We present efficient algorithms for solving systems of linear equations in weighted $1$-Laplacians of well-shaped simplicial complexes. $1$- or higher-dimensional Laplacians generalize graph Laplacians to higher-dimensional simplicial complexes. Previously, nearly-linear time solvers were designed for unweighted simplicial complexes that triangulate a three-ball in $\mathbb{R}^3$ (Cohen, Fasy, Miller, Nayyeri, Peng, and Walkington [SODA'2013]) and their sub-complexes (Black, Maxwell, Nayyeri, and Winkelman [SODA'2022], Black and Nayyeri [ICALP'2022]). Additionally, quadratic time solvers by Nested Dissection exist for more general systems whose nonzero structures encode well-shaped simplicial complexes embedded in $\mathbb{R}^3$. We generalize the specialized solvers for $1$-Laplacians to weighted simplicial complexes with additional geometric structures and improve the runtime of Nested Dissection. Specifically, we consider simplicial complexes embedded in $\mathbb{R}^3$ such that: (1) the complex triangulates a convex ball in $\mathbb{R}^3$, (2) the underlying topological space of the complex is convex and has a bounded aspect ratio, and (3) each tetrahedron has a bounded aspect ratio and volume. We say such a simplicial complex is stable. We can approximately solve weighted $1$-Laplacian systems in a stable simplicial complex with $n$ simplexes up to high precision in time $\tilde{O} (n^{3/2})$ if the ratio between the maximum and minimum simplex weights is $\tilde{O}(n^{1/6})$. In addition, we generalize this solver to a union of stable simplicial complex chunks. As a result, our solver has a comparable runtime, parameterized by the number of chunks and the number of simplexes shared by more than one chunk. Our solvers are inspired by the Incomplete Nested Dissection designed by Kyng, Peng, Schwieterman, and Zhang [STOC'2018] for stiffness matrices of well-shaped trusses.

翻译：我们提出了高效算法，用于求解良形单纯复形的加权$1$-拉普拉斯算子中的线性方程组。$1$维或更高维的拉普拉斯算子将图拉普拉斯算子推广到更高维单纯复形。此前，近线性时间求解器已设计用于三角化$\mathbb{R}^3$中三维球体的无权重单纯复形（Cohen, Fasy, Miller, Nayyeri, Peng, and Walkington [SODA'2013]）及其子复形（Black, Maxwell, Nayyeri, and Winkelman [SODA'2022], Black and Nayyeri [ICALP'2022]）。此外，嵌套剖分法以二次时间求解更一般的系统，其非零结构编码了嵌入$\mathbb{R}^3$中的良形单纯复形。我们将面向$1$-拉普拉斯算子的专用求解器推广到具有额外几何结构的加权单纯复形，并改进了嵌套剖分法的运行时间。具体而言，我们考虑嵌入$\mathbb{R}^3$中的单纯复形满足：(1) 该复形三角化$\mathbb{R}^3$中的凸球体，(2) 复形的底层拓扑空间是凸的且具有有界纵横比，(3) 每个四面体具有有界纵横比和体积。我们称此类单纯复形为稳定的。对于包含$n个单形、最大与最小单形权重之比为$\tilde{O}(n^{1/6})$的稳定单纯复形，我们可在$\tilde{O}(n^{3/2})$时间内高精度近似求解加权$1$-拉普拉斯系统。此外，我们将此求解器推广到稳定单纯复形块的并集。因此，我们的求解器具有可比的运行时间，其参数取决于块的数量和被多个块共享的单形数量。我们的求解器受Kyng, Peng, Schwieterman, and Zhang [STOC'2018]针对良形桁架刚度矩阵设计的不完全嵌套剖分法启发。