This work studies how the choice of the representation for parametric, spatially distributed inputs to elliptic partial differential equations (PDEs) affects the efficiency of a polynomial surrogate, based on Taylor expansion, for the parameter-to-solution map. In particular, we show potential advantages of representations using functions with localized supports. As model problem, we consider the steady-state diffusion equation, where the diffusion coefficient and right-hand side depend smoothly but potentially in a \textsl{highly nonlinear} way on a parameter $y\in [-1,1]^{\mathbb{N}}$. Following previous work for affine parameter dependence and for the lognormal case, we use pointwise instead of norm-wise bounds to prove $\ell^p$-summability of the Taylor coefficients of the solution. As application, we consider surrogates for solutions to elliptic PDEs on parametric domains. Using a mapping to a nominal configuration, this case fits in the general framework, and higher convergence rates can be attained when modeling the parametric boundary via spatially localized functions. The theoretical results are supported by numerical experiments for the parametric domain problem, illustrating the efficiency of the proposed approach and providing further insight on numerical aspects. Although the methods and ideas are carried out for the steady-state diffusion equation, they extend easily to other elliptic and parabolic PDEs.

翻译：本文研究了对椭圆偏微分方程中参数化空间分布输入的不同表示方式如何影响基于泰勒展开的参数-解映射多项式代理效率。特别地，我们展示了使用具有局部支撑函数的表示方法的潜在优势。以稳态扩散方程作为模型问题，其中扩散系数和右端项光滑但可能以高度非线性方式依赖于参数 $y\in [-1,1]^{\mathbb{N}}$。在先前针对仿射参数依赖和对数正态情况的工作基础上，我们采用逐点而非范数型界来证明解泰勒系数的 $\ell^p$ 可和性。作为应用，我们考虑了参数化域上椭圆偏微分方程解的代理模型。通过映射到标称构型，该情形适用于通用框架，当采用空间局部函数建模参数化边界时可获得更高收敛速率。数值实验验证了参数化域问题的理论结果，展示了所提方法的有效性并提供了数值层面的深入见解。虽然方法和思想围绕稳态扩散方程展开，但它们可轻松推广至其他椭圆和抛物型偏微分方程。