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A PRIORI ERROR ESTIMATES FOR SPACETIME FINITE ELEMENT DISCRETIZATION OF PARABOLIC OPTIMAL CONTROL PROBLEMS PART I: PROBLEMS WITHOUT CONTROL CONSTRAINTS
"... Abstract. In this paper we develop a priori error analysis for Galerkin finite element discretizations of optimal control problems governed by linear parabolic equations. The space discretization of the state variable is done using usual conforming finite elements, whereas the time discretization is ..."
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Cited by 32 (6 self)
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Abstract. In this paper we develop a priori error analysis for Galerkin finite element discretizations of optimal control problems governed by linear parabolic equations. The space discretization of the state variable is done using usual conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. For different types of control discretizations we provide error estimates of optimal order with respect to both space and time discretization parameters. The paper is divided into two parts. In the first part we develop some stability and error estimates for spacetime discretization of the state equation and provide error estimates for optimal control problems without control constraints. In the second part of the paper, the techniques and results of the first part are used to develop a priori error analysis for optimal control problems with pointwise inequality constraints on the control variable.
Semismooth Newton methods for variational inequalities of the first kind
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"... Abstract. Semi–smooth Newton methods are analyzed for a class of variational inequalities in infinite dimensions. It is shown that they are equivalent to certain active set strategies. Global and local superlinear convergence are proved. To overcome the phenomenon of finite speed of propagation of ..."
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Cited by 30 (10 self)
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Abstract. Semi–smooth Newton methods are analyzed for a class of variational inequalities in infinite dimensions. It is shown that they are equivalent to certain active set strategies. Global and local superlinear convergence are proved. To overcome the phenomenon of finite speed of propagation of discretized problems a penalty version is used as the basis for a continuation procedure to speed up convergence. The choice of the penalty parameter can be made on the basis of an L ∞ estimate for the penalized solutions. Unilateral as well as bilateral problems are considered.
Efficient Numerical Solution of Parabolic Optimization Problems by Finite Element Methods, in "Optim. Methods Softw
, 2007
"... We present an approach for efficient numerical solution of optimization problems governed by parabolic partial differential equations. The main ingredients are: spacetime finite element discretization, second order optimization algorithms and storage reduction techniques. We discuss the combination ..."
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Cited by 29 (8 self)
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We present an approach for efficient numerical solution of optimization problems governed by parabolic partial differential equations. The main ingredients are: spacetime finite element discretization, second order optimization algorithms and storage reduction techniques. We discuss the combination of these components for the solution of large scale optimization problems.
OPTIMAL CONTROL OF THE CONVECTIONDIFFUSION EQUATION USING STABILIZED FINITE ELEMENT METHODS
"... Abstract. In this paper we analyze discretization of optimal control problems governed by convectiondiffusion equations which are subject to pointwise control constraints. We present a stabilization scheme which leads to improved approximate solutions even on corse meshes in the convection dominate ..."
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Cited by 27 (1 self)
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Abstract. In this paper we analyze discretization of optimal control problems governed by convectiondiffusion equations which are subject to pointwise control constraints. We present a stabilization scheme which leads to improved approximate solutions even on corse meshes in the convection dominated case. Moreover, the in general different approaches “optimizethendiscretize” and “discretizethenoptimize ” coincide for the proposed discretization scheme. This allows for a symmetric optimality system on the discrete level and optimal order of convergence.
Error estimates for linearquadratic control problems with control constraints
"... An abstract linearquadratic optimal control problem with pointwise control constraints is investigated. This paper is concerned with the discretization of the control by piecewise linear functions. Under the assumption that the optimal control and the optimal adjoint state are Lipschitz continuous ..."
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Cited by 26 (1 self)
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An abstract linearquadratic optimal control problem with pointwise control constraints is investigated. This paper is concerned with the discretization of the control by piecewise linear functions. Under the assumption that the optimal control and the optimal adjoint state are Lipschitz continuous and piecewise of class C 2 an approximation of order h 3/2 is proved for the solution of the control discretized problem with respect to the solution of the continuous one. The theoretical results are illustrated by numerical tests.
A framework for the adaptive finite element solution of large inverse problems. I. Basic techniques
, 2004
"... Abstract. Since problems involving the estimation of distributed coefficients in partial differential equations are numerically very challenging, efficient methods are indispensable. In this paper, we will introduce a framework for the efficient solution of such problems. This comprises the use of a ..."
