Numerical stochastic differential equation

The model equation for particle position, (7.27), is a stochastic differential equation (SDE). The numerical solution of SDEs is discussed in detail by Kloeden and Platen (1992).28 Using a fixed time step At, the most widely used numerical scheme for advancing the particle position is the Euler approximation ... 

Kloeden, P. E. and E. Platen (1992). Numerical Solution of Stochastic Differential Equations. Berlin Springer-Verlag. 

In this section, we consider the description of Brownian motion by Markov diffusion processes that are the solutions of corresponding stochastic differential equations (SDEs). This section contains self-contained discussions of each of several possible interpretations of a system of nonlinear SDEs, and the relationships between different interpretations. Because most of the subtleties of this subject are generic to models with coordinate-dependent diffusivities, with or without constraints, this analysis may be more broadly useful as a review of the use of nonlinear SDEs to describe Brownian motion. Because each of the various possible interpretations of an SDE may be defined as the limit of a discrete jump process, this subject also provides a useful starting point for the discussion of numerical simulation algorithms, which are considered in the following section. 

Polystochastic models are used to characterize processes with numerous elementary states. The examples mentioned in the previous section have already shown that, in the establishment of a stochastic model, the strategy starts with identifying the random chains (Markov chains) or the systems with complete connections which provide the necessary basis for the process to evolve. The mathematical description can be made in different forms such as (i) a probability balance, (ii) by modelling the random evolution, (iii) by using models based on the stochastic differential equations, (iv) by deterministic models of the process where the parameters also come from a stochastic base because the random chains are present in the process evolution. 

Chapter 4 is devoted to the description of stochastic mathematical modelling and the methods used to solve these models such as analytical, asymptotic or numerical methods. The evolution of processes is then analyzed by using different concepts, theories and methods. The concept of Markov chains or of complete connected chains, probability balance, the similarity between the Fokker-Plank-Kolmogorov equation and the property transport equation, and the stochastic differential equation systems are presented as the basic elements of stochastic process modelling. Mathematical models of the application of continuous and discrete polystochastic processes to chemical engineering processes are discussed. They include liquid and gas flow in a column with a mobile packed bed, mechanical stirring of a liquid in a tank, solid motion in a liquid fluidized bed, species movement and transfer in a porous media. Deep bed filtration and heat exchanger dynamics are also analyzed. 

In a previous work [56] we deduced an alternative expression for the error (11) by performing extensive numerical simulations of the stochastic differential equation (19). The metadynamics parameters, w/tq, Sa, and the system-dependent parameters, f3, D and S were systematically varied, and for each choice of the parameters the error (11) was computed by repeating several metadynamics reconstructions. Fitting the results, we obtained that the data were reproduced within an accuracy of 20% by... 

It is important to emphasize, however, that our model is different from the Langevin equation, which is a stochastic differential equation. Our model has no noise in the limit of small time steps in which the numerical errors approach zero. The noise we introduce is numerical. Once we filter the rapid oscillations, it is impossible for us to recover the tme trajectory using only the low-frequency modes. The noise in the SDE approach is introduced when we approximate a differential equation by a finite difference formula and filter out high-frequency motions. 

Giona et al. (1995) studied diffusion in the presence of a constant convective field in percolation clusters with stochastic differential equations and a coupled exit-time equation. On the basis of numerical studies on percolation clusters near the percolation threshold, they found that the volume-averaged exit time as a function ofPn did not follow the normal relationship (in which it is proportional to 1 /Pn) but instead increased monotonically with Pn. Their approach needs generalization to more realistic convective fields. They also present exit-time analyses for transport on diffusion limited aggregates and in deterministic fractals... 

At its most basic level, molecular dynamics is about mapping out complicated point sets using trajectories of a system of ordinary differential equations (or, in Chaps. 6-8, a stochastic-differential equation system). The sets are typically defined as the collection of probable states for a certain system. In the case of Hamiltonian dynamics, they are directly associated to a region of the energy landscape. The trajectories are the means by which we efficiently explore the energy surface. In this chapter we address the design of numerical methods to calculate trajectories. 

We briefly present some standard material on the development of numerical schemes for SDEs. The interested reader is pointed to the standard textbooks [200,270] for a more thorough discussion of foundational concepts. Let us begin with a discussion of the first order stochastic differential equation... 

