Highly Scalable Stencil-Based Matrix-Free Stochastic Estimator for the Diagonal of the Inverse

Abstract

Selected inversion problems must be addressed in several research fields like physics, genetics, weather forecasting, and finance, in order to extract selected entries from the inverse of large, sparse matrices. State-of-the-art algorithms are either based on the LU factorization or on an iterative process. Both approaches present computational bottlenecks related to prohibitive memory requirements or extremely high running time for large-scale matrices. In recent years, in order to overcome such limitations, an alternative approach for computing stochastic estimates of the inverse entries has been developed. In this work, we present a stochastic estimator for the diagonal of the inverse and test its performance on a dataset of symmetric, positive semidefinite matrices coming from the field of atomistic quantum transport simulations with nonequilibrium Green's functions (NEGF) formalism. In such a framework, it is required to solve the Schrödinger equation thousands of times, demanding the computation of the diagonal of the retarded Green's function, i.e., the inverse of a large, sparse matrix including open boundary conditions. Given the nature and the structure of the NEGF matrices, our stochastic estimation framework exploits the capabilities of a stencil-based, matrix-free code, avoiding the fill-in and lack of scalability that the LV-based methods present for three-dimensional nanoelectronic devices. We also illustrate the impact of the stochastic estimator by comparing its accuracy against existing methods and demonstrate its scalability performance on the “Piz Daint” cluster at the Swiss National Supercomputing Center, preparing for postpetascale three-dimensional nanoscale calculations.