
Tyrone Rees; Martin Stoll; Andy Wathen
The optimization of functions subject to partial differential equations (PDE) plays an important role in many areas of science and industry. In this paper we introduce the basic concepts of PDEconstrained optimization and show how the allatonce approach will lead to linear systems in saddle point form. We will discuss implementation details and different boundary conditions. We then show how these system can be solved efficiently and discuss methods and preconditioners also in the case when bound constraints for the control are introduced. Numerical results will illustrate the competitiveness of our techniques.

Martin Stoll; John W. Pearson; Philip K. Maini
The modelling of pattern formation in biological systems using various models of reactiondiffusion type has been an active research topic for many years. We here look at a parameter identification (or PDEconstrained optimization) problem where the Schnakenberg and GiererMeinhardt equations, two wellknown pattern formation models, form the constraints to an objective function. Our main focus is on the efficient solution of the associated nonlinear programming problems via a LagrangeNewton scheme. In particular we focus on the fast and robust solution of the resulting large linear systems, which are of saddle point form. We illustrate this by considering several two and...

J. C. De Los Reyes; C. Meyer; B. Vexler
An optimal control problem for 2d and 3d Stokes equations is investigated with pointwise inequality constraints on the state and the control. The paper is concerned with the full discretization of the control problem allowing for different types of discretization of both the control and the state. For instance, piecewise linear and continuous approximations of the control are included in the present theory. Under certain assumptions on the L∞error of the finite element discretization of the state, error estimates for the control are derived which can be seen to be optimal since their order of convergence coincides with the one...

K. Krumbiegel; I. Neitzel; A. Rösch
We develop sufficient optimality conditions for a MoreauYosida regularized optimal control problem governed by a semilinear elliptic PDE with pointwise constraints on the state and the control. We make use of the equivalence of a setting of MoreauYosida regularization to a special setting of the virtual control concept, for which standard second order sufficient conditions have been shown. Moreover, we present a numerical example, solving a MoreauYosida regularized model problem with an SQP method.

Ira Neitzel; UWE PRÜFERT; Thomas Slawig
In [17] we have shown how timedependent optimal control for partial differential equations can be realized in a modern highlevel modeling and simulation package. In this article we extend our approach to (state) constrained problems. Pure state constraints in a function space setting lead to nonregular Lagrange multipliers (if they exist), i.e. the Lagrange multipliers are in general Borel measures. This will be overcome by different regularization techniques. To implement inequality constraints, active set methods and barrier methods are widely in use. We show how these techniques can be realized in a modeling and simulation package. We implement a projection...

J. Frédéric Bonnans; Francisco J. Silva

Eduardo Casas; Fredi Tröltzsch
A class of optimal control problems for quasilinear elliptic equations is considered, where the coefficients of the elliptic differential operator depend on the state function. First and secondorder optimality conditions are discussed for an associated controlconstrained optimal control problem. Main emphasis is laid on secondorder sufficient optimality conditions. To this aim, the regularity of the solutions to the state equation and its linearization is studied in detail and the Pontryagin maximum principle is derived. One of the main difficulties is the nonmonotone character of the state equation.

Eduardo Casas; Juan Carlos De Los Reyes; Fredi Tröltzsch
Secondorder sufficient optimality conditions are established for the optimal control of semilinear elliptic and parabolic equations with pointwise constraints on the control and the state. In contrast to former publications on this subject, the cone of critical directions is the smallest possible in the sense that the secondorder sufficient conditions are the closest to the associated necessary ones. The theory is developed for elliptic distributed controls in domains up to dimension three. Moreover, problems of elliptic boundary control and parabolic distributed control are discussed in spatial domains of dimension two and one, respectively.

J. C. De Los Reyes; P. Merino; J. Rehberg; F. Tröltzsch
The paper deals with optimal control problems for semilinear elliptic and parabolic PDEs subject to pointwise state constraints. The main issue is that the controls are taken from a restricted control space. In the parabolic case, they are Rmvectorvalued functions of the time, while the are vectors of Rm in elliptic problems. Under natural assumptions, first and secondorder sufficient optimality conditions are derived. The main result is the extension of secondorder sufficient conditions to semilinear parabolic equations in domains of arbitrary dimension. In the elliptic case, the problems can be handled by known results of semiinfinite optimization. Here, different examples...

Ira Neitzel; Fredi Tröltzsch
MoreauYosida and Lavrentiev type regularization methods are considered for nonlinear optimal control problems governed by semilinear parabolic equations with bilateral pointwise control and state constraints. The convergence of optimal controls of the regularized problems is studied for regularization parameters tending to infinity or zero, respectively. In particular, the strong convergence of global and local solutions is addressed. Moreover, it is shown that, under certain assumptions, locally optimal solutions of the Lavrentiev regularized problems are locally unique. This analysis is based on a secondorder sufficient optimality condition and a separation assumption on almost active sets.

Irwin Yousept
An optimal control problem arising in the context of 3D electromagnetic induction heating is investigated. The state equation is given by a quasilinear stationary heat equation coupled with a semilinear timeharmonic eddy current equation. The temperaturedependent electrical conductivity and the presence of pointwise inequality stateconstraints represent the main challenge of the paper. In the first part of the paper, the existence and regularity of the state are addressed. The second part of the paper deals with the analysis of the corresponding linearized equation. Some sufficient conditions are presented which guarantee the solvability of the linearized system. The final part of...

F. Tröltzsch, et al.

Michael Hinze; Christian Meyer

J. Frédéric Bonnans; Francisco J. Silva

F. Tröltzsch, et al.

Tran The Truyen

Dominique Guégan; Bertrand K. Hassani

Dominique Guégan; Bertrand K. Hassani
The Advanced Measurement Approach requires financial institutions to develop internal

Liang Liu
The desire to infer the evolutionary history of a group of species should be more viable now that a considerable amount of multilocus molecular data is available. However, the current molecular phylogenetic paradigm still reconstructs gene trees to represent the species tree. Further, commonly used methods to combine data, such as the concatenation method, the consensus tree method, or the gene tree parsimony method may be biased. In this dissertation, I propose a Bayesian hierarchical model to estimate the phylogeny of a group of species using multiple estimated gene tree distributions such as those that arise in a Bayesian analysis...

Kasper Kabell