DFG-Forschungszentrum Berlin

Project D10: Entropy decay and shape design for nonlinear drift diffusion systems


DFG Forschungszentrum Berlin Leerraum Technische Universität Berlin
Duration: Since May 2003
Project leader: A. Unterreiter
Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
Tel: +49 (0)30 - 314 24884 (office) / - 314 24351 (secretary)
email: unterreiter@math.tu-berlin.de
Project member: R. Plato
Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
Tel: +49 (0)30 - 314 25743
email: plato@math.tu-berlin.de
Cooperations: A. Jüngel, Universität Mainz
R. Pinnau, TU Darmstadt
S. Volkwein, TU Graz
Support: DFG Research Center "Mathematics for Key Technologies"


Description

Many important biological, chemical, hydro-dynamical or physical phenomena are successfully modeled by nonlinear drift diffusion equations What makes these models so popular is - amongst others - their ability to describe the underlying system's transient phase, that is, to identify equilibrium states and to estimate the system's "decay rate" which measures how "fast" the equilibrium state is approached as time increases.

Recently developed analytical tools have made it possible to explain and to estimate these decay rates for several vital nonlinear drift diffusion systems, e.g., porous medium and fast diffusion equations. The crucial step is played by the so-called "entropy method" - occasionally also known as "energy method" or "Lyapunoff's method". The main idea of this method is to investigate the long-time behavior of the system's entropy E, which is a real-valued function depending on the limiting equilibrium state us and on the time t. The entropy measures in a deep - though not "metric" - sense the distance of the system's actual state from us. This is literally true when measuring distances in norms: For entropies arising in many important situations, well-known Csiszar-Kullback type inequalities allow to deduce corresponding decay rates in more familiar L1-norms, in particular, in the L1-norm. Once the system's equilibrium state us and the system's entropy E are identified, one derives an (ordinary) differential inequality for E. From this inequality estimates of the decay rate of E as t increases emerge.

Research program

Bullet It is the main aim of the proposed research project to exploit and to extend entropy methods to simulate and to optimize transient performances of semiconductor devices. In particular it is planned to derive decay rates for "discrete" entropies associated with numerical schemes and to optimize semiconductor's doping profiles with respect to desired equilibrium states and with respect to entropy decay rates. First results on that topic are given in reference [4]. Investigations are in progress.
Bullet Another important feature is the calibration of drift-diffusions models, i.e., the determination of the coefficients of the system. Applications are, e.g., the determination of prices for options in the financial markets. Those problems can be decomposed into two subproblems. One of these two subproblems is the solution of Volterra integral equations of the first kind. Those problems can be efficiently and stably solved by multistep methods as it shown in reference [3].


References

[1] A. Arnold and A. Unterreiter
Entropy decay of discretized Fokker-Planck equations - temporal semi-discretization. , "Journal of Computational and Applied Mathematics". Abstract
[2] M. Ramaswami and A. Unterreiter
Generalized Hardy-Sobolev inequalities and exponential decay of the Entropy of degenerated parabolic equations, to appear in "Monatshefte für Mathematik".
[3] R. Plato
Fractional multistep methods for weakly singular Volterra integral equations of the first kind with perturbed data. Preprint DFG FZT No. 4.
[4] A. Jüngel and A. Unterreiter
Discrete Minimum and Maximum Principles for Finite Element Approximations of Non-Monotone Elliptic Equations. Preprint DFG FZT No. 42.
[5] R. Plato
Large time asymptotics for a fully discretized Fokker-Planck type equation. Preprint DFG FZT No. 137.



Related Talks

[1]R. Plato Regularization of weakly singular Volterra equations by fractional multistep methods 20th Biennial Conference on Numerical Analysis, Dundee, June 2003.
[2] R. Plato Entropie estimates for Fokker-Planck equations. Gamm conference, Dresden, March 2004.
[3] A. Unterreiter Optimal Control of the Stationary Quantum Drift-Diffusion Model EUCCO conference, Dresden, March 2004.
[4] R. Plato Large time asymptotics for Fokker-Planck equations. Conference Modern Computational Methods in Applied Mathematics, Bedlewo, June 2004.



Other activities

Book publications

R. Plato Concise Numerical Mathematics , AMS, Rhode Island, 460 pages, 2003.
R. Plato Numerische Mathematik kompakt, 2. edition , Vieweg Verlag, Wiesbaden, 410 pages, appears in August 2004.
R. Plato Übungsbuch zur Numerischen Mathematik , Vieweg Verlag, Wiesbaden, 220 pages, appears in September 2004.



mailbox plato@math.tu-berlin.de