10 research outputs found
A Class of Embedded DG Methods for Dirichlet Boundary Control of Convection Diffusion PDEs
We investigated an hybridizable discontinuous Galerkin (HDG) method for a
convection diffusion Dirichlet boundary control problem in our earlier work
[SIAM J. Numer. Anal. 56 (2018) 2262-2287] and obtained an optimal convergence
rate for the control under some assumptions on the desired state and the
domain. In this work, we obtain the same convergence rate for the control using
a class of embedded DG methods proposed by Nguyen, Peraire and Cockburn [J.
Comput. Phys. vol. 302 (2015), pp. 674-692] for simulating fluid flows. Since
the global system for embedded DG methods uses continuous elements, the number
of degrees of freedom for the embedded DG methods are smaller than the HDG
method, which uses discontinuous elements for the global system. Moreover, we
introduce a new simpler numerical analysis technique to handle low regularity
solutions of the boundary control problem. We present some numerical
experiments to confirm our theoretical results
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]
A priori error analysis for state constrained boundary control problems : Part I: Control discretization
This is the first of two papers concerned with a state-constrained
optimal control problems with boundary control, where the state constraints
are only imposed in an interior subdomain. We apply the virtual control
concept introduced in [20] to regularize the problem. The arising regularized
optimal control problem is discretized by finite elements and linear and
continuous ansatz functions for the boundary control. In the first part of
the work, we investigate the errors induced by the regularization and the
discretization of the boundary control. The second part deals with the error
arising from discretization of the PDE. Since the state constraints only
appear in an inner subdomain, the obtained order of convergence exceeds the
known results in the field of a priori analysis for state-constrained
problem
Optimal boundary control of a simplified Ericksen--Leslie system for nematic liquid crystal flows in
In this paper, we investigate an optimal boundary control problem for a two
dimensional simplified Ericksen--Leslie system modelling the incompressible
nematic liquid crystal flows. The hydrodynamic system consists of the
Navier--Stokes equations for the fluid velocity coupled with a convective
Ginzburg--Landau type equation for the averaged molecular orientation. The
fluid velocity is assumed to satisfy a no-slip boundary condition, while the
molecular orientation is subject to a time-dependent Dirichlet boundary
condition that corresponds to the strong anchoring condition for liquid
crystals. We first establish the existence of optimal boundary controls. Then
we show that the control-to-state operator is Fr\'echet differentiable between
appropriate Banach spaces and derive first-order necessary optimality
conditions in terms of a variational inequality involving the adjoint state
variables
Numerische Konzepte und Fehleranalysis zu elliptischen Randsteuerungsproblemen mit punktweisen Zustands- und KontrollbeschrÀnkungen
Optimization in technical applications described by partial differential equations plays a more and more important role. By means of the control the solution of a partial differential equation called state is influenced. Simultaneously a cost functional has to be minimized. In many technical applications pointwise constraints to the state or the control are reasonable. It is well known that the Lagrange multipliers with respect to pure state constraints are in general only regular Borel measures. This fact implies a lower regularity of the optimal solution of the problem.
In this dissertation a linear quadratic optimal control problem governed by an elliptic partial differential equation an Neumann boundary control is investigated. Furthermore, we consider pointwise state constraints in an inner subdomain and bilateral constraints on the boundary control. Despite the above mentioned problems, we benefit from the localization of the Lagrange multiplier in the inner subdomain such that a higher regularity of the optimal control is shown. However, the so called dual variables of the optimal control problem are not unique. Hence, the application of well known and efficient optimization algorithms becomes difficult.
Presenting a regularization concept, we will avoid these problems. We introduce an additional distributed control ("virtual control") which appears in the cost functional, the right hand side of the partial differential equation and in the regularized state constraints. The effect of regularization is influenced by several parameter functions. We derive an error estimate for the error between the optimal solution of the original problem and the regularized one. Moreover, under some assumptions on the parameter functions we obtain certain convergence rates of the regularization error.
In the following a finite element based approximation of the regularized optimal control problems is established. Based on appropriate feasible test functions, we derive an error estimate between the optimal solution of the unregularized original problem and the regularized and discretized one. Thereby, we consider the regularization and discretization simultaneously and we propose a suitable coupling of the parameter functions and the mesh size.
