Recursive sparse LU decomposition based on nested dissection and low rank approximations

TL;DR

A recursive sparse LU decomposition for matrices arising from the discretization of linear PDEs is proposed based on the nested dissection and low rank approximations, yielding a fast direct solver for sparse matrices, applicable to both symmetric and non-symmetric cases.

Abstract

When solving partial differential equations (PDEs) using finite difference or finite element methods, efficient solvers are required for handling large sparse linear systems. In this paper, a recursive sparse LU decomposition for matrices arising from the discretization of linear PDEs is proposed based on the nested dissection and low rank approximations. The matrix is reorganized based on the nested structure of the associated graph. After eliminating the interior vertices at the finest level, dense blocks on the separators are hierarchically sparsified using low rank approximations. To efficiently skeletonize these dense blocks, we split the separators into segments and introduce a hybrid algorithm to extract the low rank structures based on a randomized algorithm and the fast multipole method. The resulting decomposition yields a fast direct solver for sparse matrices, applicable to both symmetric and non-symmetric cases. Under a mild assumption on the compression rate of dense blocks, we prove an $Ø(N)$ complexity for the fast direct solver. Several numerical experiments are provided to verify the effectiveness of the proposed method.

Figures (14)

• Figure 1: A reordering of matrix elements. (a): In the arrow-like matrix, a large number of fill-ins will be introduced in the LU elimination. (b): In the reordered reverse-arrow-like matrix, the LU elimination creates a minimal number of fill-ins.
• Figure 2: An illustration of nested dissection.
• Figure 3: Splitting a separator into segments. (a): Separator $S_i$ is split into 3 segments by the separator $S_j$. (b): Segment $S_i^1$ is split into smaller segments at the lower level.
• Figure 4: Sparsification of segments. (a): Two segments $S_i^l$ and $S_i^k$ belong to the same separator $S_i$ but have disjoint edge relations. (b): After the sparsification, the skeleton $V_s$ of $S_i^k$ still connects to $V_n$, the neighbor of $S_i^k$, but its remainder $V_r$ does not connect to $V_n$ anymore.
• Figure 5: A sketch of generating separators. Given the separator direction $\vec{d}$ in each subgraph, the separator is generated by expanding from the center vertex (in red) along the direction $\vec{d}$ (in blue) and $-\vec{d}$ (in green).

Theorems & Definitions (1)

• Remark 5.1