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

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" through 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 [1-3], from now on abbreviated as IFMM). We rely on our new Nested Cross Approximation [4] (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.