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In this work, we consider optimal control problems constrained by ellipticpartial differential equations (PDEs) with lognormal random coefficients, whichare represented by a countably infinite-dimensional random parameter withi.i.d. normal distribution. We approximate the optimal solution by a suitabletruncation of its Hermite polynomial chaos expansion, which is known as asparse polynomial approximation. Based on the convergence analysis in\cite{BachmayrCohenDeVoreEtAl2017} for elliptic PDEs with lognormal randomcoefficients, we establish the dimension-independent convergence rate of thesparse polynomial approximation of the optimal solution. Moreover, we present apolynomial-based sparse quadrature for the approximation of the expectation ofthe optimal solution and prove its dimension-independent convergence rate basedon the analysis in \cite{Chen2018}. Numerical experiments demonstrate that theconvergence of the sparse quadrature error is independent of the activeparameter dimensions and can be much faster than that of a Monte Carlo method.

Sparse polynomial approximation for optimal control problems constrained by elliptic PDEs with lognormal random coefficients