Solvers and precondtioners based on Gauss-Seidel and Jacobi algorithms for non-symmetric stochastic Galerkin system of equations
This work provides efficient preconditioners for solving non-symmetric stochastic Galerkin systems, which are computationally challenging in uncertainty quantification.
The authors developed Gauss-Seidel and Jacobi-based solvers and preconditioners for stochastic Galerkin systems arising from PDEs with random inputs. Numerical results show that an approximate Gauss-Seidel preconditioner effectively accelerates GMRES for non-symmetric systems, outperforming traditional mean-based and Kronecker-product preconditioners.
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.