Non-intrusive reduced-order modeling for uncertainty quantification of space–time-dependent parameterized problems

Abstract

We propose a non-intrusive reduced-order modeling method for space–time-dependent parameterized problems in the context of uncertainty quantification. In the offline stage, proper orthogonal decomposition (POD) is used to extract the spatial modes based on a set of high-fidelity time-parameter-dependent snapshots and then to extract the temporal modes of the projection coefficients of the spatial modes. Finally, parameter-dependent combination coefficients are approximated using polynomial chaos expansions (PCEs). Cubic spline interpolation is used to evaluate the temporal modes at any other given time. In the online stage, a fast evaluation is provided at any given time and parameter by simply estimating the values of the polynomials and temporal modes. To validate the numerical performance of the proposed method, three time-dependent parameterized problems are tested: a one-dimensional (1-D) forced Burger’s equation with a random force term, a 1-D diffusion–reaction equation with a random field force term, and a two-dimensional incompressible fluid flow over a cylinder with a random inflow boundary condition. The results indicate that the proposed method is able to approximate the full-order model very inexpensively with a reasonable loss of accuracy for problems with uncorrelated or correlated input parameters. Furthermore, the proposed method is effective in estimating low-order moments, indicating that it has great potential for use in uncertainty quantification analysis of space–time-dependent problems.