Finite element approximations of an optimal control problem associated with the scalar Ginzburg-Landau equation

Gunzburger, Max; Hou, Lihong; Svobodny, Thomas P.

doi:10.1016/0898-1221(91)90090-q

Cited by 17 publications

(12 citation statements)

References 1 publication

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“…Such problems arise in many applications (see e.g., [35,41,47]) and, perhaps more importantly, they arise as subproblems in Newton-type or sequential quadratic programming (SQP) methods for many nonlinear elliptic control problems (see, e.g., [9,18,33,34,43]). After a discretization, a linear-quadratic elliptic optimal control problems leads to a large-scale quadratic programming problems whose solution is, under suitable conditions, characterized through the linear system of optimality conditions.…”

Section: Introductionmentioning

confidence: 99%

Neumann--Neumann Domain Decomposition Preconditioners for Linear-Quadratic Elliptic Optimal Control Problems

Heinkenschloss¹,

Nguyen²

2006

SIAM J. Sci. Comput.

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ABSTRACT. We develop and analyze a class of overlapping domain decomposition (DD) preconditioners for linear-quadratic elliptic optimal control problems. Our preconditioners utilize the structure of the optimal control problems. Their execution requires the parallel solution of subdomain linear-quadratic elliptic optimal control problems, which are essentially smaller subdomain copies of the original problem. This work extends to optimal control problems the application and analysis of overlapping DD preconditioners, which have been used successfully for the solution of single PDEs.We prove that for a class of problems the performance of the two-level versions of our preconditioners is independent of the mesh size and of the subdomain size.

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Section: Introductionmentioning

confidence: 99%

Neumann--Neumann Domain Decomposition Preconditioners for Linear-Quadratic Elliptic Optimal Control Problems

Heinkenschloss¹,

Nguyen²

2006

SIAM J. Sci. Comput.

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“…The nonlinearity involves y and u d through the function g : R n y +n d −→ R n y . The examples tested involve nonlinearities of the form e y (modified Bratu problem [21]) and y 3 + y (the simplified Ginzburg-Landau model for superconductivity [22], [5,Ex. 4.8]).…”

Section: Numerical Resultsmentioning

confidence: 99%

Solving discretized degenerate optimal control problems with state constraints

Avelino

2009

Optim Control Appl Methods

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In this paper, we consider a class of nonlinear optimization problems that arise from the discretization of optimal control problems with bounds on both state and control variables. We are particularly interested in degenerate cases, i.e. when the linear independence constraint qualification is not satisfied. For these problems, we analyse the basic global convergence properties and the numerical behaviour of a multiplier method that updates multipliers corresponding to inequality constraints instead of dealing with multipliers associated with equality constraints. Numerical results obtained for several instances of a discretized optimal control problem governed by a semi-linear elliptic equation are included and indicate that this method is robust on degenerate cases, compared with other nonlinear optimization solvers.

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“…Variational methods, and in particular ÿnite element methods, are especially appropriate for these types of problems. Such methods have been recently proposed in Reference [4] for the optimal control of the Ginzburg-Landau equations, in Reference [5] for plastic deformation problems in metal forming, in Reference [6] for the Von Karman elastic plate equations, in Reference [7] for a class of parabolic equations, in Reference [8] for viscoelasticity, in References [9; 10] for incompressible viscous ows, and in Reference [11] for the linear heat and wave equations. In Reference [12], the static stabilization of a cracked linear elastic structure through optimal control of the prestress reinforcement was considered.…”

Section: Introductionmentioning

confidence: 99%

Solution of static optimal control problems in non‐linear elasticity via quadratic programming

Givoli

Patlashenko

2000

Commun. Numer. Meth. Engng.

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A class of static (elliptic) optimal control problems in non-linear elasticity is considered. A new direct ÿnite element formulation is used which results in a sequence of quadratic programming (QP) problems. The latter are solved using a standard QP algorithm. The method is applied to the problems of minimizing the lateral de ection of a membrane on a nonlinear elastic foundation and minimizing the in-plane large deformation of a hyperelastic plate.

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Finite element approximations of an optimal control problem associated with the scalar Ginzburg-Landau equation

Cited by 17 publications

References 1 publication

Neumann--Neumann Domain Decomposition Preconditioners for Linear-Quadratic Elliptic Optimal Control Problems

Neumann--Neumann Domain Decomposition Preconditioners for Linear-Quadratic Elliptic Optimal Control Problems

Solving discretized degenerate optimal control problems with state constraints

Solution of static optimal control problems in non‐linear elasticity via quadratic programming

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