Stochastic Variational Partitioned Runge-Kutta Integrators for Constrained Systems

Stochastic variational integrators for constrained, stochastic mechanical systems are developed in this paper. The main results of the paper are twofold: an equivalence is established between a stochastic Hamilton-Pontryagin (HP) principle in generalized coordinates and constrained coordinates via Lagrange multipliers, and variational partitioned Runge-Kutta (VPRK) integrators are extended to this class of systems. Among these integrators are first and second-order strongly convergent RATTLE-type integrators. We prove order of accuracy of the methods provided. The paper also reviews the deterministic treatment of VPRK integrators from the HP viewpoint.Comment: 26 pages, 2 figure

New Langevin and Gradient Thermostats for Rigid Body Dynamics

We introduce two new thermostats, one of Langevin type and one of gradient (Brownian) type, for rigid body dynamics. We formulate rotation using the quaternion representation of angular coordinates; both thermostats preserve the unit length of quaternions. The Langevin thermostat also ensures that the conjugate angular momenta stay within the tangent space of the quaternion coordinates, as required by the Hamiltonian dynamics of rigid bodies. We have constructed three geometric numerical integrators for the Langevin thermostat and one for the gradient thermostat. The numerical integrators reflect key properties of the thermostats themselves. Namely, they all preserve the unit length of quaternions, automatically, without the need of a projection onto the unit sphere. The Langevin integrators also ensure that the angular momenta remain within the tangent space of the quaternion coordinates. The Langevin integrators are quasi-symplectic and of weak order two. The numerical method for the gradient thermostat is of weak order one. Its construction exploits ideas of Lie-group type integrators for differential equations on manifolds. We numerically compare the discretization errors of the Langevin integrators, as well as the efficiency of the gradient integrator compared to the Langevin ones when used in the simulation of rigid TIP4P water model with smoothly truncated electrostatic interactions. We observe that the gradient integrator is computationally less efficient than the Langevin integrators. We also compare the relative accuracy of the Langevin integrators in evaluating various static quantities and give recommendations as to the choice of an appropriate integrator.Comment: 16 pages, 4 figure

A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods

In this article we compare the mean-square stability properties of the Theta-Maruyama and Theta-Milstein method that are used to solve stochastic differential equations. For the linear stability analysis, we propose an extension of the standard geometric Brownian motion as a test equation and consider a scalar linear test equation with several multiplicative noise terms. This test equation allows to begin investigating the influence of multi-dimensional noise on the stability behaviour of the methods while the analysis is still tractable. Our findings include: (i) the stability condition for the Theta-Milstein method and thus, for some choices of Theta, the conditions on the step-size, are much more restrictive than those for the Theta-Maruyama method; (ii) the precise stability region of the Theta-Milstein method explicitly depends on the noise terms. Further, we investigate the effect of introducing partially implicitness in the diffusion approximation terms of Milstein-type methods, thus obtaining the possibility to control the stability properties of these methods with a further method parameter Sigma. Numerical examples illustrate the results and provide a comparison of the stability behaviour of the different methods.Comment: 19 pages, 10 figure

Maximum-likelihood estimation for diffusion processes via closed-form density expansions

This paper proposes a widely applicable method of approximate maximum-likelihood estimation for multivariate diffusion process from discretely sampled data. A closed-form asymptotic expansion for transition density is proposed and accompanied by an algorithm containing only basic and explicit calculations for delivering any arbitrary order of the expansion. The likelihood function is thus approximated explicitly and employed in statistical estimation. The performance of our method is demonstrated by Monte Carlo simulations from implementing several examples, which represent a wide range of commonly used diffusion models. The convergence related to the expansion and the estimation method are theoretically justified using the theory of Watanabe [Ann. Probab. 15 (1987) 1-39] and Yoshida [J. Japan Statist. Soc. 22 (1992) 139-159] on analysis of the generalized random variables under some standard sufficient conditions.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1118 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Variational Integrators and Generating Functions for Stochastic Hamiltonian Systems

In this work, the stochastic version of the variational principle is established, important for stochastic symplectic integration, and for structure-preserving algorithms of stochastic dynamical systems. Based on it, the stochastic variational integrators in formulation of stochastic Lagrangian functions are proposed, and some applications to symplectic integrations are given. Three types of generating functions in the cases of one and two noises are discussed for constructing new schemes

