Techniques based on $k$-th order Hodge Laplacian operators $L_k$ are widely used to describe the topology as well as the governing dynamics of high-order systems modeled as simplicial complexes. In all of them, it is required to solve a number of least square problems with $L_k$ as coefficient matrix, for example in order to compute some portions of the spectrum or integrate the dynamical system. In this work, we introduce the notion of optimal collapsible subcomplex and we present a fast combinatorial algorithm for the computation of a sparse Cholesky-like preconditioner for $L_k$ that exploits the topological structure of the simplicial complex. The performance of the preconditioner is tested for conjugate gradient method for least square problems (CGLS) on a variety of simplicial complexes with different dimensions and edge densities. We show that, for sparse simplicial complexes, the new preconditioner reduces significantly the condition number of $L_k$ and performs better than the standard incomplete Cholesky factorization.

翻译：基于$k$阶Hodge拉普拉斯算子$L_k$的技术被广泛用于描述建模为单纯复形的高阶系统的拓扑结构及主导动力学。在这类应用中，需要在以$L_k$为系数矩阵的情况下求解多个最小二乘问题，例如计算谱的某些部分或对动力系统进行积分。本文引入了最优可坍缩子复的概念，并提出了一种快速组合算法，用于计算$L_k$的稀疏类Cholesky预处理器，该算法利用了单纯复形的拓扑结构。我们在不同维度和边密度的多种单纯复形上，针对最小二乘问题的共轭梯度法（CGLS）测试了该预处理器的性能。结果表明，对于稀疏单纯复形，新预处理器显著降低了$L_k$的条件数，且其性能优于标准的不完全Cholesky分解。