Uncertainty Quantification for Numerical Solutions of the Nonlinear Partial Differential Equations by Using the Multi-Fidelity Monte Carlo Method

The Monte Carlo simulation is a popular statistical method to estimate the effect of uncertainties on the solutions of nonlinear partial differential equations, but it requires a huge computational cost of the deterministic model, and the convergence may become slow. For this reason, we developed the multi-fidelity Monte Carlo (MFMC) methods based on data-driven low-fidelity models for uncertainty analysis of nonlinear partial differential equations. Firstly, the nonlinear partial differential equations are transformed into ordinary differential equations (ODEs) by using finite difference discretization or Fourier transformation. Then, the reduced dimension model and discrete empirical interpolation method (DEIM) are coupled to construct effective nonlinear low-fidelity models in ODEs system. Finally, the MFMC method is used to combine the output information of the high-fidelity model and the low-fidelity models to give the optimal estimation of the statistics. Experimental results of the nonlinear Schrodinger equation and the Burgers' equation show that, compared with the standard Monte Carlo method, the MFMC method based on the data-driven low-fidelity model in this paper can improve the calculation efficiency significantly.