Accelerating the Bayesian inference of inverse problems by using data-driven compressive sensing method based on proper orthogonal decomposition

In Bayesian inverse problems, using the Markov Chain Monte Carlo method to sample from the posterior space of unknown parameters is a formidable challenge due to the requirement of evaluating the forward model a large number of times. For the purpose of accelerating the inference of the Bayesian inverse problems, in this work, we present a proper orthogonal decomposition (POD) based data-driven compressive sensing (DCS) method and construct a low dimensional approximation to the stochastic surrogate model on the prior support. Specifically, we first use POD to generate a reduced order model. Then we construct a compressed polynomial approximation by using a stochastic collocation method based on the generalized polynomial chaos expansion and solving an \begin{document}$ l_1 $\end{document} -minimization problem. Rigorous error analysis and coefficient estimation was provided. Numerical experiments on stochastic elliptic inverse problem were performed to verify the effectiveness of our POD-DCS method.