We establish summability results for coefficient sequences of Wiener-Hermite polynomial chaos expansions for countably-parametric solutions of linear elliptic and parabolic divergence-form partial differential equations with Gaussian random field inputs. The novel proof technique developed here is based on analytic continuation of parametric solutions into the complex domain. It differs from previous works that used bootstrap arguments and induction on the differentiation order of solution derivatives with respect to the parameters. The present holomorphy-based argument allows a unified, "differentiation-free" sparsity analysis of Wiener-Hermite polynomial chaos expansions in various scales of function spaces. The analysis also implies corresponding results for posterior densities in Bayesian inverse problems subject to Gaussian priors on uncertain inputs from function spaces. Our results furthermore yield dimension-independent convergence rates of various constructive high-dimensional deterministic numerical approximation schemes such as single-level and multi-level versions of anisotropic sparse-grid Hermite-Smolyak interpolation and quadrature in both forward and inverse computational uncertainty quantification.
翻译:我们为Wiener-Hermite 多元混杂现象的系数序列设定了可测量的系数序列的可比较性结果,以便用高斯随机的实地输入来测量线性椭圆形和parbolic 多元混乱形态的偏差方程式的可测量的参数解决方案。这里开发的新的证据技术是基于对复杂域的参数解决方案的分析性延续。它不同于以前使用靴子捕捉参数的参数参数和关于溶解衍生物差异顺序的引言的工程。目前基于全貌的论证允许对不同功能空间范围内的Wiener-Hermite 线性椭圆形和parblical 多元混乱形态的宽度进行统一、“无差异性”的宽度分析。 分析还意味着,在功能空间的不确定投入上,根据古斯先前的参数,对巴伊斯的后方问题的密度问题得出相应的结果。我们的结果还进一步产生了各种建设性的高度确定性数字接近率,例如,前方和方位和方位数字的单度、方位微阵阵阵阵阵阵阵阵阵阵阵阵阵阵阵阵式间间和方计算中,等的单一和多级定型组合。