Incomplete Nested Dissection

TL;DR

This paper introduces a faster algorithm for solving linear systems in 3D truss stiffness matrices, leveraging a novel combination of nested dissection and support theory to improve computational efficiency over traditional methods.

Contribution

The paper presents a new algorithm that combines nested dissection and support theory to solve 3D truss linear systems more efficiently than existing methods.

Findings

Achieves faster solution times for well-structured 3D truss systems.

Improves over Nested Dissection for systems with small k relative to n.

Provides bounds on the spectrum of decomposed regions to enhance solver performance.

Abstract

We present an asymptotically faster algorithm for solving linear systems in well-structured 3-dimensional truss stiffness matrices. These linear systems arise from linear elasticity problems, and can be viewed as extensions of graph Laplacians into higher dimensions. Faster solvers for the 2-D variants of such systems have been studied using generalizations of tools for solving graph Laplacians [Daitch-Spielman CSC'07, Shklarski-Toledo SIMAX'08]. Given a 3-dimensional truss over $n$ vertices which is formed from a union of $k$ convex structures (tetrahedral meshes) with bounded aspect ratios, whose individual tetrahedrons are also in some sense well-conditioned, our algorithm solves a linear system in the associated stiffness matrix up to accuracy $\epsilon$ in time $O(k^{1/3} n^{5/3} \log (1 / \epsilon))$. This asymptotically improves the running time $O(n^2)$ by Nested Dissection for all $k \ll n$. We also give a result that improves on Nested Dissection even when we allow any aspect ratio for each of the $k$ convex structures (but we still require well-conditioned individual tetrahedrons). In this regime, we improve on Nested Dissection for $k \ll n^{1/44}$. The key idea of our algorithm is to combine nested dissection and support theory. Both of these techniques for solving linear systems are well studied, but usually separately. Our algorithm decomposes a 3-dimensional truss into separate and balanced regions with small boundaries. We then bound the spectrum of each such region separately, and utilize such bounds to obtain improved algorithms by preconditioning with partial states of separator-based Gaussian elimination.