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20
Efficient Numerical Solution of Parabolic Optimization Problems by Finite Element Methods
, 2008
"... 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 26 (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 SNAPSHOT LOCATION FOR COMPUTING POD BASIS FUNCTIONS
, 2008
"... The construction of reduced order models for dynamical systems using proper orthogonal decomposition (POD) is based on the information contained in socalled snapshots. These provide the spatial distribution of the dynamical system at discrete time instances. This work is devoted to optimizing the ..."
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Cited by 21 (3 self)
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The construction of reduced order models for dynamical systems using proper orthogonal decomposition (POD) is based on the information contained in socalled snapshots. These provide the spatial distribution of the dynamical system at discrete time instances. This work is devoted to optimizing the choice of these time instances in such a manner that the error between the PODsolution and the trajectory of the dynamical system is minimized. First and second order optimality systems are given. Numerical examples illustrate that the proposed criterion is sensitive with respect to the choice of the time instances and further they demonstrate the feasibility of the method in determining optimal snapshot locations for concrete diffusion equations.
A hierarchical spacetime solver for distributed control of the Stokes equation
, 2008
"... We present a spacetime hierarchical multigrid solution concept for optimisation problems governed by the timedependent Stokes system. Discretisation is carried out with finite elements in space and a onestep θscheme in time. It is a keyfeature of our multigrid solver that it shows a convergence ..."
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Cited by 13 (2 self)
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We present a spacetime hierarchical multigrid solution concept for optimisation problems governed by the timedependent Stokes system. Discretisation is carried out with finite elements in space and a onestep θscheme in time. It is a keyfeature of our multigrid solver that it shows a convergence behaviour which is independent of the degrees of freedom of the discrete problem, and that the solver performs robust with regard to the considered flow configuration. A set of numerical tests confirms this expectation and shows the efficiency of this approach for various problem settings.
Optimal ramp metering strategy with an extended LWR model: Analysis and computational methods
 in Proc. 16th IFAC World Congr., 2005
"... Abstract: This paper treats the problem of optimizing a coordinated ramp metering strategy based on a distributed macroscopic traffic model. The Lighthill, Whitham and Richards (LWR) model is extended to include on/offramps with onramp flow saturations and an optimal control formulation is propose ..."
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Cited by 7 (2 self)
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Abstract: This paper treats the problem of optimizing a coordinated ramp metering strategy based on a distributed macroscopic traffic model. The Lighthill, Whitham and Richards (LWR) model is extended to include on/offramps with onramp flow saturations and an optimal control formulation is proposed. Using optimization techniques in Banach spaces, a solution is shown to exist and the necessary optimality system is stated using formal adjoint calculus. An iterative descent algorithm and numerical methods for gradient evaluations are proposed to compute the optimal strategy with reasonable effort. The effectiveness of the approach is illustrated through a study case with field data. Copyright 2005 IFAC.
Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE’s
 M2AN Math. Model. Numer. Anal
"... Abstract. A discontinuous Galerkin finite element method for an optimal control problem related to semilinear parabolic PDE’s is examined. The schemes under consideration are discontinuous in time but conforming in space. Convergence of discrete schemes of arbitrary order is proven. In addition, th ..."
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Cited by 5 (2 self)
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Abstract. A discontinuous Galerkin finite element method for an optimal control problem related to semilinear parabolic PDE’s is examined. The schemes under consideration are discontinuous in time but conforming in space. Convergence of discrete schemes of arbitrary order is proven. In addition, the convergence of discontinuous Galerkin approximations of the associated optimality system to the solutions of the continuous optimality system is shown. The proof is based on stability estimates at arbitrary time points under minimal regularity assumptions, and a discrete compactness argument for discontinuous Galerkin schemes (see Walkington [42, Sections 3,4]).
Numerical sensitivity analysis for the quantity of interest in PDEconstrained optimization
 SIAM J. Sci. Comput
"... Abstract. In this paper, we consider the efficient computation of derivatives of a functional (the quantity of interest) which depends on the solution of a PDEconstrained optimization problem with inequality constraints and which may be different from the cost functional. The optimization problem i ..."
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Cited by 5 (3 self)
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Abstract. In this paper, we consider the efficient computation of derivatives of a functional (the quantity of interest) which depends on the solution of a PDEconstrained optimization problem with inequality constraints and which may be different from the cost functional. The optimization problem is subject to perturbations in the data. We derive conditions under with the quantity of interest possesses first and second order derivatives with respect to the perturbation parameters. An algorithm for the efficient evaluation of these derivatives is developed, with considerable savings over a direct approach, especially in the case of highdimensional parameter spaces. The computational cost is shown to be small compared to that of the overall optimization algorithm. Numerical experiments involving a parameter identification problem for Navier–Stokes flow and an optimal control problem for a reactiondiffusion system are presented which demonstrate the efficiency of the method.
B.: Adaptive finite element methods for PDEconstrained optimal control problems
 Numer. Math
, 2007
"... Summary. We present a systematic approach to error control and mesh adaptation in the numerical solution of optimal control problems governed by partial differential equations. By the Lagrangian formalism the optimization problem is reformulated as a saddlepoint boundary value problem which is disc ..."
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Cited by 4 (0 self)
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Summary. We present a systematic approach to error control and mesh adaptation in the numerical solution of optimal control problems governed by partial differential equations. By the Lagrangian formalism the optimization problem is reformulated as a saddlepoint boundary value problem which is discretized by a finite element Galerkin method. The accuracy of the discretization is controlled by residualbased a posteriori error estimates. The main features of this method are illustrated by examples from optimal control of heat transfer, fluid flow and parameter estimation. The contents of this article is as follows: Preliminary thoughts A general framework for a posteriori error estimation Solution process and mesh adaptation Examples of optimal control problems
Differential Stability of Control Constrained Optimal Control Problems for the NavierStokes Equations
, 2005
"... Distributed optimal control problems for the timedependent and the stationary NavierStokes equations subject to pointwise control constraints are considered. Under a coercivity condition on the Hessian of the Lagrange function, optimal solutions are shown to be directionally differentiable functio ..."
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Cited by 2 (1 self)
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Distributed optimal control problems for the timedependent and the stationary NavierStokes equations subject to pointwise control constraints are considered. Under a coercivity condition on the Hessian of the Lagrange function, optimal solutions are shown to be directionally differentiable functions of perturbation parameters such as the Reynolds number, the desired trajectory, or the initial conditions. The derivative is characterized as the solution of an auxiliary linearquadratic optimal control problem. Thus, it can be computed at relatively low cost. Taylor expansions of the minimum value function are provided as well. 1
SYMMETRIC ERROR ESTIMATES FOR DISCONTINUOUS GALERKIN APPROXIMATIONS FOR AN OPTIMAL CONTROL PROBLEM ASSOCIATED TO SEMILINEAR PARABOLIC PDE’S
"... (Communicated by Jie Shen) Abstract. A discontinuous Galerkin finite element method for an optimal control problem having states constrained to semilinear parabolic PDE’s is examined. The schemes under consideration are discontinuous in time but conforming in space. It is shown that under suitable a ..."
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Cited by 1 (1 self)
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(Communicated by Jie Shen) Abstract. A discontinuous Galerkin finite element method for an optimal control problem having states constrained to semilinear parabolic PDE’s is examined. The schemes under consideration are discontinuous in time but conforming in space. It is shown that under suitable assumptions, the error estimates of the corresponding optimality system are of the same order to the standard linear (uncontrolled) parabolic problem. These estimates have symmetric structure and are also applicable for higher order elements.