A new bi-fidelity model reduction method for Bayesian inverse problems

This work presents a new bi-fidelity model reduction approach to the inverse problem under the framework of Bayesian inference. A low-rank approximation is introduced to the solution of the corresponding forward problem and admits a variable-separation form in terms of stochastic basis functions and physical basis functions. The calculation of stochastic basis functions is computationally predominant for the low-rank expression. To significantly improve the efficiency of constructing the low-rank approximation, we propose a bi-fidelity model reduction based on a novel variable-separation method, where a low-fidelity model is used to compute the stochastic basis functions and a high-fidelity model is used to compute the physical basis functions. The low-fidelity model has lower accuracy but efficient to evaluate compared with the high-fidelity model; it accelerates the derivative of recursive formulation for the stochastic basis functions. The high-fidelity model is computed in parallel for a few samples scattered in the stochastic space when we construct the high-fidelity physical basis functions. The required number of forward model simulations in constructing the basis functions is very limited. The bi-fidelity model can be constructed efficiently while retaining good accuracy simultaneously. In the proposed approach, both the stochastic basis functions and physical basis functions are calculated using the model information. This implies that a few basis functions may accurately represent the model solution in high-dimensional stochastic spaces. The bi-fidelity model reduction is applied to Bayesian inverse problems to accelerate posterior exploration. A few numerical examples in time-fractional derivative diffusion models are carried out to identify the smooth field and channel-structured field in porous media in the framework of Bayesian inverse problems.