Convergence of Probability Densities using Approximate Models for Forward and Inverse Problems in Uncertainty Quantification: Extensions to $L^p$

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 $L^\infty$. This work generalizes the analysis to cases where the approximate maps converge in $L^p$ for any $1\leq p < \infty$. Specifically, 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 $L^q$ where the value of $1\leq q<\infty$ 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 $1\leq p<\infty$. Several numerical examples are also included along with numerical diagnostics of solutions and verification of assumptions made in the analysis.