Multistep methods for SDEs and their application to problems with small noise

Buckwar, Evelyn; Winkler, Renate

doi:10.1137/040602857

Cited by 73 publications

(88 citation statements)

References 23 publications

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“…However, the noise densities given in Section 1 contain small parameters and the error behaviour is much better. In fact, the errors are dominated by the deterministic terms as long as the step-size is large enough [6,7]. In more detail, the error of the given methods behaves like O(h 2 + εh + ε 2 h 1/2 ), when ε is used to measure the smallness of the noise, i.e., g r (x,t) = εĝ r (x,t), r = 1,...,m where ε ≪ 1.…”

Section: Adaptive Numerical Methodsmentioning

confidence: 99%

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Simultaneous Step-Size and Path Control for Efficient Transient Noise Analysis

Römisch

Sickenberger

Winkler

2010

Scientific Computing in Electrical Engineering SCEE 2008

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Noise in electronic components is a random phenomenon that can adversely affect the desired operation of a circuit. Transient noise analysis is designed to consider noise effects in circuit simulation. Taking noise into account by means of Gaussian white noise currents, mathematical modelling leads to stochastic differential algebraic equations (SDAEs) with a large number of small noise sources. Their simulation requires an efficient numerical time integration by mean-square convergent numerical methods. As efficient approaches for their integration we discuss adaptive linear multi-step methods, together with a new step-size and path selection control strategy. Numerical experiments on industrial real-life applications illustrate the theoretical findings.

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Section: Adaptive Numerical Methodsmentioning

confidence: 99%

“…The variable step-size BDF 2 Maruyama method for the SDAE (2) has the form (see [6] and, for constant step-sizes, e.g. [7])…”

Section: Adaptive Numerical Methodsmentioning

confidence: 99%

Simultaneous Step-Size and Path Control for Efficient Transient Noise Analysis

Römisch

Sickenberger

Winkler

2010

Scientific Computing in Electrical Engineering SCEE 2008

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“…For β 2 = 0, the stochastic multi-step scheme (1.2) is explicit, otherwise it is drift-implicit. See also [3,4,7,8,9,13,14,17,18].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Mean-Square Stability of Two-Step Methods for Stochastic Ordinary Differential Equations

2006

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Abstract.We deal with linear multi-step methods for SDEs and study when the numerical approximation shares asymptotic properties in the mean-square sense of the exact solution.As in deterministic numerical analysis we use a linear time-invariant test equation and perform a linear stability analysis. Standard approaches used either to analyse deterministic multi-step methods or stochastic one-step methods do not carry over to stochastic multi-step schemes. In order to obtain sufficient conditions for asymptotic mean-square stability of stochastic linear two-step-Maruyama methods we construct and apply Lyapunov-type functionals. In particular we study the asymptotic meansquare stability of stochastic counterparts of two-step Adams-Bashforth-and AdamsMoulton-methods, the Milne-Simpson method and the BDF method.

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“…However, the noise densities given in Section 1 contain small parameters and the error behaviour is much better. In fact, the errors are dominated by the deterministic terms as long as the step-size is large enough [2,11]. In more detail, the error of the given methods behaves like O(h 2 + εh + ε 2 h 1/2 ), when ε is used to measure the smallness of the noise, i.e., g r (x, t) = εĝ r (x, t), r = 1,...,m where ε 1.…”

Section: Adaptive Numerical Methodsmentioning

confidence: 99%

“…The variable step-size BDF 2 Maruyama method for the SDAE (3) has the form (see [11] and, for constant step-sizes, e.g. [2])…”

Section: Adaptive Numerical Methodsmentioning

confidence: 99%

Efficient Transient Noise Analysis in Circuit Simulation

Denk¹,

Römisch

Sickenberger

et al.

Mathematics – Key Technology for the Future

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Transient noise analysis means time domain simulation of noisy electronic circuits. We consider mathematical models where the noise is taken into account by means of sources of Gaussian white noise that are added to the deterministic network equations, leading to systems of stochastic differential algebraic equations (SDAEs). A crucial property of the arising SDAEs is the large number of small noise sources that are included. As efficient means of their integration we discuss adaptive linear multi-step methods, in particular stochastic analogues of the trapezoidal rule and the two-step backward differentiation formula, together with a new step-size control strategy. Test results including real-life problems illustrate the performance of the presented methods. Transient noise analysis in circuit simulationThe increasing scale of integration, high clock frequencies and low supply voltages cause smaller signal-to-noise ratios. Reduced signal-to-noise ratio means that the difference between the wanted signal and noise is getting smaller. A consequence of this is that the circuit simulation has to take noise into account. In several applications the noise influences the system behaviour in an essentially nonlinear way such that linear noise analysis is no longer satisfactory and transient noise analysis, i.e., the simulation of noisy systems in the time domain, becomes necessary (see [4,16]). For an implementation of an efficient transient noise analysis in an analog simulator, both an appropriate modelling and integration scheme is necessary (see [3]).Here we deal with the thermal noise of resistors as well as the shot noise of semiconductors that are modelled by additional sources of additive or multiplicative Gaussian white noise currents that are shunt in parallel to the noise-free elements. Thermal noise i th of resistors is caused by the thermal motion of electrons and is described by Nyquist's theorem. Shot noise i shot of

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Multistep methods for SDEs and their application to problems with small noise

Cited by 73 publications

References 23 publications

Simultaneous Step-Size and Path Control for Efficient Transient Noise Analysis

Simultaneous Step-Size and Path Control for Efficient Transient Noise Analysis

Asymptotic Mean-Square Stability of Two-Step Methods for Stochastic Ordinary Differential Equations

Efficient Transient Noise Analysis in Circuit Simulation

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