Elliptic Distributed Optimal Control Problems Research Articles

C0 interior penalty methods for an elliptic distributed optimal control problem with general tracking and pointwise state constraints

We consider C0 interior penalty methods for a linear-quadratic elliptic distributed optimal control problem with pointwise state constraints in two spatial dimensions, where the cost function tracks the state at points, curves and regions of the domain. Error estimates and numerical results that illustrate the performance of the methods are presented.

A Symmetric Interior Penalty Method for an Elliptic Distributed Optimal Control Problem with Pointwise State Constraints

Abstract We construct a symmetric interior penalty method for an elliptic distributed optimal control problem with pointwise state constraints on general polygonal domains. The resulting discrete problems are quadratic programs with simple box constraints that can be solved efficiently by a primal-dual active set algorithm. Both theoretical analysis and corroborating numerical results are presented.

Multigrid methods for an elliptic optimal control problem with pointwise state constraints

We design multigrid methods for an elliptic distributed optimal control problem with pointwise state constraints. They are based on the P1 finite element method from Brenner et al. (2021), where the optimal control problem was reformulated as a variational inequality for the state variable. The performance of the multigrid methods is illustrated by numerical experiments.

Multiscale Finite Element Methods for an Elliptic Optimal Control Problem with Rough Coefficients

We investigate multiscale finite element methods for an elliptic distributed optimal control problem with rough coefficients. They are based on the (local) orthogonal decomposition methodology of Målqvist and Peterseim.

Optimal convergence of finite element approximation to an optimization problem with PDE constraint* *Wei Gong was supported by the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB 41000000), the National Key Basic Research Program (Grant No. 2018YFB0704304) and the National Natural Science Foundation of China (Grant No. 12071468, 11671391). Zhaojie Zhou was supported by the National

We study in this paper the optimal convergence of finite element approximation to an optimization problem with PDE constraint. Specifically, we consider an elliptic distributed optimal control problem without control constraints, which can also be viewed as a regularized inverse source problem. The main contributions are two-fold. First, we derive a priori and a posteriori error estimates for the optimization problems, under an appropriately chosen norm that allows us to establish an isomorphism between the solution space and its dual. These results yield error estimates with explicit dependence on the regularization parameter α so that the constants appeared in the derivation are independent of α. Second, we prove the contraction property and rate optimality for the adaptive algorithm with respect to the error estimator and solution errors between the adaptive finite element solutions and the continuous solutions. Extensive numerical experiments are presented that confirm our theoretical results.

A C1 virtual element method for an elliptic distributed optimal control problem with pointwise state constraints

We design and analyze a [Formula: see text] virtual element method for an elliptic distributed optimal control problem with pointwise state constraints. Theoretical estimates and corroborating numerical results are presented.

A finite element method for an elliptic optimal control problem with integral state constraints

In this article, we discuss a priori error estimates of a bubble enriched nonconforming finite element method for a linear quadratic elliptic distributed optimal control problem with integral state constraints. Therein, using the state equation we reduce the state-control constrained minimization problem to a pure state constrained minimization problem. We have obtained the optimal order (with respect to regularity) error estimates of finite element approximation in two cases: the optimal control problem with integral state constraints together with (i) integral control constraints (ii) pointwise control constraints. Numerical results are presented to confirm the theoretical findings.

A 𝑃1 Finite Element Method for a Distributed Elliptic Optimal Control Problem with a General State Equation and Pointwise State Constraints

Abstract We investigate a P 1 P_{1} finite element method for an elliptic distributed optimal control problem with pointwise state constraints and a state equation that includes advective/convective and reactive terms. The convergence of this method can be established for general polygonal/polyhedral domains that are not necessarily convex. The discrete problem is a strictly convex quadratic program with box constraints that can be solved efficiently by a primal-dual active set algorithm.

A One Dimensional Elliptic Distributed Optimal Control Problem with Pointwise Derivative Constraints

We consider a one dimensional elliptic distributed optimal control problem with pointwise constraints on the derivative of the state. By exploiting the variational inequality satisfied by the derivative of the optimal state, we obtain higher regularity for the optimal state under appropriate assumptions on the data. We also solve the optimal control problem as a fourth order variational inequality by a C 1 finite element method, and present the error analysis together with numerical results.

