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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 advantagesof representations using functions with localized supports. As model problem,we consider the steady-state diffusion equation, where the diffusioncoefficient 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 lognormalcase, we use pointwise instead of norm-wise bounds to prove$\ell^p$-summability of the Taylor coefficients of the solution. Asapplication, we consider surrogates for solutions to elliptic PDEs onparametric domains. Using a mapping to a nominal configuration, this case fitsin the general framework, and higher convergence rates can be attained whenmodeling the parametric boundary via spatially localized functions. Thetheoretical results are supported by numerical experiments for the parametricdomain problem, illustrating the efficiency of the proposed approach andproviding further insight on numerical aspects. Although the methods and ideasare carried out for the steady-state diffusion equation, they extend easily toother elliptic and parabolic PDEs.

Exploiting locality in sparse polynomial approximation of parametric elliptic PDEs and application to parameterized domains