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Cited by 24 (7 self)
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Abstract. Since problems involving the estimation of distributed coefficients in partial differential equations are numerically very challenging, efficient methods are indispensable. In this paper, we will introduce a framework for the efficient solution of such problems. This comprises the use of adaptive finite element schemes, solvers for the large linear systems arising from discretization, and methods to treat additional information in the form of inequality constraints on the parameter to be recovered. The methods to be developed will be based on an allatonce approach, in which the inverse problem is solved through a Lagrangian formulation. The main feature of the paper is the use of a continuous (function space) setting to formulate algorithms, in order to allow for discretizations that are adaptively refined as nonlinear iterations proceed. This entails that steps such as the description of a Newton step or a line search are first formulated on continuous functions and only then evaluated for discrete functions. On the other hand, this approach avoids the dependence of finite dimensional norms on the mesh size, making individual steps of the algorithm comparable even if they used differently refined meshes. Numerical examples will demonstrate the applicability and efficiency of the method for problems with several million unknowns and more than 10,000 parameters. Key words. Adaptive finite elements, inverse problems, Newton method on function spaces. AMS subject classifications. 65N21,65K10,35R30,49M15,65N50 1. Introduction. Parameter
Optimization Techniques for Solving Elliptic Control Problems with Control and State Constraints. Part 2: Distributed Control
 Comp. Optim. Applic
"... : Part 2 continues the study of optimization techniques for elliptic control problems subject to control and state constraints and is devoted to distributed control. Boundary conditions are of mixed Dirichlet and Neumann type. Necessary conditions of optimality are formally stated in form of a local ..."
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Cited by 17 (3 self)
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: Part 2 continues the study of optimization techniques for elliptic control problems subject to control and state constraints and is devoted to distributed control. Boundary conditions are of mixed Dirichlet and Neumann type. Necessary conditions of optimality are formally stated in form of a local Pontryagin minimum principle. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. The problems are formulated as AMPL [13] scripts and several optimization codes are applied. In particular, it is shown that a recently developed interior point method is able to solve theses problems even for high discretizations. Several numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for dierent types of controls including bang{bang controls. The necessary conditions of optimality are checked numerically in the presence of active control and state constraints....
Error estimates for the finiteelement approximation of an elliptic control problem with pointwise state and control constraints
 CONTROL AND CYBERNETICS
, 2008
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A MULTIGRID METHOD FOR CONSTRAINED OPTIMAL CONTROL PROBLEMS
"... We consider the fast and efficient numerical solution of linearquadratic optimal control problems with additional constraints on the control. Discretization of the firstorder conditions leads to an indefinite linear system of saddle point type with additional complementarity conditions due to the ..."
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Cited by 15 (0 self)
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We consider the fast and efficient numerical solution of linearquadratic optimal control problems with additional constraints on the control. Discretization of the firstorder conditions leads to an indefinite linear system of saddle point type with additional complementarity conditions due to the control constraints. The complementarity conditions are treated by a primaldual activeset strategy that serves as outer iteration. At each iteration step, a KKT system has to be solved. Here, we develop a multigrid method for its fast solution. To this end, we use a smoother which is based on an inexact constraint preconditioner. We present numerical results which show that the proposed multigrid method possesses convergence rates of the same order as for the underlying (elliptic) PDE problem. Furthermore, when combined with a nested iteration, the solver is of optimal complexity and achieves the solution of the optimization problem at only a small multiple of the cost for the PDE solution.
CONSTRAINED DIRICHLET BOUNDARY CONTROL IN L 2 FOR A CLASS OF EVOLUTION EQUATIONS
"... Abstract. Optimal Dirichlet boundary control based on the very weak solution of a parabolic state equation is analysed. This approach allows to consider the boundary controls in L2 which has advantages over approaches which consider control in Sobolev involving (fractional) derivatives. Pointwise c ..."
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Cited by 15 (3 self)
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Abstract. Optimal Dirichlet boundary control based on the very weak solution of a parabolic state equation is analysed. This approach allows to consider the boundary controls in L2 which has advantages over approaches which consider control in Sobolev involving (fractional) derivatives. Pointwise constraints on the boundary are incorporated by the primaldual active set strategy. Its global and local superlinear convergence are shown. A discretization based on spacetime finite elements is proposed and numerical examples are included. Key words. Dirichlet boundary control, inequality constraints, parabolic equations, very weak solution 1. Introduction. In