Efficient $1$-Laplacian Solvers for Well-Shaped Simplicial Complexes: Beyond Betti Numbers and Collapsing Sequences

TL;DR

This paper introduces efficient algorithms for approximately solving systems involving 1-Laplacians of well-shaped simplicial complexes, extending previous methods to more general structures without collapsing sequences and improving runtime.

Contribution

It generalizes specialized solvers for 1-Laplacians to broader classes of simplicial complexes with geometric structures, removing the need for collapsing sequences and bounded Betti numbers, and enhances Nested Dissection runtime.

Findings

Achieved nearly-linear time approximate solvers for new classes of complexes.

Extended Nested Dissection to improve runtime for general systems.

Demonstrated applicability to well-shaped simplicial complexes in computational topology.

Abstract

We present efficient algorithms for approximately solving systems of linear equations in $1$-Laplacians of well-shaped simplicial complexes up to high precision. $1$-Laplacians, or higher-dimensional Laplacians, generalize graph Laplacians to higher-dimensional simplicial complexes and play a key role in computational topology and topological data analysis. Previously, nearly-linear time approximate solvers were developed for simplicial complexes with known collapsing sequences and bounded Betti numbers, such as those triangulating a three-ball in $\mathbb{R}^3$ (Cohen, Fasy, Miller, Nayyeri, Peng, and Walkington [SODA'2014], Black, Maxwell, Nayyeri, and Winkelman [SODA'2022], Black and Nayyeri [ICALP'2022]). Furthermore, Nested Dissection provides quadratic time exact solvers for more general systems with nonzero structures representing well-shaped simplicial complexes embedded in $\mathbb{R}^3$. We generalize the specialized solvers for $1$-Laplacians to simplicial complexes with additional geometric structures but without collapsing sequences and bounded Betti numbers, and we improve the runtime of Nested Dissection. We focus on simplicial complexes that meet two conditions: (1) each individual simplex has a bounded aspect ratio, and (2) they can be divided into "disjoint" and balanced regions with well-shaped interiors and boundaries. Our solvers draw inspiration from the Incomplete Nested Dissection for stiffness matrices of well-shaped trusses (Kyng, Peng, Schwieterman, and Zhang [STOC'2018]).