Publications Details

Convergence of Probability Densities Using Approximate Models for Forward and Inverse Problems in Uncertainty Quantification

Butler, T.; Jakeman, John D.; Wildey, Timothy M.

A previous study analyzed the convergence of probability densities for forward and inverse problems when a sequence of approximate maps between model inputs and outputs converges in L^∞. Our report generalizes the analysis to cases where the approximate maps converge in L^P for any 1 ≤ p < ∞. In particular, under the assumption that the approximate maps converge in L^P, the convergence of probability density functions solving either forward or inverse problems is proven in V where the value of 1 ≤ q < ∞ may even be greater than p in certain cases. This greatly expands the applicability of the previous results to commonly used methods for approximating models (such as polynomial chaos expansions) that only guarantee L^P convergence for some 1 ≤ p < ∞. Severalnumerical examples are also included along with numerical diagnostics of solutions and verification of assumptions made in the analysis.