Split-step forward methods for stochastic differential equations

AbstractIn this paper we discuss split-step forward methods for solving Itô stochastic differential equations (SDEs). Eight fully explicit methods, the drifting split-step Euler (DRSSE) method, the diffused split-step Euler (DISSE) method and the three-stage Milstein (TSM 1a–TSM 1f) methods, are constructed based on Euler–Maruyama method and Milstein method, respectively, in this paper. Their order of strong convergence is proved. The analysis of stability shows that the mean-square stability properties of the methods derived in this paper are improved on the original methods. The numerical results show the effectiveness of these methods in the pathwise approximation of Itô SDEs

Stochastic modelling and numerical simulation of fatigue damage

In continuum damage mechanics, fatigue is a phenomenon associated with a continuous material stiffness reduction. Numerically, it can be simulated as an accumulation of damage process. Since the resistance of concrete material reduces drastically after the initiation of macroscopic cracks, fatigue life can be approximated using damage models as the number of cycles by which the material continuity vanishes. The fatigue scatter is an interpretation of material heterogeneity and uncertain external influences. It can be reproduced by treating the damage evolution as a stochastic process. Inspired by the application of the stochastic process in molecular physics, the deterministic damage evolution rate of the Lemaitre model is modified as a stochastic differential equation to characterise the random damage increment. The implicit Euler scheme associated with Monte-Carlo simulation is demonstrated as a practical approach to solve the stochastic integration problem. The stochastic damage model is designed carefully to obey the thermodynamic principles and the deterministic damage law. Particular efforts are addressed to determine suitable random distributions, avoiding negative random damage increments in individual realisations, to have a statistically unbiased mean. To adequately approximate the high-cycle fatigue damage with random noise, the "jumping-cycle" algorithms with different extrapolation strategies are investigated. This damage model is further implemented in the simulation of four-point flexural fatigue of concrete beam, solved by the finite element method. The numerically reproduced fatigue data closely fit the published experimental results and the empirical solution, both in the mean and standard deviation. Compared to the Gaussian white noise, the Weibull random variable has broad applicability to simulate random fatigue damage and other physical processes.Um die Streuung der Messdaten in der Materialermüdung zu beschreiben, wird basierend auf Zufallsprozessen ein phenomenologische Modellierung vorgestellt. Erprobt wird die Modellierung an einem Betonbalken mit ebener Finite Element Diskretisierung, wobei die stochastischen Ermüdungsgleichungen mit der Monte Carlo Methode gelöst werden. Die simulierten Ermüdungsprozesse unter Biegebeanspruchung des quasi-spröden Materialswerden mit experimentellen Daten und etablierten empirischen Gleichungen vergleichen. Um hochzyklische Beanspruchungen zu behandeln, wird ein „jumping-cycle“ Algorithmus angewendet, mit dem die Rechenzeiten stark reduziert werden. Dieser Modellansatz ermöglicht die Simulation von Ermüdungsprozessen mit probabilistischen Information in einem sehr langen Zeitintervall. In derKontinuums-Modellierung geht der Prozess der Materialermüdung mit einer Degeneration der materiellen Integrität einher, die sich z.B. in der Abnahme des elastischen Moduls niederschlägt. Numerisch wird dies als ein kumulativer Schädigungsprozess modelliert. Weil der Materialwiderstand von Beton nach der Entstehung makroskopischer Risse drastisch abnimmt, kann die Ermüdungslebensdauer unter zyklischer Beanspruchung durch ein Schädigungsmodell praktisch sehr gut abgeschätzt werden, sobald das Auftreten makroskopischer Risse prognostiziert wird. Die Streuung in experimentell ermittelten Ermüdungskurven kann durch die mikro-Heterogenität der Materialien und Unsicherheiten in weiteren externen Faktoren verstanden werden, mittels einer Modellierung der Schädigungsentwicklung als stochastische Prozessgleichungen kann diese gut reproduziert werden. In Anlehnung an die Beschreibung stochastischer Prozesse in der theoretischen Physik werden die volutionsgleichungen für die Schädigungsentwicklung des Lemaitre-Modells als stochastische Differentialgleichungen dargestellt. Diese werden mittels impliziter Euler-Verfahren und Monte-Carlo Methoden effizient gelöst. Um die thermodynamische Konsistenz sicherzustellen, insbesondere negative Inkremente der Schädigungsentwicklung zu vermeiden, und unverzerrte statistische Mittel-werte zu erhalten, werden klassische Gaußsche Prozesse durch Weibull-Verteilungen substituiert. Für hochzyklische Belastungen werden „jumping-cycle“ Algorithmen hinsichtlich der Extrapolations-strategien systematisch untersucht. Am Beispiel eines Betonträgers unter Biegebeanspruchung wird das Ermüdungsverhalten simuliert und mit experimentellen Ergebnissen aus der Literatur und empirischen Formeln vergleichen. Der vorgeschlagene Modellierungsansatz zeigt eine gute Übereinstimmung der Mittelwerte und Standardabweichungen mit den publizierten Erkenntnissen. Wenngleich die hier verwendeteWeibull-Statistik im strengen mathematischen Sinne nicht konsistent sein sollte, hat sich diese jedoch als physikalisch konsistent erwiesen, um streuende Ermüdungsschädigung effizient zu beschreiben

