DFG-Forschungszentrum Berlin

Model reduction for large-scale systems in control and circuit simulation
DFG-Forschungszentrum Technische Universität Berlin

Duration: August 2002 - May 2006
Project directors: Prof. Dr. P. Benner (*) (* leaves Matheon to 2004/09/30)
Fakultät für Mathematik, Technische Universität Chemnitz, Reichenhainer Straße 41, 09126 Chemnitz, Germany
Tel: +49 (0)371 - / 531 8367
email: benner@mathematik.tu-chemnitz.de
Prof. Dr. V. Mehrmann
Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
Tel: +49 (0)30 - / 314 25736
email: mehrmann@math.tu-berlin.de
Prof. Dr. F. Tröltzsch
Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
Tel: +49 (0)30 - /314 79688
email: troeltzsch@math.tu-berlin.de
Researcher: U. Baur
Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
Tel: +49 (0)30 - / 314 79177 
email: baur@math.tu-berlin.de
Cooperation: A.C. Antoulas (Rice University, USA)
E. Quintana-Ortí (U Castellón, Spain)
D. Sorensen (Rice University, USA)
T. Stykel (Technische Universität Berlin, Germany)
Support: DFG Research Center "Mathematics for Key Technologies"

Project description Publications Guests Talks Organized Conferences Software

Project description.

Background.

The simulation of electronic circuits and the control of dynamical processes described by coupled systems of ODEs, DAEs or PDEs leads to models of ever-increasing complexity. In circuit simulation, in particular the further miniaturization in VLSI chip design and the resulting need for modeling sidewall effects and interconnect leads to large-scale ODE or DAE systems [4].

In optimal control of PDEs, a trend towards the coupling of several types of PDEs describing the system dynamics can be observed. For instance, the modelling of non-isothermal viscoelastic fluid flows requires to couple the equations of Navier-Stokes and heat transfer [5].

In chip design or optimal control of PDEs, the associated very large scale systems must be solved very often. A model reduction is absolutely necessary to attack these problems in reasonable time. In the ODE/DAE case, Padé approximations [4] are frequently used, but they do not preserve per se important properties of electronic circuits such as stability and passivity. Often, POD methods are the method of choice in PDE control [6]. However, they depend on a fairly artificial choice of reference controls. In both cases, the approximation error is hard to quantify.

Expertise.

Special factorizations of the system Gramians are used to develop stochastic and balanced truncation methods for large-scale dense systems in [2,3]. Initial tests based on applying similar ideas to sparse problems are reported in [7]. This project continues research in the project A8 of the SFB 393 at TU Chemnitz (End December 31, 2001).

Research program.

We will develop model reduction methods based on (approximate) balanced truncation. These methods were successfully applied to compute reduced-order models for linear control systems of high state-space dimensions [1,2,7]. These methods do not depend on single reference controls, permit to obtain error estimates [1,2,7] and preserve stability. Balanced truncation does not preserve passivity. Therefore, the stochastic truncation method will be adapted to large-scale (DAE) systems. Moreover, balanced truncation is designed for linear systems and we aim to extend this method to the case of nonlinear coupled PDE systems by integrating it in SQP iterations.

Cooperation.

Within the DFG Research Center, we will employ methods from "Solution of large unstructured linear systems in circuit simulation" (M. Bollhöfer, V. Mehrmann) in the innermost iteration loops of the model reduction methods. We will also cooperate with "Structure analysis for simulation and control problems of differential algebraic equations" (R. März, V. Mehrmann), "Numerical Simulation of Integrated Circuits for Future Chip Generations" (R. März, C. Tischendorf) in circuit simulation, "Efficient simulation of flows in semiconductor melts" (E. Bänsch), "Optimal control of sublimation growth of SiC bulk single crystals" (J. Sprekels, F. Tröltzsch, O. Klein, A. Rösch) in optimization of crystal growth and "Shape optimization and control of curved mechanical structures" (J. Sprekels) on shape optimization.

References.


[1] A.C. Antoulas and D. C. Sorensen: Approximation of large-scale dynamical systems: An overview. Int. J. Appl. Math. Comp. Sci., vol. 11, pp. 1093--1121, 2001.
[2] P. Benner, E.S. Quintana-Ortí, and G. Quintana-Ortí: Balanced Truncation Model Reduction of Large-Scale Dense Systems on Parallel Computers. Math. Comp. Model. Dyn. Syst., vol. 6, no. 4, pp. 383--405, 2000. 
[3] P. Benner, E.S. Quintana-Ortí, and G. Quintana-Ortí: Efficient Numerical Algorithms for Balanced Stochastic Truncation. Int. J. Appl. Math. Comp. Sci., vol. 11, pp. 1123--1150, 2001. 
[4] R. Freund: Reduced-order modeling techniques based on Krylov subspaces and their use in circuit simulation. In B.N. Datta, ed., Appl. Comp. Control, Signals and Circuits, vol. 1, pp. 435--498. Birkhäuser, Boston, MA, 1999. 
[5] K. Kunisch and X. Marduel: Optimal control of non-isothermal viscoelastic fluid flow. J. of Non-Newtonian Fluid Mechanics, pp. 261--301, 2000.
[6] K. Kunisch and S. Volkwein: Control of Burgers' equation by a reduced order approach using proper orthogonal decomposition. J. Optim. Theory Appl., vol. 102, pp. 345--371, 1999. 
[7] V. Mehrmann, T. Penzl, and F. Tröltzsch: Control of heterogeneous systems of partial differential equations and differential algebraic equations. Invited presentation: 9th Seminar on Num. Sol. of Diff. and Diff.-Alg. Eqs., Halle, 4.9.--8.9.2000.

Publications.


Guests.


Talks.


Organized Conferences.


Software.