A cubic [formula omitted] interior penalty method for elliptic distributed optimal control problems with pointwise state and control constraints

We design and analyze a cubic C0 interior penalty method for linear–quadratic elliptic distributed optimal control problems with pointwise state and control constraints. Numerical results that corroborate the theoretical error estimates are also presented.

Finite element methods for one dimensional elliptic distributed optimal control problems with pointwise constraints on the derivative of the state

We investigate $C^1$ finite element methods for one dimensional elliptic distributed optimal control problems with pointwise constraints on the derivative of the state formulated as fourth order variational inequalities for the state variable. For the problem with Dirichlet boundary conditions, we use an existing $H^{\frac52-\epsilon}$ regularity result for the optimal state to derive $O(h^{\frac12-\epsilon})$ convergence for the approximation of the optimal state in the $H^2$ norm. For the problem with mixed Dirichlet and Neumann boundary conditions, we show that the optimal state belongs to $H^3$ under appropriate assumptions on the data and obtain $O(h)$ convergence for the approximation of the optimal state in the $H^2$ norm.

Multigrid methods for saddle point problems: Optimality systems

We develop multigrid methods for an elliptic distributed optimal control problem on convex domains that are robust with respect to a regularization parameter. We prove the uniform convergence of the W-cycle algorithm and demonstrate the performance of V-cycle and W-cycle algorithms through numerical experiments.

Formula omitted] finite element methods for an elliptic state-constrained distributed optimal control problem with Neumann boundary conditions

We investigate two P1 finite element methods for an elliptic state-constrained distributed optimal control problem with Neumann boundary conditions on general polygonal domains.

P 1 finite element methods for an elliptic optimal control problem with pointwise state constraints

Abstract We present theoretical and numerical results for two $P_1$ finite element methods for an elliptic distributed optimal control problem on general polygonal/polyhedral domains with pointwise state constraints.

$C^0$ Interior Penalty Methods for an Elliptic Distributed Optimal Control Problem on Nonconvex Polygonal Domains with Pointwise State Constraints

We design and analyze $C^0$ interior penalty methods for an elliptic distributed optimal control problem on nonconvex polygonal domains with pointwise state constraints.MSC codeselliptic distributed optimal control problemspointwise state constraintsnonconvex domainsvariational inequalitiesdiscontinuous Galerkin methodsMSC codes49J2065K1565N30

A morley finite element method for an elliptic distributed optimal control problem with pointwise state and control constraints

We design and analyze a Morley finite element method for an elliptic distributed optimal control problem with pointwise state and control constraints on convex polygonal domains. It is based on the formulation of the optimal control problem as a fourth order variational inequality. Numerical results that illustrate the performance of the method are also presented.

A New Convergence Analysis of Finite Element Methods for Elliptic Distributed Optimal Control Problems with Pointwise State Constraints

We consider finite element methods for elliptic distributed optimal control problems with pointwise state constraints on two and three dimensional convex polyhedral domains formulated as fourth order variational inequalities. We develop a new convergence analysis that is applicable to $C^1$ finite element methods, classical nonconforming finite element methods and discontinuous Galerkin methods.

Preconditioners for reduced saddle point systems arising in elliptic PDE-constrained optimization problems

In this paper, we propose some preconditioning techniques for reduced saddle point systems arising from linear elliptic distributed optimal control problems. The eigenvalues of preconditioned matrices are analyzed. Moreover, the bounds of these eigenvalues with respect to the mesh size h are also obtained. Some numerical tests are presented to validate the theoretical analysis.

Post-processing procedures for an elliptic distributed optimal control problem with pointwise state constraints

We consider an elliptic distributed optimal control problem with state constraints and compare three post-processing procedures that compute approximations of the optimal control from the approximation of the optimal state obtained by a quadratic C0 interior penalty method.

A partition of unity method for a class of fourth order elliptic variational inequalities

We consider a partition of unity method (PUM) for a class of fourth order elliptic variational inequalities on convex polygonal domains that include obstacle problems of simply supported Kirchhoff plates and elliptic distributed optimal control problems with pointwise state constraints as special cases. By including singular functions in the local approximation spaces we are able to show that the partition of unity method converges optimally. Numerical results that corroborate the theoretical estimates are also presented.