Sparse Adaptive Tensor Galerkin Approximations of Stochastic PDE-Constrained Control Problems

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 [SIAM J. Control Optim., 51 (2013), pp. 2442--2471] the 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 $0<\mathfrak{p}\leq1$. 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 our previous work serve as the analytical foundation for the development of adaptive Galerkin approximation methods in the present paper. Following the ideas and their realizations in several recent papers from Gittelson, Schwab, and coworkers 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 our previous paper. 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.