Stochastic spectral methods for efficient Bayesian solution of inverse problems

The Bayesian setting for inverse problems provides a rigorous foundation for inference from noisy data and uncertain forward models, a natural mechanism for incorporating prior information, and a quantitative assessment of uncertainty in the inferred results. Obtaining useful information from the posterior density—e.g., computing expectations via Markov Chain Monte Carlo (MCMC)—may be a computationally expensive undertaking, however. For complex and high‐dimensional forward models, such as those that arise in inverting systems of PDEs, the cost of likelihood evaluations may render MCMC simulation prohibitive.We explore the use of polynomial chaos (PC) expansions for spectral representation of stochastic model parameters in the Bayesian context. The PC construction employs orthogonal polynomials in i.i.d. random variables as a basis for the space of square‐integrable random variables. We use a Galerkin projection of the forward operator onto this basis to obtain a PC expansion for the outputs of the forwar...