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Sparse adaptive tensor Galerkin approximations of stochastic PDE-constrained control problems

by A. Kunoth and Ch. Schwab

(Report number 2015-37)

Abstract
For control problems constrained by linear elliptic or parabolic PDEs (partial differential equations) depending on countably many parameters, i.e., on $\sigma_j$ with $j\in\mathbb{N}$, we proved in \cite{KS} analytic parameter dependence of the state, the co-state and the control. Moreover, we established that these functions allow expansions in terms of sparse tensorized generalized polynomial chaos (gpc) bases. Their sparsity was quantified in terms of ${\mathfrak p}$-summability of the coefficient sequences for some $0 < {\mathfrak p} \le 1$. Resulting a-priori estimates established the existence of an index set $\Lambda$, allowing for concurrent approximations of state, co-state and control for which the gpc approximations attain rates of best $N$-term approximation. The regularity and N-term approximation results of \cite{KS} serve as the analytical foundation for the development of adaptive Galerkin approximation methods in the present paper. Following the ideas in \cite{CJG11,SG11} and the realizations in \cite{EGSZ13,EGSZ15,GAS} for a single PDE, we construct deterministic adaptive Galerkin approximations of state, co-state and control on the entire, possibly infinite-dimensional, parameter space. The starting point for these constructions are control problems formulated as abstract symmetric saddle point problems as in \cite{KS}. Specifying this to adaptive wavelet based schemes in space and time, we prove convergence as well as optimal complexity estimates, when compared to best $N$-term approximations.

Keywords: Linear-quadratic optimal control problems, stochastic or parametric coefficients, linear elliptic or parabolic PDE, analyticity, polynomial chaos approximation, symmetric saddle point problems, tensor Galerkin discretization, adaptivity, wavelets, convergence, optimal complexity

@Techreport{KS15_627,
  author = {A. Kunoth and Ch. Schwab},
  title = {Sparse adaptive tensor Galerkin approximations of stochastic PDE-constrained control problems
},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2015-37},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2015/2015-37.pdf },
  year = {2015}
}

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