8,141 research outputs found
Parallel iterative solvers for discretized reduced optimality systems
We propose, analyze, and test new iterative solvers for large-scale systems
of linear algebraic equations arising from the finite element discretization of
reduced optimality systems defining the finite element approximations to the
solution of elliptic tracking-type distributed optimal control problems with
both the standard and the more general energy regularizations. If we aim
at an approximation of the given desired state by the computed finite
element state that asymptotically differs from in the order of the
best approximation under acceptable costs for the control, then the
optimal choice of the regularization parameter is linked to the
mesh-size by the relations and for the
and the energy regularization, respectively. For this setting, we can construct
efficient parallel iterative solvers for the reduced finite element optimality
systems. These results can be generalized to variable regularization parameters
adapted to the local behavior of the mesh-size that can heavily change in case
of adaptive mesh refinement. Similar results can be obtained for the space-time
finite element discretization of the corresponding parabolic and hyperbolic
optimal control problems
A Frame Work for the Error Analysis of Discontinuous Finite Element Methods for Elliptic Optimal Control Problems and Applications to IP methods
In this article, an abstract framework for the error analysis of
discontinuous Galerkin methods for control constrained optimal control problems
is developed. The analysis establishes the best approximation result from a
priori analysis point of view and delivers reliable and efficient a posteriori
error estimators. The results are applicable to a variety of problems just
under the minimal regularity possessed by the well-posed ness of the problem.
Subsequently, applications of interior penalty methods for a boundary
control problem as well as a distributed control problem governed by the
biharmonic equation subject to simply supported boundary conditions are
discussed through the abstract analysis. Numerical experiments illustrate the
theoretical findings. Finally, we also discuss the variational discontinuous
discretization method (without discretizing the control) and its corresponding
error estimates.Comment: 23 pages, 5 figures, 1 tabl
A residual based snapshot location strategy for POD in distributed optimal control of linear parabolic equations
In this paper we study the approximation of a distributed optimal control
problem for linear para\-bolic PDEs with model order reduction based on Proper
Orthogonal Decomposition (POD-MOR). POD-MOR is a Galerkin approach where the
basis functions are obtained upon information contained in time snapshots of
the parabolic PDE related to given input data. In the present work we show that
for POD-MOR in optimal control of parabolic equations it is important to have
knowledge about the controlled system at the right time instances. For the
determination of the time instances (snapshot locations) we propose an
a-posteriori error control concept which is based on a reformulation of the
optimality system of the underlying optimal control problem as a second order
in time and fourth order in space elliptic system which is approximated by a
space-time finite element method. Finally, we present numerical tests to
illustrate our approach and to show the effectiveness of the method in
comparison to existing approaches
Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs
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 state equation depends on a countable number of parameters i.e., on with , 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 . We
establish sparsity of generalized polynomial chaos (gpc) expansions of both, state and control, in terms of the stochastic coordinate sequence of the random inputs, and prove convergence rates of best -term truncations of these expansions. Such truncations are the key for subsequent computations since they do {\em not} assume that the stochastic input data has a finite expansion. In the follow-up paper [KS2], we explain two methods how such best -term truncations can practically be computed, by greedy-type algorithms
as in [SG, Gi1], or by multilevel Monte-Carlo methods as in
[KSS]. The sparsity result allows in conjunction with adaptive wavelet Galerkin schemes for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [DK, GK, K], see [KS2]
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