Winnifried Wollner

Adaptive Finite Elements for Elliptic Optimization Problems with Control Constraints

In this paper we develop a posteriori error estimates for finite element discretization of elliptic optimization problems with pointwise inequality constraints on the control variable. We derive error estimators for assessing the discretization error with respect to the cost functional as well as with respect to a given quantity of interest. These error estimators provide quantitative information about the discretization error and guide an adaptive mesh refinement algorithm allowing for substantial saving in degrees of freedom. The behavior of the method is demonstrated on numerical examples.

A posteriori error estimates for a finite element discretization of interior point methods for an elliptic optimization problem with state constraints

In this paper we are concerned with a posteriori error estimates for the solution of some state constraint optimization problem subject to an elliptic PDE. The solution is obtained using an interior point method combined with a finite element method for the discretization of the problem. We will derive separate estimates for the error in the cost functional introduced by the interior point parameter and by the discretization of the problem. Finally we show numerical examples to illustrate the findings for pointwise state constraints and pointwise constraints on the gradient of the state.

Adaptive finite element solution of eigenvalue problems: Balancing of discretization and iteration error

Abstract This paper develops a combined a posteriori analysis for the discretization and iteration errors in the solution of elliptic eigenvalue problems by the finite element method. The emphasis is on the iterative solution of the discretized eigenvalue problem by a Krylov-space method. The underlying theoretical framework is that of the Dual Weighted Residual (DWR) method for goal-oriented error estimation. On the basis of computable a posteriori error estimates the algebraic iteration can be adjusted to the discretization within a successive mesh adaptation process. The functionality of the proposed method is demonstrated by numerical examples.

Barrier Methods for Optimal Control Problems with Convex Nonlinear Gradient State Constraints

In this paper we are concerned with the application of interior point methods in function space to gradient constrained optimal control problems, governed by partial differential equations. We will derive existence of solutions together with first order optimality conditions. Afterwards we show continuity of the central path, together with convergence rates depending on the interior point parameter.

FINITE-RANK ADI ITERATION FOR OPERATOR LYAPUNOV EQUATIONS ∗

We give an algorithmic approach to the approximative solution of operator Lyapunov equations for controllability. Motivated by the successfully applied alternating direction implicit (ADI) iteration for matrix Lyapunov equations, we consider this method for the determination of Gramian operators of infinite-dimensional control systems. In the case where the input space is finite-dimensional, this method provides approximative solutions of finite rank. Under the assumption of infinite-time admissibility and boundedness of the semigroup, we analyze convergence in several operator norms. We show that under a mild assumption on the shift parameters, convergence to the Gramian is obtained. Particular emphasis is placed on systems governed by a heat equation with boundary control. We present that ADI iteration for the heat equation consists of solving a sequence of Helmholtz equations. Two numerical examples are presented; the first showing the benefit of adaptive finite elements and the second illustrating con...

A priori error estimates for optimal control problems with pointwise constraints on the gradient of the state

We analyze a finite element approximation of an elliptic optimal control problem with pointwise bounds on the gradient of the state variable. We derive convergence rates if the control space is discretized implicitly by the state equation. In contrast to prior work we obtain these results directly from classical results for the W1,∞-error of the finite element projection, without using adjoint information. If the control space is discretized directly, we first prove a regularity result for the optimal control to control the approximation error, based on which we then obtain analogous convergence rates.

The Length of the Primal-Dual Path in Moreau--Yosida-Based Path-Following Methods for State Constrained Optimal Control

A priori estimates of the length of the primal-dual path resulting from a Moreau--Yosida approximation of the feasible set for state constrained optimal control problems are derived. These bounds d...

Adaptive Optimal Control of the Obstacle Problem

This article is concerned with the derivation of a posteriori error estimates for optimization problems subject to an obstacle problem. To circumvent the nondifferentiability inherent to this type of problem, we introduce a sequence of penalized but differentiable problems. We show differentiability of the central path and derive separate a posteriori dual weighted residual estimates for the errors due to penalization, discretization, and iterative solution of the discrete problems. The effectivity of the derived estimates and of the adaptive algorithm is demonstrated on two numerical examples.

The Damped Crank–Nicolson Time-Marching Scheme for the Adaptive Solution of the Black–Scholes Equation

This paper is concerned with the derivation of a residual-based a posteriori error estimator and mesh-adaptation strategies for the space-time finite element approximation of parabolic problems with irregular data. Typical applications arise in the field of mathematical finance, where the Black–Scholes equation is used for modeling the pricing of European options. A conforming finite element discretization in space is combined with second-order time discretization by a damped Crank–Nicolson scheme for coping with data irregularities in the model. The a posteriori error analysis is developed within the general framework of the dual weighted residual method for sensitivity-based, goal-oriented error estimation and mesh optimization. In particular, the correct form of the dual problem with damping is considered.

A Posteriori Error Estimation in PDE-constrained Optimization with Pointwise Inequality Constraints

This article summarizes several recent results on goal-oriented error estimation and mesh adaptation for the solution of elliptic PDE-constrained optimization problems with additional inequality constraints. The first part is devoted to the control constrained case. Then some emphasis is given to pointwise inequality constraints on the state variable and on its gradient. In the last part of the article regularization techniques for state constraints are considered and the question is addressed, how the regularization parameter can adaptively be linked to the discretization error.