论文标题
具有高斯随机场输入的PDE的不确定性定量中的分析性和稀疏性
Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs
论文作者
论文摘要
我们为维纳 - 热矿多项式混乱的系数序列建立了稀疏性和可总结性结果，该系数的线性椭圆形和抛物线差异形式的偏置溶液具有高斯随机场输入。 此处开发的新型证明技术基于将参数解决方案延续到复杂域中的分析。它不同于以前使用引导性参数的作品和有关参数的溶液衍生物的分化顺序的诱导。目前的基于Holomorphy的论点允许统一的``无分化''稀疏证明（用$ \ ell^p $ - summability或加权$ \ ell^2 $ - 2 $ - summability）的wiener-hermite系数序列在各种功能空间的各个尺度中的多项式混沌扩展中的序列。该分析还意味着相应的分析性和稀疏性结果，贝叶斯反问题的后验密度在功能空间不确定的投入方面受到高斯先验。 我们的结果此外，各种\ emph {建设性}的高维确定性数值近似方案（例如单级和多级版本的Hermite-smolyak Anisotropic稀疏稀疏网格插入和Quadration in Compartitional Comporty Intervication Intervicational Interationalitionalition tationalitionalitivation tositionalitionalitive ventical Interation vationalitivation tationalitive andivalitionalition factortional offertional of Expertional notiveration formention opentive}的收敛速率。
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.
