A Stochastic Collocation Approach to Bayesian Inference in Inverse Problems (Communications in Computational Physics, 2009)
We present an efficient numerical strategy for the Bayesian solution of inverseproblems. Stochastic collocation methods, based on generalized polynomialchaos (gPC), are used to construct a polynomial approximation of the forward solutionover the support of the prior distribution. This approximation then defines a surrogateposterior probability density that can be evaluated repeatedly at minimal computationalcost. The ability to simulate a large number of samples from the posteriordistribution results in very accurate estimates of the inverse solution and its associateduncertainty. Combined with high accuracy of the gPC-based forward solver, thenew algorithm can provide great efficiency in practical applications. A rigorous erroranalysis of the algorithm is conducted, where we establish convergence of the approximateposterior to the true posterior and obtain an estimate of the convergence rate. Itis proved that fast (exponential) convergence of the gPC forward solution yields similarlyfast (exponential) convergence of the posterior. The numerical strategy and thepredicted convergence rates are then demonstrated on nonlinear inverse problems ofvarying smoothness and dimension.