Work

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

Year: 2021

Type: preprint

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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2105.11901

Authors: Feng, Xiaobing, Luo, Yan, Vo, Liet, Wang, Zhu

Institutions: University of Tennessee at Knoxville, University of Electronic Science and Technology of China, University of South Carolina

Language: English

Topic: Matrix Theory and Algorithms

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, National Natural Science Foundation of China, Natural Science Foundation for Young Scientists of Shanxi Province, Young Scientists Fund, Fundamental Research Funds for the Central Universities

Awards: Novel numerical methods for fully nonlinear second order elliptic and parabolic Monge-Ampere and Hamilton-Jacobi-Bellman equations, Efficient Numerical Simulations of Oceanic Flows with Application to Coastal Modeling, G602113614, Efficient Numerical Methods and Algorithms for Nonlinear Stochastic Partial Differential Equations

OpenAlex ID: W3205536016

Cites: 20

Cited by: 1

Related to: 10