An Efficient Iteration Based on Reduced Basis Method for Time-Dependent Problems with Random Inputs (CSIAM Transactions on Applied Mathematics, 2025)
In this paper, we propose an efficient iterative method called RB-iteration,based on reduced basis (RB) techniques, for addressing time-dependent problems withrandom input parameters. This method reformulates the original model such that theleft-hand side is parameter-independent, while the right-hand side remains parameterdependent, facilitating the application of fixed-point iteration for solving the system.High-fidelity simulations for time-dependent problems often demand considerablecomputational resources, rendering them impractical for many applications. RB-iteration enhances computational efficiency by executing iterations in a reduced orderspace. This approach results in significant reductions in computational costs. We conduct a rigorous convergence analysis and present detailed numerical experiments forthe RB-iteration method. Our results clearly demonstrate that RB-iteration achievessuperior efficiency compared to the direct fixed-point iteration method and providesenhanced accuracy relative to the classical proper orthogonal decomposition (POD)greedy method.
Keywords: Reduced order model; time-dependent problems; random inputs; reduced basis; iteration method.