An efficient iterative method for solving parameter-dependent and random convention-diffusion problems

TL;DR

This paper introduces a unified iterative framework for efficiently solving parameter-dependent and random convection-diffusion problems, leveraging reformulation and fixed-point iteration to enable reuse of solvers and reduce computational costs.

Contribution

It extends existing multi-modes and ensemble methods into a general framework with proven convergence, enabling efficient reuse of LU factorizations and matrix operations for parameter-dependent problems.

Findings

The method achieves significant computational savings.

Convergence is established at both continuous and discrete levels.

Numerical experiments validate efficiency and theoretical results.

Abstract

This paper develops and analyzes a general iterative framework for solving parameter-dependent and random convection-diffusion problems. It is inspired by the multi-modes method of [7,8] and the ensemble method of [20] and extends those methods into a more general and unified framework. The main idea of the framework is to reformulate the underlying problem into another problem with parameter-independent convection and diffusion coefficients and a parameter-dependent (and solution-dependent) right-hand side, a fixed-point iteration is then employed to compute the solution of the reformulated problem. The main benefit of the proposed approach is that an efficient direct solver and a block Krylov subspace iterative solver can be used at each iteration, allowing to reuse the $LU$ matrix factorization or to do an efficient matrix-matrix multiplication for all parameters, which in turn results in significant computation saving. Convergence and rates of convergence are established for the iterative method both at the variational continuous level and at the finite element discrete level under some structure conditions. Several strategies for establishing reformulations of parameter-dependent and random diffusion and convection-diffusion problems are proposed and their computational complexity is analyzed. Several 1-D and 2-D numerical experiments are also provided to demonstrate the efficiency of the proposed iterative method and to validate the theoretical convergence results.