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, particularly in cases where the size of the matrix $N$ becomes very large. Examples of preconditioners include incomplete LU (ILU) preconditioners that sparsify the matrix based on some threshold, algebraic multigrid preconditioners, and specialized preconditioners, e.g., Calderon 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 $\mathcal{O}(N)$ for dense matrices, when the matrix is given by a “smooth” (as opposed to a highly oscillatory) kernel. Extending the method using the same framework as general $\mathcal{H}^2$-matrices (i.e., algebraic instead of defined, in terms of an analytical kernel) is relatively straightforward, but...