DFG-Forschungszentrum Matheon

Application Area D: Electronic circuits and optical technologies
Project D6

Numerical methods for stochastic differential-algebraic equations 
applied to transient noise analysis in circuit simulation

Infineon Technologies AG DFG Forschungszentrum Berlin Humboldt-Universität zu Berlin
Duration: October 2003 - May 2006
Project directors: Prof. Dr. R. März, Prof. Dr. W. Römisch

Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany

Tel: +49 (0)30 - 2093 2353 (secretary) / - 2093 2861 (office März) / - 2093 2561 (office Römisch)

email: maerz@mathematik.hu-berlin.de, romisch@mathematik.hu-berlin.de
Responsible: Dr. Renate Winkler, Humboldt-Universität Berlin, Tel: 030 - 2093 5448 / Sekr. 2353 , 

email: winkler@mathematik.hu-berlin.de
Cooperation: Infineon Technologies AG Mnchen
Support: DFG Research Center "Mathematics for Key Technologies"

Description :

In several fields of applications including electric circuit simulation, chemical engineering and mechanical multibody systems one is confronted with differential-algebraic equations driven by noise processes. Much research has been devoted to the numerical solution of stochastic differential equations. Yet, only first attempts have been made towards a numerical analysis of stochastic differential-algebraic equations (SDAEs). This work  extends research that was performed within two BMBF-projects by R. März and W. Römisch.

This project is concerned with the analysis and development of numerical methods for SDAEs as they arise in transient noise analysis in circuit simulation. There, thermal noise of resistors as well as  shot noise of semiconductors is modelled by additional sources of additive or multiplicative Gaussian white noise currents that are shunt in parallel to the ideal, noise-free elements. Typical circuit simulation models form specially structured nonlinear SDAEs of index 1 or 2, which contain a  large number of equations as well as of small noise sources. Recognizing their special DAE- structure is crucial for stable and efficient simulations even in the deterministic case.
In this application one is interested in solution paths, since they supply more insight in characteristics like the phase-noise than only the time-evaluation of the  distribution of the solution can do. Therefore the strong solution concept is applied.

Initiated by the industrial partners at Infineon AG , first results concerning the analysis and simulation of  SDAEs have been achieved in the past. In particular, the existence and uniqueness of solutions as well as  stability, consistency and convergence of certain drift-implicit schemes have been proven for SDAEs of index~1 with noisefree constraints [9-12]. In this project we aim at an analysis of SDAEs  of index~2 and the development of suitable numerical schemes with higher deterministic order including error-estimation and stepsize-control. 

Results :

Structural analysis of  index 2  Ito- SDAEs :

Index 2 DAEs are characterized by hidden constraints and  differentiation tasks.  A theoretical decoupling of index 2 SDAEs in its qualitatively different parts forms the basis of an understanding of such problems. Here, it is essential to find constant transformations that are compatible with the  Ito-calculus. This can be  done  using structural properties given in circuit simulation. Using the charge-oriented version of the modified nodal analysis is essential for a problem-adapted stable numerical simulation as is shown in [13-16] for deterministic circuit models.

Integration schemes:

Schemes that are  specially suited for  SDAEs with a large number of small noise sources and low smoothness of the  coefficients are needed. In particular, stochastic versions of those order 2 schemes are of interest that are well established in  deterministic circuit simulation. There, the BDF-schemes are widely used. As they are multi-step schemes, they only need one evaluation of the residuum per Newton-step. A convergence analysis of multi-step schemes for SDEs  is done in [3]. Introducing the concepts of stability, consistency und convergence for multi-step schemes for SDEs in a suitable way we related local and global errors. This formed the basis for the analysis of local errors and lead to suitable proposals for schemes in case of small noise [3,4,5]. Moreover, an analysis of the  asymptotic  stability properties of two-step schemes  for SDEs  [6], and a generalization of the   convergence analysis to stochastic delay equations [7] has been done. The  implementation of the  two-step BDF-Maruyama-scheme for SDAEs in circuit simulation is presented  in [8].

