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.
A fast direct solver based neural network for solving PDEs
Department of Computer Science & Engineering, International Institute of Information Technology Bangalore (IIIT-B), India（计算机科学与工程系，国际信息学院班加罗尔（IIIT-B），印度）；Department of Data Science and Artificial Intelligence, International Institute of Information Technology Bangalore (IIIT-B), India（数据科学与人工智能系，国际信息学院班加罗尔（IIIT-B），印度）
本文介绍了一种新的线性方程组快速直接求解器，它产生于积分方程、多元统计、径向基插值等广泛的应用领域，仅举几例。\emph{The highlight of this new fast direct solver is that the solver scales linearly in the number of unknowns in all dimensions.}该求解器称为逆快速多极方法（缩写为IFMM），其工作原理与快速多极方法（缩写为FMM）相同。更一般地，求解器可以立即扩展到分层矩阵类，表示为具有强容许准则（弱低秩结构）的$\mathcal{H}^2$矩阵，即，\emph{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}。该算法从现有的方法出发，在整个算法中，对应于相邻集群的相互作用总是被视为满秩相互作用。我们的方法依赖于两个主要思想：（i）由FMM产生的$N \times N$矩阵（从现在起称为FMM矩阵）可以…
