Analytic regularity and GPC approximation for control problems constrained by linear parametric elliptic and parabolic PDEs
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Date
2011-09Type
- Report
ETH Bibliography
yes
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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]. Show more
Permanent link
https://doi.org/10.3929/ethz-a-010397666Publication status
publishedExternal links
Journal / series
SAM Research ReportVolume
Publisher
Seminar for Applied Mathematics, ETH ZurichSubject
Linear-quadratic optimal control; Linear parametric or stochastic PDE; Distributed or boundary control; Elliptic or parabolic PDE; Analyticity; Polynomial chaos approximationOrganisational unit
02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics
03435 - Schwab, Christoph / Schwab, Christoph
Funding
247277 - Automated Urban Parking and Driving (EC)
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Is previous version of: http://hdl.handle.net/20.500.11850/70043
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ETH Bibliography
yes
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