Abstract
In this paper we describe how stochastic differential-algebraic equations (SDAEs) arise as a mathematical model for network equations that are influenced by additional sources of Gaussian white noise. We give the necessary analytical theory for the existence and uniqueness of strong solutions, provided that the systems have noise-free constraints and are uniformly of DAE-index 1. We express these conditions in terms of the network-topology for reasons of use within a circuit simulator. In the second part we analyze discretization methods. Due to the differential-algebraic structure, implicit methods will be necessary. By the examples of the drift-implicit Euler and Milstein schemes we show how drift-implicit schemes for SDEs can be adapted to become directly applicable to stochastic DAEs and prove that the convergence properties of these methods known for SDEs are preserved. For illustration we apply the drift-implicit Euler scheme to an oscillator circuit.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
U. Ascher and L. Petzold. Computer methods for ordinary differential equations and differential-algebraic equations. SIAM, 1998.
A. Blum. au]Elektronisches_Rauschen. Teubner, 1996.
A. Demir. Analysis and simulation of noise in nonlinear electronic circuits and systems. PhD thesis, University of California, Berkeley, 1997.
A. Demir and A. Sangiovanni-Vincentelli. Analysis and simulation of noise in nonlinear electronic circuits and systems. Kluwer Academic Publishers, 1998.
D. Estevez Schwarz and C. Tischendorf. Structural analysis for electronic circuits and consequences for MNA. Int. J. Circ. Theor. Appl., 28:131–162, 2000.
M. Günther and U. Feldmann. CAD-based electric-circuit modeling in industry I. mathematical structure and index of network equations. Surv. Math. Ind., 8:97–129, 1999.
D.J. Higham. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review, 43:525–546, 2001.
G. Maruyama. Continuous Markov processes and stochastic equations. Rend. Circolo Math. Palermo, 4:48–90, 1955.
R. März. Numerical methods for differential-algebraic equations. Acta Numérica 141–198, 1992.
G.N. Milstein and M.V. Tretyakov. Mean-square numerical methods for stochastic differential equations with small noise. SIAM J.Sci. Corn-put., 18:1067–1087, 1997.
C. Penski. Analysis and numerical integration of stochastic differential-algebraic equations and applications in electronic circuit simulation. Preprint TUM-M9907, Technische Universität, München, 1999, (www-lit.ma.turn.de/bb/bb).
C. Penski. A new numerical method for SDEs and its application in circuit simulation. J. Comput. Appl. Math., 115:461–470, 2000.
O. Schein. Stochastic differential algebraic equations in circuit simulation. PhD thesis, Technische Universität Darmstadt, 1999.
O. Schein and G. Denk. Numerical solution of stochastic differential-algebraic equations with applications to transient noise simulation of microelectronic circuits. J. Comput. Appl. Math., 100:77–92, 1998.
J. Sieber. Local error control for general index-1 and index-2 differential algebraic equations. Preprint 97-21, Humboldt-Universität, Berlin, 1997.
C. Tischendorf. Topological index calculation of DAEs in circuit simulation. Surv. Math. Ind., 8(3-4):187–199, 1999.
L. Weiß. Rauschen in nichtlinearen elektronischen Schaltungen und Bauelementen — ein thermodynamischer Zugang. PhD thesis, Otto-von-Guericke Universität Magdeburg, 1999.
L. Weiß and W. Mathis. A thermodynamical approach to noise in nonlinear networks. Int. J. Circ. Theor. Appl., 26:147–165, 1998.
R. Winkler. Stochastic differential algebraic equations and applications in circuit simulation. Preprint 01-13, Humboldt-Universität, Berlin, 2001, submitted to J. Comput. Appl. Math. (www.mathematik.hu-berlin.de/publ/pre/2001/p-list-01.html/publ/pre/2001/p-list-01.html).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Winkler, R. (2004). Stochastic DAEs in Transient Noise Simulation. In: Schilders, W.H.A., ter Maten, E.J.W., Houben, S.H.M.J. (eds) Scientific Computing in Electrical Engineering. Mathematics in Industry, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55872-6_45
Download citation
DOI: https://doi.org/10.1007/978-3-642-55872-6_45
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-21372-7
Online ISBN: 978-3-642-55872-6
eBook Packages: Springer Book Archive