Effective simulation techniques for biological systems

In this paper we give an overview of some very recent work on the stochastic simulation of systems involving chemical reactions. In many biological systems (such as genetic regulation and cellular dynamics) there is a mix between small numbers of key regulatory proteins, and medium and large numbers of molecules. In addition, it is important to be able to follow the trajectories of individual molecules by taking proper account of the randomness inherent in such a system. We describe different types of simulation techniques (including the stochastic simulation algorithm, Poisson Runge-Kutta methods and the Balanced Euler method) for treating simulations in the three different reaction regimes: slow, medium and fast. We then review some recent techniques on the treatment of coupled slow and fast reactions for stochastic chemical kinetics and discuss how novel computing implementations can enhance the performance of these simulations

A brief analysis of certain numerical methods used to solve stochastic differential equations

Stochastic differential equations (SDE’s) are used to describe systems which are influenced by randomness. Here, randomness is modelled as some external source interacting with the system, thus ensuring that the stochastic differential equation provides a more realistic mathematical model of the system under investigation than deterministic differential equations. The behaviour of the physical system can often be described by probability and thus understanding the theory of SDE’s requires the familiarity of advanced probability theory and stochastic processes. SDE’s have found applications in chemistry, physical and engineering sciences, microelectronics and economics. But recently, there has been an increase in the use of SDE’s in other areas like social sciences, computational biology and finance. In modern financial practice, asset prices are modelled by means of stochastic processes. Thus, continuous-time stochastic calculus plays a central role in financial modelling. The theory and application of interest rate modelling is one of the most important areas of modern finance. For example, SDE’s are used to price bonds and to explain the term structure of interest rates. Commonly used models include the Cox-Ingersoll-Ross model; the Hull-White model; and Heath-Jarrow-Morton model. Since there has been an expansion in the range and volume of interest rate related products being traded in the international financial markets in the past decade, it has become important for investment banks, other financial institutions, government and corporate treasury offices to require ever more accurate, objective and scientific forms for the pricing, hedging and general risk management of the resulting positions. Similar to ordinary differential equations, many SDE’s that appear in practical applications cannot be solved explicitly and therefore require the use of numerical methods. For example, to price an American put option, one requires the numerical solution of a free-boundary partial differential equation. There are various approaches to solving SDE’s numerically. Monte Carlo methods could be used whereby the physical system is simulated directly using a sequence of random numbers. Another method involves the discretisation of both the time and space variables. However, the most efficient and widely applicable approach to solving SDE’s involves the discretisation of the time variable only and thus generating approximate values of the sample paths at the discretisation times. This paper highlights some of the various numerical methods that can be used to solve stochastic differential equations. These numerical methods are based on the simulation of sample paths of time discrete approximations. It also highlights how these methods can be derived from the Taylor expansion of the SDE, thus providing opportunities to derive more advanced numerical schemes.Dissertation (MSc (Mathematics of Finance))--University of Pretoria, 2007.Mathematics and Applied MathematicsMScunrestricte