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
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 Schroedinger 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
LU-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
9th Workshop on Applications for Multi-Core Architectures
30th IEEE International Symposium on Computer, Architecture and High Performance Computing (SBAC-PAD 2018)