Algebraic Inverse Fast Multipole Method: A fast direct solver that is better than HODLR based fast direct solver

TL;DR

This paper introduces the Algebraic Inverse Fast Multipole Method (AIFMM), a more efficient algebraic fast direct solver for N-body problems that outperforms existing methods like HODLR and can serve as a preconditioner.

Contribution

The paper presents a new algebraic AIFMM using Nested Cross Approximation, improving efficiency over existing IFMMs by updating fill-ins once, and compares it with HODLR and GMRES solvers.

Findings

AIFMM is more efficient than existing IFMMs.

AIFMM outperforms HODLR based solvers.

AIFMM can be used as an effective preconditioner.

Abstract

This article presents a fast direct solver, termed Algebraic Inverse Fast Multipole Method (from now on abbreviated as AIFMM), for linear systems arising out of $N$-body problems. AIFMM relies on the following three main ideas: (i) Certain sub-blocks in the matrix corresponding to $N$-body problems can be efficiently represented as low-rank matrices; (ii) The low-rank sub-blocks in the above matrix are leveraged to construct an extended sparse linear system; (iii) While solving the extended sparse linear system, certain fill-ins that arise in the elimination phase are represented as low-rank matrices and are "redirected" though other variables maintaining zero fill-in sparsity. The main highlights of this article are the following: (i) Our method is completely algebraic (as opposed to the existing Inverse Fast Multipole Method~\cite{ arXiv:1407.1572,doi:10.1137/15M1034477,TAKAHASHI2017406}, from now on abbreviated as IFMM). We rely on our new Nested Cross Approximation~\cite{arXiv:2203.14832} (from now on abbreviated as NNCA) to represent the matrix arising out of $N$-body problems. (ii) A significant contribution is that the algorithm presented in this article is more efficient than the existing IFMMs. In the existing IFMMs, the fill-ins are compressed and redirected as and when they are created. Whereas in this article, we update the fill-ins first without affecting the computational complexity. We then compress and redirect them only once. (iii) Another noteworthy contribution of this article is that we provide a comparison of AIFMM with Hierarchical Off-Diagonal Low-Rank (from now on abbreviated as HODLR) based fast direct solver and NNCA powered GMRES based fast iterative solver. (iv) Additionally, AIFMM is also demonstrated as a preconditioner.