3,892 research outputs found
A multigrid method for constrained optimal control problems
AbstractWe consider the fast and efficient numerical solution of linear–quadratic optimal control problems with additional constraints on the control. Discretization of the first-order 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 primal–dual active set 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
Multigrid Methods for Elliptic Optimal Control Problems
In this dissertation we study multigrid methods for linear-quadratic elliptic distributed optimal control problems.
For optimal control problems constrained by general second order elliptic partial differential equations, we design and analyze a finite element method based on a saddle point formulation. We construct a -cycle algorithm for the discrete problem and show that it is uniformly convergent in the energy norm for convex domains. Moreover, the contraction number decays at the optimal rate of , where is the number of smoothing steps. We also prove that the convergence is robust with respect to a regularization parameter. The robust convergence of -cycle and -cycle algorithms on general domains are demonstrated by numerical results.
For optimal control problems constrained by symmetric second order elliptic partial differential equations together with pointwise constraints on the state variable, we design and analyze symmetric positive definite finite element methods based on a reformulation of the optimal control problem as a fourth order variational inequality. We develop a multigrid algorithm for the reduced systems that appear in a primal-dual active set method for the discrete variational inequalities. The performance of the algorithm is demonstrated by numerical results
A multigrid method for PDE-constrained optimization with uncertain inputs
We present a multigrid algorithm to solve efficiently the large saddle-point
systems of equations that typically arise in PDE-constrained optimization under
uncertainty. The algorithm is based on a collective smoother that at each
iteration sweeps over the nodes of the computational mesh, and solves a reduced
saddle-point system whose size depends on the number of samples used to
discretized the probability space. We show that this reduced system can be
solved with optimal complexity. We test the multigrid method on three
problems: a linear-quadratic problem for which the multigrid method is used to
solve directly the linear optimality system; a nonsmooth problem with box
constraints and -norm penalization on the control, in which the multigrid
scheme is used within a semismooth Newton iteration; a risk-adverse problem
with the smoothed CVaR risk measure where the multigrid method is called within
a preconditioned Newton iteration. In all cases, the multigrid algorithm
exhibits very good performances and robustness with respect to all parameters
of interest.Comment: 24, 2 figure
Preconditioners for state constrained optimal control problems with Moreau-Yosida penalty function
Optimal control problems with partial differential equations as constraints play an important role in many applications. The inclusion of bound constraints for the state variable poses a significant challenge for optimization methods. Our focus here is on the incorporation of the constraints via the Moreau-Yosida regularization technique. This method has been studied recently and has proven to be advantageous compared to other approaches. In this paper we develop robust preconditioners for the efficient solution of the Newton steps associated with solving the Moreau-Yosida regularized problem. Numerical results illustrate the efficiency of our approach
Preconditioners for state constrained optimal control problems\ud with Moreau-Yosida penalty function tube
Optimal control problems with partial differential equations play an important role in many applications. The inclusion of bound constraints for the state poses a significant challenge for optimization methods. Our focus here is on the incorporation of the constraints via the Moreau-Yosida regularization technique. This method has been studied recently and has proven to be advantageous compared to other approaches. In this paper we develop preconditioners for the efficient solution of the Newton steps associated with the fast solution of the Moreau-Yosida regularized problem. Numerical results illustrate the competitiveness of this approach. \ud
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Copyright c 2000 John Wiley & Sons, Ltd
Optimal solvers for PDE-Constrained Optimization
Optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering, in particular in problems of design. The solution of such PDE-constrained optimization problems is usually a major computational task. Here we consider simple problems of this type: distributed control problems in which the 2- and 3-dimensional Poisson problem is the PDE. The large dimensional linear systems which result from discretization and which need to be solved are of saddle-point type. We introduce two optimal preconditioners for these systems which lead to convergence of symmetric Krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem. These preconditioners are block structured and involve standard multigrid cycles. The optimality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases. The theoretical proof indicates that these approaches may have much broader applicability for other partial differential equations
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