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Analytic regularity and GPC approximation for control problems constrained by linear parametric elliptic and parabolic PDEs

by A. Kunoth and Ch. Schwab

(Report number 2011-54)

Abstract
This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic PDE. We address the (difficult) case that the number of parameters may be countable infinite, i.e., $\sigma_j$ with $j\in N$, and that the PDE operator may depend non-affinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [CDS1,CDS2], we show that the state and the control are analytic as functions depending on these parameters $\sigma_j$. Polynomial approximations of state and control in terms of the possibly countably many stochastic coordinates $\sigma_j$ will be used to establish sparsity of polynomial "generalized polynomial chaos (gpc)" expansions of the state and the control with respect to the parameter sequence $(\sigma_j)_{j\geq 1}$. These imply, in particular, convergence rates of best $N$-term truncations of these expansions. The sparsity result allows in conjunction with adaptive wavelet Galerkin schemes as in [SG11,CJG11] for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [DK,GK,K].

Keywords: Linear-quadratic optimal control, linear parametric or stochastic PDE, distributed or boundary control, elliptic or parabolic PDE, analyticity, polynomial chaos approximation

@Techreport{KS11_106,
  author = {A. Kunoth and Ch. Schwab},
  title = {Analytic regularity and GPC approximation for control problems constrained by linear parametric elliptic and parabolic PDEs},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-54},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-54.pdf },
  year = {2011}
}

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