An ensemble-based efficient iterative method for uncertainty quantification of partial differential equations with random inputs

In this paper, an ensemble-based efficient iterative method is used to solve the partial differential equations (PDEs) with random inputs. The aim of the efficient iterative method is to get a good approximation of the Galerkin solution for PDEs with random inputs. An essential ingredient of the proposed method is to construct the decomposition of stochastic functions, involving parameter-independent and parameter-dependent. The parameter-dependent term can affect the computation efficiency and approximation accuracy. In order to decrease the computation cost, the efficient iterative method by the decomposition is performed by a fixed-point iterative manner. The computation of the efficient iterative method decomposes into offline phase and online phase. The parameter-independent matrices can be precomputed and stored in offline stage. At online stage, a group of numerical simulations is simultaneously calculated in each iterative step. For the parameter identification, the proposed inversion method combines the advantages of the ensemble-based efficient iterative method and ensemble filtering. Then four models with random inputs are considered to formulate the details and methodologies of the proposed method. To illustrate the computation efficiency and approximation accuracy, the results of the efficient iterative method are compared with the model order reduction methods.