Forthcoming, we present the primal-dual active set strategy as a optimization method for solving the regularized optimal control problems. Moreover, we derive an error estimate between the current iterates of the algorithm and the optimal solution. Based on this, we construct an error estimator, which is reliable as an alternative stopping criterion for the primal-dual active set strategy.
Finally, the theoretical results of this work are illustrated by several numerical examples.Physikalische und technische Anwendungen werden hĂ€ufig durch partielle Differentialgleichungen beschrieben. Die Optimierung solcher Prozesse fĂŒhrt auf sogenannte Optimalsteuerprobleme mit partiellen Differentialgleichungen. Mit Hilfe einer Steuerungsvariable wird die Lösung der Differentialgleichung, welche Zustand genannt wird, beeinflusst. Gleichzeitig soll ein Zielfunktional minimiert werden. Bei vielen technischen Anwendungen sind punktweise BeschrĂ€nkungen an den Zustand oder die Steuerung sinnvoll. Es ist bekannt, dass die zu den ZustandsbeschrĂ€nkungen gehörigen Lagrangsche Multiplikatoren im allgemeinen nur regulĂ€re Borel-MaĂe sind. Dies fĂŒhrt zu einer geringeren RegularitĂ€t der optimalen Lösung des Problems.
In dieser Dissertationsschrift wird ein linear-quadratisches Optimalsteuerproblem mit elliptischer partieller Differentialgleichung und Neumann-Randsteuerung untersucht. Wir betrachten punkteweise Zustandsschranken in einem inneren Teilgebiet und bilaterale Schranken an die Randsteuerung. Die rÀumliche Trennung der ZustandsbeschrÀnkungen von dem Wirkungsgebiet der Steuerung gestattet an vielen Stellen den Einsatz von speziell konstruierten mathematischen Techniken. Dies betrifft sowohl RegularitÀtsaussagen als auch FehlerabschÀtzungen. Allerdings sind die sogenannten dualen Variablen des Problems nicht eindeutig. Dies macht die Anwendung bekannter effizienter Optimierungsalgorithmen unmöglich.
Es wird ein Regularisierungskonzept vorgestellt, um dieses Problem zu vermeiden. Dabei wird eine zusĂ€tzliche verteilte Steuerung ("virtuelle Steuerung") eingefĂŒhrt, welche im Zielfunktional, in der rechten Seite der Differentialgleichungen und in den regularisierten ZustandsbeschrĂ€nkungen auftaucht. Die Regularisierung wird durch verschiedene Parameterfunktionen beeinflusst. Wir leiten AbschĂ€tzungen fĂŒr den Fehler zwischen der optimalen Lösung des Ausgangsproblems und der des regularisierten Problems her. Bei Verwendung geschickt gewĂ€hlter Parameterfunktionen ergeben sich aus diesen AbschĂ€tzungen direkt Konvergenzraten fĂŒr die Regularisierung.
Im weiteren betrachten wir auch eine Diskretisierung des regularisierten Problems mit Hilfe von finiten Elementen. Basierend auf geeignet konstruierten zulÀssigen Testfunktionen wird eine FehlerabschÀtzung der optimalen Lösung des unregularisierten Problems zur diskretisierten und regularisierten Lösung hergeleitet. Da der Regularisierungs- und der Diskretisierungsfehler gleichzeitig auftreten, wird eine geeignete Kopplung des Regularisierungsparameters mit der Gitterweite angegeben.
Eine primal-duale aktive Mengenstrategie wird als Optimierungsalgorithmus zur Lösung der regularisierten Probleme vorgestellt. Weiterhin wird eine FehlerabschĂ€tzung der aktuellen Iterierten dieses Algorithmus zur optimalen Lösung bewiesen. Basierend auf diesem Resultat wird ein FehlerschĂ€tzer konstruiert, welcher als alternatives Abbruchkriterium fĂŒr die aktive Mengenstrategie benutzt werden kann.
Die Resultate der Arbeit werden durch verschiedene numerische Beispiele bestÀtigt