The Inverse Fast Multipole Method: Using a Fast Approximate Direct Solver as a Preconditioner for Dense Linear Systems

Abstract

Although some preconditioners are available for solving dense linear systems, there are still many matrices for which preconditioners are lacking, in particular in cases where the size of the matrix N becomes very large. Examples of preconditioners include ILU preconditioners that sparsify the matrix based on some threshold, algebraic multigrid, and specialized preconditioners, e.g., Calderón and other analytical approximation methods when available. Despite these methods, there remains a great need to develop general purpose preconditioners whose cost scales well with the matrix size N . In this paper, we propose a preconditioner, with broad applicability and with cost O(N ), for dense matrices, when the matrix is given by a smooth kernel. Extending the method using the same framework to general H 2 -matrices (i.e., algebraic instead of defined in terms of an analytical kernel) is relatively straightforward, but this won't be discussed here. These preconditioners have a controlled accuracy (e.g., machine accuracy can be achieved if needed) and scale linearly with N . They are based on an approximate direct solve of the system. The linear scaling of the algorithm is achieved by means of two key ideas. First, the H 2 -structure of the dense matrix is exploited to obtain an extended sparse system of equations. Second, fill-ins arising when performing the elimination of the latter are compressed as low-rank matrices if they correspond to well-separated interactions. This ensures that the sparsity pattern of the extended sparse matrix is preserved throughout the elimination, hence resulting in a very efficient algorithm with O(N log(1/ε) 2 ) computational cost and O(N log 1/ε) memory requirement, for an error tolerance 0 < ε < 1. The solver is inexact, although the error can be controlled and made as small as needed. These solvers are related to ILU in the sense that the fill-in is controlled. However, in ILU, most of the fill-in (i.e., below a certain tolerance) is simply discarded, whereas here it is approximated using low-rank blocks, with a prescribed tolerance. Numerical examples are discussed to demonstrate the linear scaling of the method and to illustrate its effectiveness as a preconditioner. ). 1 2. The inverse fast multipole method: key ideas and algorithm.2.1. Extended sparsification. It is the aim to obtain a fast solver for the dense linear system Ax = b, where A is an H 2 -matrix, i.e., a matrix constructed based on strong admissibility criteria (well-separated clusters are assumed to be low-rank only), and with a nested basis for the low-rank representations.The first step in obtaining such a solver is to exploit the H 2 -structure of the matrix in order to convert the original dense system into an extended sparse sys-2 The term fill-in is used here to denote the non-zero matrix entries. 3 The term elimination refers to removing variables from a system of equations and updating the coefficients of the remaining variables appropriately.

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