CHOLESKY-LIKE PRECONDITIONER FOR HODGE LAPLACIANS VIA HEAVY COLLAPSIBLE SUBCOMPLEX

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-squares problems with L k as coefficient matrix, for example, in order to compute some portions of the spectrum or integrate the dynamical system, thus making a fast and efficient solver for the least-squares problems highly desirable. To this aim, we introduce the notion of an optimal weakly collapsible sub complex used to construct an effective sparse Cholesky-like preconditioner for L k that exploits the topological structure of the simplicial complex. The performance of the preconditioner is tested for the conjugate gradient method for least-squares problems (CGLS) on a variety of simplicial complexes with different dimensions and edge densities. We show that, for sparse simplicial complexes, the new preconditioner significantly reduces the condition number of L k and performs better than the standard incomplete Cholesky factorization.