We establish sparsity and summability results for coefficient sequences of Wiener-Hermite polynomial chaos expansions of 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'' proof of sparsity (expressed in terms of $\ell^p$-summability or weighted $\ell^2$-summability) of sequences of Wiener-Hermite coefficients in polynomial chaos expansions in various scales of function spaces. The analysis also implies corresponding analyticity and sparsity 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 \emph{constructive} high-dimensional deterministic numerical approximation schemes such as single-level and multi-level versions of Hermite-Smolyak anisotropic sparse-grid interpolation and quadrature in both forward and inverse computational uncertainty quantification.

翻译：我们针对含高斯随机场输入的线性椭圆型与抛物型散度形式偏微分方程的计数参数化解，建立了维纳-埃尔米特多项式混沌展开系数序列的稀疏性和可和性结果。本文发展的新型证明技术基于参数解向复域的解析延拓，与以往采用靴带论证和参数导数阶数归纳法的研究不同。本全纯性论证方法实现了维纳-埃尔米特多项式混沌展开系数序列在不同函数空间尺度下稀疏性（以$\ell^p$可和性或加权$\ell^2$可和性表征）的统一“无微分”证明。该分析还推导了以函数空间不确定输入高斯先验为条件的贝叶斯反问题后验密度的相应分析性与稀疏性结果。我们的研究进一步证明了正向与计算不确定度量化中多种构造性高维确定性数值逼近方案（如埃尔米特-斯莫利亚克各向异性稀疏网格插值与求积的单层与多层版本）的维数无关收敛速率。