scholarly journals MATHEMATICAL MODELING OF THE PROCESS OF DRAWING AN OPTICAL FIBER USING THE LANGEVIN EQUATION

2019 ◽  
Vol 9 (2) ◽  
pp. 64-67
Author(s):  
Aliya Tergeussizova
Keyword(s):  
Mathematical Modeling ◽  
Differential Equations ◽  
Optical Fiber ◽  
Automatic Control ◽  
Stochastic Differential Equations ◽  
Differential System ◽  
Langevin Equation ◽  
Pulse Frequency ◽  
Dynamic Compensation

In order to design stochastic pulse frequency systems for automatic control of objects with delay, this article shows how we obtained their models in the form of stochastic differential equations. The method of dynamic compensation of objects with delay is considered. A stochastic differential system in the Langevin form is obtained.

scholarly journals Order Conditions for Sampling the Invariant Measure of Ergodic Stochastic Differential Equations on Manifolds

2021 ◽  
Author(s):  
Adrien Laurent ◽  
Gilles Vilmart
Keyword(s):  
Invariant Measure ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Linear Group ◽  
Langevin Equation ◽  
Numerical Experiments ◽  
Special Linear Group ◽  
Order Conditions ◽  
High Order ◽  
Differential Equations On Manifolds

AbstractWe derive a new methodology for the construction of high-order integrators for sampling the invariant measure of ergodic stochastic differential equations with dynamics constrained on a manifold. We obtain the order conditions for sampling the invariant measure for a class of Runge–Kutta methods applied to the constrained overdamped Langevin equation. The analysis is valid for arbitrarily high order and relies on an extension of the exotic aromatic Butcher-series formalism. To illustrate the methodology, a method of order two is introduced, and numerical experiments on the sphere, the torus and the special linear group confirm the theoretical findings.

scholarly journals Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation

2015 ◽  
Vol 471 (2176) ◽  
pp. 20140679 ◽  
Author(s):  
Eike H. Müller ◽  
Rob Scheichl ◽  
Tony Shardlow
Keyword(s):  
Monte Carlo ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Langevin Equation ◽  
Operator Splitting ◽  
Random Variables ◽  
Weak Approximation ◽  
Multilevel Monte Carlo ◽  
Discrete Random Variables ◽  
Modified Equations

This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis as an alternative to strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy.

Reconstruction of Complex Dynamical Systems Using Stochastic Differential Equations

2020 ◽  
Author(s):  
Forough Hassanibesheli ◽  
Niklas Boers ◽  
Jürgen Kurths
Keyword(s):  
Time Series ◽  
Differential Equations ◽  
Complex Systems ◽  
Stochastic Differential Equations ◽  
Langevin Equation ◽  
Degrees Of Freedom ◽  
Evolution Equations ◽  
Ice Core ◽  
Dynamical Behavior ◽  
Generalized Langevin Equation

<p>A complex system is a system composed of highly interconnected components in which the collective property of an underlying system cannot be described by dynamical behavior of the individual parts. Typically, complex systems are governed by nonlinear interactions and intricate fluctuations, thus to retrieve dynamics of a system, it is required to characterize and asses interactions between deterministic tendencies and random fluctuations. </p><p>For systems with large numbers of degrees of freedom, interacting across various time scales, deriving time-evolution equations from data is computationally expensive. A possible way to circumvent this problem is to isolate a small number of relatively slow degrees of freedom that may suffice to characterize the underlying dynamics and solve the governing motion equation for the reduced-dimension system in the framework of stochastic differential equations(SDEs).  For some specific example settings, we have studied the performance of three stochastic dimension-reduction methods (Langevin equation(LE), generalized Langevin Equation(GLE) and Empirical Model Reduction(EMR)) to model various synthetic and real-world time series. In this study corresponding numerical simulations of all models have been examined by probability distribution function(PDF) and Autocorrelation function(ACF) of the average simulated time series as statistical benchmarks for assessing the differnt models' performance. </p><p>First we reconstruct the Niño-3 monthly sea surface temperature (SST) indices averages across (5°N–5°S, 150°–90°W) from 1891 to 2015 using the three aforementioned stochastic models. We demonstrate that all these considered models can reproduce the same skewed and heavy-tailed distributions of Niño-3 SST, comparing ACFs, GLE exhibits a tendency towards achieving a higher accuracy than LE and EMR. A particular challenge for deriving the underlying dynamics of complex systems from data is given by situations of abrupt transitions between alternative states. We show how the Kramers-Moyal approach to derive drift and diffusion terms for LEs can help in such situations. A prominent example of such 'Tipping Events' is given by the Dansgaard-Oeschger events during previous glacial intervals. We attempt to obtain the statistical properties of high-resolution, 20yr average, δ<sup>18</sup>O and Ca<sup>+</sup><sup>2</sup> collected from the same ice core from the NGRIP on the GICC05 time scale. Through extensive analyses of various systems, our results signify that stochastic differential equation models considering memory effects are comparatively better approaches for understanding  complex systems.</p><p> </p>

Generalized Langevin Equation: An Introductory Review for Biophysicists

2019 ◽  
Vol 14 (04) ◽  
pp. 171-196
Author(s):  
Shin-Ho Chung ◽  
Michael Roper
Keyword(s):  
Molecular Dynamics ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Langevin Equation ