Analytic regularity and nonlinear approximation of a class of parametric semilinear elliptic PDEs

We investigate existence and regularity of a class of semilinear, parametric elliptic PDEs with affine dependence of the principal part of the differential operator on countably many parameters. We establish a priori estimates and analyticity of the parametric solutions. We establish summability results of coefficient sequences of polynomial chaos type expansions of the parametric solutions in terms of tensorized Taylor‐, Legendre‐ and Chebyshev polynomials on the infinite‐dimensional parameter domain. We deduce rates of convergence for N term truncated approximations of expansions of the parametric solution. We also deduce spatial regularity of the solution, and establish convergence rates of N‐term discretizations of the parametric solutions with respect to these polynomials in parameter space and with respect to a multilevel hierarchy of finite element spaces in the spatial domain of the PDE.

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