Incomplete Nested Dissection
Rasmus Kyng, Richard Peng, Robert Schwieterman, Peng Zhang

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 vertices which is formed from a union of 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 in time . This asymptotically improves the running time by Nested Dissection…
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Markov Chains and Monte Carlo Methods · Topological and Geometric Data Analysis
