can be immediately extended to the class of hierarchical matrices, denoted as H2 matrices with strong admissibility criteria (weak low-rank structure), i.e., the interaction between neighboring cluster of particles is full-rank whereas the interaction between particles corresponding to well-separated clusters can be efficiently represented as a low-rank matrix. The algorithm departs from existing approaches in the fact that throughout the algorithm the interaction corresponding to neighboring clusters are always treated as full-rank interactions. Our approach relies on two major ideas: (i) The N×N matrix arising out of FMM (from now on termed as FMM matrix) can be represented as an extended sparser matrix of size M×M , where M ≈ 3N . (ii) While solving the larger extended sparser matrix, the fill-in’s that arise in the matrix blocks corresponding to well-separated clusters are hierarchically compressed. The ordering of the equations and the unknowns in the extended sparser matrix is strongly related to the local and multipole coefficients in the FMM [33] and the order of elimination is different from the usual nested dissection approach. Numerical benchmarks on 2D manifold confirm the linear scaling of the algorithm.