Publications

Stochastic spectral methods for efficient Bayesian solution of inverse problems

Marzouk, Youssef M.; Najm, H.N.; Rahn, Larry A.

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 forward problem. Evaluation of integrals over the parameter space is recast as Monte Carlo sampling of the random variables underlying the PC expansion. We evaluate the utility of this technique on a transient diffusion problem arising in contaminant source inversion. The accuracy of posterior estimates is examined with respect to the order of the PC representation and the decomposition of the support of the prior. We contrast the computational cost of the new scheme with that of direct sampling. © 2005 American Institute of Physics.