@Article{CiCP-6-826,
author = {Youssef Marzouk and Dongbin Xiu},
title = {A Stochastic Collocation Approach to Bayesian Inference in Inverse Problems},
journal = {Communications in Computational Physics},
year = {2009},
volume = {6},
number = {4},
pages = {826--847},
abstract = {<p style="text-align: justify;">We present an efficient numerical strategy for the Bayesian solution of inverse
problems. Stochastic collocation methods, based on generalized polynomial
chaos (gPC), are used to construct a polynomial approximation of the forward solution
over the support of the prior distribution. This approximation then defines a surrogate
posterior probability density that can be evaluated repeatedly at minimal computational
cost. The ability to simulate a large number of samples from the posterior
distribution results in very accurate estimates of the inverse solution and its associated
uncertainty. Combined with high accuracy of the gPC-based forward solver, the
new algorithm can provide great efficiency in practical applications. A rigorous error
analysis of the algorithm is conducted, where we establish convergence of the approximate
posterior to the true posterior and obtain an estimate of the convergence rate. It
is proved that fast (exponential) convergence of the gPC forward solution yields similarly
fast (exponential) convergence of the posterior. The numerical strategy and the
predicted convergence rates are then demonstrated on nonlinear inverse problems of
varying smoothness and dimension.</p>},
issn = {1991-7120},
doi = {https://doi.org/},
url = {http://global-sci.org/intro/article_detail/cicp/7708.html}
}