论文标题
使用近似模型在不确定性量化中使用近似模型的概率密度收敛：扩展到$ l^p $
Convergence of Probability Densities using Approximate Models for Forward and Inverse Problems in Uncertainty Quantification: Extensions to $L^p$
论文作者
论文摘要
先前的研究分析了当模型输入和输出之间的一系列近似地图收敛于$ l^\ infty $时，概率密度的概率密度收敛。这项工作将分析概括为近似地图在任何$ 1 \ leq p <\ infty $的$ l^p $中收敛的情况。具体而言，在假设近似地图收敛于$ l^p $的假设下，在$ l^q $中证明了概率密度函数的收敛性在$ 1 \ leq q <\ iffty $中甚至可能大于$ p $，在某些情况下，$ l^q $证明了。这大大扩展了以前的结果的适用性，以用于近似模型（例如多项式混乱扩展）的常用方法，该方法仅保证$ 1 \ leq p <\ infty $。还包括了几个数值示例，以及解决方案的数值诊断以及分析中做出的假设的验证。
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.
