Distributed optimal control of a nonstandard system of phase field equations

We investigate a distributed optimal control problem for a phase field model of Cahn-Hilliard type. The model describes two-species phase segregation on an atomic lattice under the presence of diffusion; it has been recently introduced by the same authors in arXiv:1103.4585v1 [math.AP] and consists of a system of two highly nonlinearly coupled PDEs. For this reason, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.Comment: Key words: distributed optimal control, nonlinear phase field systems, first-order necessary optimality condition

Local Error Analysis of Discontinuous Galerkin Methods for Advection-Dominated Elliptic Linear-Quadratic Optimal Control Problems

This paper analyzes the local properties of the symmetric interior penalty upwind discontinuous Galerkin (SIPG) method for the numerical solution of optimal control problems governed by linear reaction-advection-diffusion equations with distributed controls. The theoretical and numerical results presented in this paper show that for advection-dominated problems the convergence properties of the SIPG discretization can be superior to the convergence properties of stabilized finite element discretizations such as the streamline upwind Petrov Galerkin (SUPG) method. For example, we show that for a small diffusion parameter the SIPG method is optimal in the interior of the domain. This is in sharp contrast to SUPG discretizations, for which it is known that the existence of boundary layers can pollute the numerical solution of optimal control problems everywhere even into domains where the solution is smooth and, as a consequence, in general reduces the convergence rates to only first order. In order to prove the nice convergence properties of the SIPG discretization for optimal control problems, we first improve local error estimates of the SIPG discretization for single advection-dominated equations by showing that the size of the numerical boundary layer is controlled not by the mesh size but rather by the size of the diffusion parameter. As a result, for small diffusion, the boundary layers are too “weak” to pollute the SIPG solution into domains of smoothness in optimal control problems. This favorable property of the SIPG method is due to the weak treatment of boundary conditions, which is natural for discontinuous Galerkin methods, while for SUPG methods strong imposition of boundary conditions is more conventional. The importance of the weak treatment of boundary conditions for the solution of advection dominated optimal control problems with distributed controls is also supported by our numerical results

An Iterative Model Reduction Scheme for Quadratic-Bilinear Descriptor Systems with an Application to Navier-Stokes Equations

We discuss model reduction for a particular class of quadratic-bilinear (QB) descriptor systems. The main goal of this article is to extend the recently studied interpolation-based optimal model reduction framework for QBODEs [Benner et al. '16] to a class of descriptor systems in an efficient and reliable way. Recently, it has been shown in the case of linear or bilinear systems that a direct extension of interpolation-based model reduction techniques to descriptor systems, without any modifications, may lead to poor reduced-order systems. Therefore, for the analysis, we aim at transforming the considered QB descriptor system into an equivalent QBODE system by means of projectors for which standard model reduction techniques for QBODEs can be employed, including aforementioned interpolation scheme. Subsequently, we discuss related computational issues, thus resulting in a modified algorithm that allows us to construct \emph{near}--optimal reduced-order systems without explicitly computing the projectors used in the analysis. The efficiency of the proposed algorithm is illustrated by means of a numerical example, obtained via semi-discretization of the Navier-Stokes equations

Model Order Reduction for Rotating Electrical Machines

The simulation of electric rotating machines is both computationally expensive and memory intensive. To overcome these costs, model order reduction techniques can be applied. The focus of this contribution is especially on machines that contain non-symmetric components. These are usually introduced during the mass production process and are modeled by small perturbations in the geometry (e.g., eccentricity) or the material parameters. While model order reduction for symmetric machines is clear and does not need special treatment, the non-symmetric setting adds additional challenges. An adaptive strategy based on proper orthogonal decomposition is developed to overcome these difficulties. Equipped with an a posteriori error estimator the obtained solution is certified. Numerical examples are presented to demonstrate the effectiveness of the proposed method

Constrained dogleg methods for nonlinear systems with simple bounds

We focus on the numerical solution of medium scale bound-constrained systems of nonlinear equations. In this context, we consider an affine-scaling trust region approach that allows a great flexibility in choosing the scaling matrix used to handle the bounds. The method is based on a dogleg procedure tailored for constrained problems and so, it is named Constrained Dogleg method. It generates only strictly feasible iterates. Global and locally fast convergence is ensured under standard assumptions. The method has been implemented in the Matlab solver CoDoSol that supports several diagonal scalings in both spherical and elliptical trust region frameworks. We give a brief account of CoDoSol and report on the computational experience performed on a number of representative test problem

A Spatial Domain Decomposition Method for Parabolic Optimal Control Problems

We present a non-overlapping spatial domain decomposition method for the solution of linear-quadratic parabolic optimal control problems. The spatial domain is decomposed into non-overlapping subdomains. The original parabolic optimal control problem is decomposed into smaller problems posed on space-time cylinder subdomains with auxiliary state and adjoint variables imposed as Dirichlet boundary conditions on the space-time interface boundary. The subdomain problems are coupled through Robin transmission conditions. This leads to a Schur complement equation in which the unknowns are the auxiliary state adjoint variables on the space-time interface boundary. The Schur complement operator is the sum of space-time subdomain Schur complement operators. The application of these subdomain Schur complement operators is equivalent to the solution of an subdomain parabolic optimal control problem. The subdomain Schur complement operators are shown to be invertible and the application of their inverses is equivalent to the solution of a related subdomain parabolic optimal control problem. We introduce a new family of Neumann-Neumann type preconditioners for the Schur complement system including several different coarse grid corrections. We compare the numerical performance of our preconditioners with an alternative approach recently introduced by Benamou