Error-estimation and stepsize-control:

Based on the concepts of local and global errors for one-step schemes [10,11] a stepsize-control for the family of Euler-Maruyama-schemes has been developed [1]. The mean square of local errors is estimated by means of a number of simultaneously computed paths, which leads to  an adaptive stepsize-sequence that is identical for all paths. Exploiting the smallness of the noise and looking at stepsizes that are not asymptotically small, a derivative-free realization of this stepsize-control has been presented and implemented. First steps towards a generalization to multi-step schemes with deterministic order 2 for small noise  SDEs and SDAEs are taken in [2].

References :

W. Römisch and R. Winkler:
 Stepsize control for mean-square numerical methods for stochastic differential equations with small noise,
to appear in SIAM J. Sci. Comp.
T. Sickenberger, E. Weinmüller  and R. Winkler:
Local error estimates for moderately smooth ODEs and DAEs,
Preprint 06-1, Institut für Mathematik, Humboldt-Universität Berlin, 2006
E. Buckwar and R. Winkler:
 Multi-step methods for SDEs and their application to problems with small noise,
to appear in SIAM J. Num. Anal.
E. Buckwar and R.Winkler:
 On two-step schemes for SDEs with small noise,
PAMM, vol. 4 (1), 15-18,  2004.
E. Buckwar and R. Winkler:
 Improved linear multi-step methods for stochastic ordinary differential equations,
Preprint 05-10, Institut für Mathematik, Humboldt-Universität Berlin, 2005, and submitted.
E. Buckwar, R. Horvath Bokor and R. Winkler:
 Asymptotic mean-square stability of two-step methods for stochastic ordinary differential equations,
Preprint 04-25, Institut für Mathematik, Humboldt-Universität Berlin, 2004, and submitted.
E. Buckwar and R. Winkler:
Multi-step Maruyama methods for stochastic delay differential equations,
Preprint 04-15, Institut für Mathematik, Humboldt-Universität Berlin, 2004.
 Stochastic differential algebraic equations  in transient noise analysis,
Springer Series 'Mathematics in Industry' ,Proceedings of 'Scientific Computing in Electrical Engineering', September, 25th - 9th, 2004, Capo D'Orlando, to appear.
G. Denk and R.Winkler:
Modeling and simulation of transient noise in circuit simulation,

Proceedings of 4th MATHMOD, Vienna, Feb. 5-7, 2003, to appear in: Mathematical and Computer Modelling of Dynamical Systems (MCMDS).
W. Römisch and R. Winkler:
 Stochastic DAEs in circuit simulation,
in: Modeling, Simulation and Optimization of Integrated Circuits (K. Antreich, R. Bulirsch, A. Gilg and P. Rentrop eds.), Birkhäuser, Basel 2003, 303-318.
[11] R. Winkler:
Stochastic differential algebraic equations of index 1 and application in circuit simulation.
Journal of Computational and Applied Mathematics 157 (2003), 477-505.
[12] R. Winkler:
 Stochastic DAEs in transient noise simulation.,
 Springer Series 'Mathematics in Industry' ,Proceedings of 'Scientific Computing in Electrical Engineering', June, 23rd - 28th, 2002, Eindhoven, 408-415.
I. Higueras, R. März, and C. Tischendorf.
Stability preserving integration of index-1 DAE's.
Appl. Numer. Math., 45(2-3):175-200, 2003.
I. Higueras, R. März, and C. Tischendorf.
Stability preserving integration of index-2 DAEs.
Appl. Numer. Math., 45(2-3):201-229, 2003.
R. März.
Differential-algebraic systems with properly stated leading term and MNA equations.
In K. Anstreich, R. Bulirsch, A. Gilg, and P. Rentrop, editors, Modelling, Simulation and Optimization of Integrated Circuits, pages 135-151. Birkhäuser, 2003.
R. März.
Characterizing differential algebraic equations without the use of derivative arrays.
Comput. Math. Appl., 50(7):1141-1156, 2005.

Poster :   January 2006

last modified January 25, 2006