Hodge Decomposition and General Laplacian Solvers for Embedded Simplicial Complexes

TL;DR

This paper introduces a nearly-linear time algorithm for solving 1-Laplacian linear systems on embedded simplicial complexes, generalizing previous work to complexes with arbitrary first homology.

Contribution

It presents a novel nearly-linear time algorithm for computing the Hodge decomposition and solving 1-Laplacian systems on complexes with arbitrary first homology.

Findings

Efficient nearly-linear time solver for 1-Laplacian systems.

Algorithm extends to complexes with non-trivial first homology.

Implication of nearly quadratic solver and Hodge decomposition for embedded complexes.

Abstract

We describe a nearly-linear time algorithm to solve the linear system $L_1x = b$ parameterized by the first Betti number of the complex, where $L_1$ is the 1-Laplacian of a simplicial complex $K$ that is a subcomplex of a collapsible complex $X$ linearly embedded in $\mathbb{R}^{3}$. Our algorithm generalizes the work of Black et al.~[SODA2022] that solved the same problem but required that $K$ have trivial first homology. Our algorithm works for complexes $K$ with arbitrary first homology with running time that is nearly-linear with respect to the size of the complex and polynomial with respect to the first Betti number. The key to our solver is a new algorithm for computing the Hodge decomposition of 1-chains of $K$ in nearly-linear time. Additionally, our algorithm implies a nearly quadratic solver and nearly quadratic Hodge decomposition for the 1-Laplacian of any simplicial complex $K$ embedded in $\mathbb{R}^{3}$, as $K$ can always be expanded to a collapsible embedded complex of quadratic complexity.