A FE-ADMM algorithm for Lavrentiev-regularized state-constrained elliptic control problem

In this paper, Elliptic control problems with pointwise box constraints on the state is considered, where the corresponding Lagrange multipliers in general only represent regular Borel measure functions. To tackle this difficulty, the Lavrentiev regularization is employed to deal with the state constraints. To numerically discretize the resulted problem, full piecewise linear finite element discretization is employed. Estimation of the error produced by regularization and discretization is done. The error order of full discretization is not inferior to that of variational discretization because of the Lavrentiev-regularization. Taking the discretization error into account, algorithms of high precision do not make much sense. Utilizing efficient first-order algorithms to solve discretized problems to moderate accuracy is sufficient. Then a heterogeneous alternating direction method of multipliers (hADMM) is proposed. Different from the classical ADMM, our hADMM adopts two different weighted norms in two subproblems respectively. Additionally, to get more accurate solution, a two-phase strategy is presented, in which the primal-dual active set (PDAS) method is used as a postprocessor of the hADMM. Numerical results not only verify error estimates but also show the efficiency of the hADMM and the two-phase strategy.

Preconditioned Dirichlet-Dirichlet Methods for Optimal Control of Elliptic PDE

The discretization of optimal control of elliptic partial differential equations problems yields optimality conditions in the form of large sparse linear systems with block structure. Correspondingly, when the solution method is a Dirichlet-Dirichlet non-overlapping domain decomposition method, we need to solve interface problems which inherit the block structure. It is therefore natural to consider block preconditioners acting on the interface variables for the acceleration of Krylov methods with substructuring preconditioners. In this paper we describe a generic technique which employs a preconditioner block structure based on the fractional Sobolev norms corresponding to the domains of the boundary operators arising in the matrix interface problem, some of which may include a dependence on the control regularization parameter. We illustrate our approach on standard linear elliptic control problems. We present analysis which shows that the resulting iterative method converges independently of the size of the problem. We include numerical results which indicate that performance is also independent of the control regularization parameter and exhibits only a mild dependence on the number of the subdomains.

A Goal-Oriented Adaptive Moreau-Yosida Algorithm for Control- and State-Constrained Elliptic Control Problems

In this work, we develop an adaptive algorithm for solving elliptic optimal control problems with simultaneously appearing state and control constraints. The algorithm combines a Moreau-Yosida technique for handling state constraints with a semi-smooth Newton method for solving the optimality systems of the regularized sub-problems. The state and adjoint variables are discretized using continuous piecewise linear finite elements while a variational discretization concept is applied for the control. To perform the adaptive mesh refinements cycle we derive local error estimators which extend the goal-oriented error approach to our setting. The performance of the overall adaptive solver is assessed by numerical examples.