Adaptive multi-fidelity polynomial chaos approach to Bayesian inference in inverse problems

Abstract

The polynomial chaos (PC) expansion has been widely used as a surrogate model in the Bayesian inference to speed up the Markov chain Monte Carlo (MCMC) calculations. However, the use of a PC surrogate introduces the modeling error, that may severely distort the estimate of the posterior distribution. This error can be corrected by increasing the order of the PC expansion, but the cost for building the surrogate may increase dramatically. In this work, we seek to address this challenge by proposing an adaptive procedure to construct a multi-fidelity PC surrogate. This new strategy combines (a large number of) low-fidelity surrogate model evaluations and (a small number of) high-fidelity model evaluations, yielding an adaptive multi-fidelity approach. Here the low-fidelity surrogate is chosen as the prior-based PC surrogate, while the high-fidelity model refers to the true forward model. The key idea is to construct and refine the multi-fidelity approach over a sequence of samples adaptively determined form data, so that the approximation can eventually concentrate to the posterior distribution. We illustrate the performance of the proposed strategy through two nonlinear inverse problems. It is shown that the proposed adaptive multi-fidelity approach can improve significantly the accuracy, yet without a dramatic increase in the computational complexity. The numerical results also indicate that our new algorithm can enhance the efficiency by several orders of magnitude compared to a standard MCMC approach using only the true forward model.