# Solvers and precondtioners based on Gauss-Seidel and Jacobi algorithms for non-symmetric stochastic Galerkin system of equations

## TL;DR

This paper investigates the effectiveness of Gauss-Seidel and Jacobi algorithms as solvers and preconditioners for stochastic Galerkin systems arising from PDEs with random data, demonstrating their practical utility.

## Contribution

It introduces formulations of Gauss-Seidel and Jacobi algorithms tailored for stochastic Galerkin discretizations and compares their performance with existing preconditioners.

## Key findings

- Gauss-Seidel based preconditioner improves GMRES convergence.
- Approximate Gauss-Seidel is effective for non-symmetric systems.
- Preconditioners outperform traditional mean-based and Kronecker-product methods.

## Abstract

In this work, solvers and preconditioners based on Gauss-Seidel and Jacobi algorithms are explored for stochastic Galerkin discretization of partial differential equations (PDEs) with random input data. Gauss-Seidel and Jacobi algorithms are formulated such that the existing software is leveraged in the computational effort. These algorithms are also used as preconditioners to Krylov iterative methods. The solvers and preconditiners are compared with Krylov based iterative methods with the traditional mean-based preconditioner [13] and Kronecker-product preconditioner [17] by solving a steady state state advection-diffusion equation, which upon discretization, results in a non-symmetric positive definite matrix on left-hand-side. Numerical results show that an approximate version of Gauss-Seidel algorithm is a good preconditioner for GMRES to solve non-symmetric Galerkin system of equations.