René Pinnau

Optimization with PDE constraints

This project is concerned with the development, analysis and application of new, innovative mathematical techniques for the solution of constrained optimization problems where a partial differential equation (PDE) or a system of PDEs appears as an essential part of the constraints. Such optimization problems arise in a wide variety of important applications in the form of, e.g., parameter identification problems, optimal design problems, or optimal control problems. The efficient and robust solution of PDE constrained optimization problems has a strong impact on more traditional applications in, e.g., automotive and aerospace industries and chemical processing, as well as on applications in recently emerging technologies in materials and life sciences including environmental protection, bioand nanotechnology, pharmacology, and medicine. The appropriate mathematical treatment of PDE constrained optimization problems requires the integrated use of advanced methodologies from the theory of optimization and optimal control in a functional analytic setting, the theory of PDEs as well as the development and implementation of powerful algorithmic tools from numerical mathematics and scientific computing. Experience has clearly shown that the design of efficient and reliable numerical solution methods requires a fundamental understanding of the subtle interplay between optimization in function spaces and numerical discretization techniques which can only be achieved by a close cooperation between researchers from the above mentioned fields.

Global Nonnegative Solutions of a Nonlinear Fourth-Order Parabolic Equation for Quantum Systems

The existence of nonnegative weak solutions globally in time of a nonlinear fourth-order parabolic equation in one space dimension is shown. This equation arises in the study of interface fluctuations in spin systems and in quantum semiconductor modeling. The problem is considered on a bounded interval subject to initial and Dirichlet and Neumann boundary conditions. Further, the initial datum is assumed only to be nonnegative and to satisfy a weak integrability condition. The main difficulty of the existence proof is to ensure that the solutions stay nonnegative and exist globally in time. The first property is obtained by an exponential transformation of variables. Moreover, entropy-type estimates allow for the proof of the second property. Results concerning the regularity and long-time behavior are given. Finally, numerical experiments underlining the preservation of positivity are presented.

NUMERICAL METHODS AND OPTIMAL CONTROL FOR GLASS COOLING PROCESSES

ABSTRACT In this paper, we discuss numerical and analytical approximations of radiative heat transfer equations used to model cooling processes of molten glass. Simplified diffusion type approximations are discussed and investigated numerically. These approximations are also used to develop acceleration methods for the iterative solution of the full radiative heat transfer problem. Moreover, applications of the above diffusion type approximations to optimal control problems for glass cooling processes are discussed.

The Stationary Current-Voltage Characteristics of the Quantum Drift-Diffusion Model

This paper is concerned with numerical algorithms for the bipolar quantum drift-diffusion model. For the thermal equilibrium case, a quasi-gradient method minimizing the energy functional is introduced and strong convergence is proved. The computation of current-voltage characteristics is performed by means of an extended Gummel-iteration. It is shown that the involved fixed point mapping is a contraction for small applied voltages. In this case the model equations are uniquely solvable and convergence of the proposed iteration scheme follows. Numerical simulations of a one-dimensional resonant tunneling diode are presented. The computed current-voltage characteristics are in good qualitative agreement with experimental measurements. The appearance of negative differential resistances is verified for the first time in a quantum drift-diffusion model.

Fast Optimal Design of Semiconductor Devices

This paper presents a new approach to the design of semiconductor devices, which leads to fast optimization methods whose numerical effort is of the same order as a single forward simulation of the underlying model, the stationary drift-diffusion system. The design goal we investigate is to increase the outflow current on a contact for fixed applied voltage; the natural design variable is the doping profile.By reinterpreting the doping profile as a state variable and the electrostatic potential as the new design variable, we obtain a simpler optimization problem, whose Karush--Kuhn--Tucker conditions partially decouple. This property allows us to construct efficient iterative optimization algorithms, which avoid solving the fully coupled drift-diffusion system, and need only solves of the continuity equations and their adjoints. The efficiency and success of the new approach is demonstrated in several numerical examples.

AN OPTIMAL CONTROL APPROACH TO SEMICONDUCTOR DESIGN

The design problem for semiconductor devices is studied via an optimal control approach for the standard drift–diffusion model. The solvability of the minimization problem is proved. The first-order optimality system is derived and the existence of Lagrange-multipliers is established. Further, estimates on the sensitivities are given. Numerical results concerning a symmetric n–p-diode are presented.

A comparison of approximate models for radiation in gas turbines

Approximate equations for radiative heat transfer equations coupled to an equation for the temperature are stated and a comparative numerical study of the different approximations is given. The approximation methods considered here range from moment methods to simplified PN-approximations. Numerical experiments and comparisons in different space dimensions and for various physical situations are presented.

A REVIEW ON THE QUANTUM DRIFT DIFFUSION MODEL

ABSTRACT We consider the quantum drift diffusion model for semiconductor devices and collect recent results on the stationary and transient equations. The stationary model including generation–recombination terms is studied for bipolar devices and the transient equations are considered in the unipolar case. We cover several topics, such as existence and uniqueness of solutions, asymptotic limits and convergence of a nonlinear iteration scheme in the stationary case as well convergence of a positivity preserving semidiscretization of the transient equations and the linear stability of stationary states.

Regularized fixed-point iterations for nonlinear inverse problems

In this paper, we introduce a derivative-free, iterative method for solving nonlinear ill-posed problems Fu = y, where instead of y, noisy data yδ with ||y − yδ|| ≤ δ are given and F:X → Y is a nonlinear operator between Hilbert spaces X and Y. This method is defined by splitting the operator F into a linear part A and a nonlinear part G, such that F = A + G. Then iterations are organized as Auk+1 = yδ − Guk. In the context of ill-posed problems, we consider the situation when A does not have a bounded inverse, thus each iteration needs to be regularized. Under some conditions on the operators A and G, we study the behaviour of the iteration error. We obtain its stability with respect to the iteration number k as well as the optimal convergence rate with respect to the noise level δ, provided that the solution satisfies a generalized source condition. As an example, we consider an inverse problem of initial temperature reconstruction for a nonlinear heat equation, where the nonlinearity appears due to radiation effects. The obtained iteration error in the numerical results has the theoretically expected behaviour. The theoretical assumptions are illustrated by a computational experiment.

Newton's method for optimal temperature-tracking of glass cooling processes

We study Newtons method for the open-loop optimal temperature-tracking problem in glass manufacturing. This results in an optimal boundary control problem for a nonlinear system of partial differential equations. Since at high temperatures heat transfer due to radiation plays a dominant role, we employ the diffusive SP 1 -approximation for the computation of the temperature profile. The optimization algorithm relies on the introduction of the adjoint states, which significantly reduces numerical costs. We present numerical results underlining the feasibility of our approach.