# Gaussian belief propagation solvers for nonsymmetric systems of linear equations

## TL;DR

This paper extends Gaussian belief propagation to nonsymmetric linear systems, introduces a new matrix inversion algorithm, and demonstrates its advantages over classical methods in terms of robustness and efficiency.

## Contribution

It adapts Gaussian belief propagation for nonsymmetric matrices and proposes a new belief propagation-based matrix inversion algorithm.

## Key findings

- Algorithms outperform classical solvers like Gauss-Seidel and BiCGSTAB.
- Gaussian belief propagation serves as an effective multigrid smoother.
- Proposed methods show increased robustness in solving linear elliptic equations.

## Abstract

In this paper, we argue for the utility of deterministic inference in the classical problem of numerical linear algebra, that of solving a linear system. We show how the Gaussian belief propagation solver, known to work for symmetric matrices can be modified to handle nonsymmetric matrices. Furthermore, we introduce a new algorithm for matrix inversion that corresponds to the generalized belief propagation derived from the cluster variation method (or Kikuchi approximation). We relate these algorithms to LU and block LU decompositions and provide certain guarantees based on theorems from the theory of belief propagation. All proposed algorithms are compared with classical solvers (e.g., Gauss-Seidel, BiCGSTAB) with application to linear elliptic equations. We also show how the Gaussian belief propagation can be used as multigrid smoother, resulting in a substantially more robust solver than the one based on the Gauss-Seidel iterative method.