Efficient Bayesian Parameter Inversion Facilitated by Multi-Fidelity Modeling

We propose an efficient Bayesian parameter inversion technique that utilizes the implicit particle filter to characterize the posterior distribution, and a multi-scale surrogate modeling method called the proper orthogonal decomposition mapping method to provide high-fidelity solutions to the forward model by conducting only low-fidelity simulations. The proposed method is applied to the nonlinear Burgers equation, widely used to model electromagnetic waves, with stochastic viscosity and periodic solutions. We consider solving the equation with a coarsely-discretized finite difference scheme, of which the solutions are used as the low-fidelity solutions, and a Fourier spectral collocation method, which can provide high-fidelity solutions. The results demonstrate that the computational cost of characterizing the posterior distribution of viscosity is greatly reduced by utilizing the low-fidelity simulations, while the loss of accuracy is unnoticeable.