The numerical analysis of stochastic differential equations is traditionally based on the concepts of weak and strong accuracy. Let a numerical method be given for solving an SDE in the form of a discrete stochastic process X +i = 0(X , h) for n = 0, l,...,v — 1, where vh = x is, fixed. We denote the stochastic solution of the SDE by X(t). In the case of strong accuracy, our measure of the global error is the quantity [200]... 

In the previous chapter we described numerical methods for solving the equations of Langevin dynamics, a system of stochastic differential equations that sample the canonical ensemble. The methods allow us to compute averages of the form... 

Burrage, K., Lythe, G. Accurate stationary densities with partitioned numerical methods for stochastic differential equations. SIAM J. Numer. Anal. 47, 1601-1618 (2009). doi 10.1137/ 060677148... 

Faou, E., LeUevre, T Conservative stochastic differential equations Mathematical and numerical analysis. Math. Comput. 78(268), 2047-2074 (2009). doi 10.1090/S0025-5718-09-02220-0... 

Talay, D., Tubaro, L. Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8, 483-509 (1990). doi 10.1080/... 

The positions are obtained by numerically solving differential equations. Hence, these positions are connected in time. The positions reveal real dynamics of individual molecules. In other simulation methods, the molecular positions are not temporarily related. In other simulation methods, such as Monte Carlo simulations, the positions are generated stochastically such that a molecular configuration depends only on the previous configuration [7-10],... 

Springer-Verlag, 1999. Numerical Solution of Stochastic Differential Equations. 

As far as the present work is concerned, the relevance of numerical stochastic methods for polymer dynamics in micro/macro calculations resides in their ability to yield (within error bars) exact numerical solutions to dynamic models which are insoluble in the framework of polymer kinetic theory. In addition, and mainly as a consequence of the correspondence between Fokker Planck and stochastic differential equations, complex polymer dynamics can be mapped onto extremely efficient computational schemes. Another reason for the efficiency of stochastic dynamic models for polymer melts stems from the reduction of a many-chain problem to a single-chain or two-chain representation, i.e., to linear computational complexity in the number of particles. This circumstance permits the treatment of global ensembles consisting of several tens of millions of particles on current hardware, corresponding to local ensemble sizes of O(IO ) particles per element. 

Haken, H., Wunderlin, A. (1982) Slaving principle for stochastic differential equations with additive and multiplicative noise and for discrete noisy maps. Z. Phys. 47, 179 Hastings, S. P. (1976) Periodic plane waves for the Oregonator. Stud. Appl. Math. 55, 293 Herschkowitz-Kaufman, M. (1975) Bifurcation analysis of nonlinear reaction-diffusion equations II. Steady state solutions and comparison with numerical simulations. Bull. Math. Biol. 37, 589 Howard, L. N., Kopell, N. (1977) Slowly varying waves and shock structures in reaction-diffusion equations. Stud. Appl. Math. 56, 95... 

In order to numerically solve the stochastic differential equation, the constraint that p is constant is made, which is satisfactory so long as the time steps used in the simulation remain sufficiently small during interval t and t + dt. The solutions to Eq. (2.58) has been done by Kolmogorov [13] using different boundary conditions. Due to the extensive use of these Greens functions in the simulation packages, they have been reproduced below using the four most common boundary conditions." ... 

Jeffreys, H., 1961, Theory of Probability, third edition. Oxford Oxford University Press Kennedy, J. and Eberhart, R., 1995, Proa IEEE Inti. Conf. on Neural Networks, Perth, Australia. Piscataway Institute of Electrical and Electronics Engineers Kloeden, P. E. and Platen, E., 2000, Numerical Solution of Stochastic Differential Equations. Berlin Springer... 

The CLE is a multivariate Ito stochastic differential equation with multiple, multiplicative noise. We define the CLE again and present methods to solve it in Chapter 18, where we discuss numerical simulations of stochastic reaction kinetics. 

It is beyond the scope of this book to present methods for the numerical integration of stochastic differential equations. The interested reader is referred to the excellent book by Kloeden and Platen (see Further reading). 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

Another difference is related to the mathematical formulation. Equation (1) is deterministic and does not include explicit stochasticity. In contrast, the equations of motion for a Brownian particle include noise. Nevertheless, similar algorithms are adopted to solve the two differential equations as outlined below. The most common approach is to numerically integrate the above differential equations using small time steps and preset initial values. 

---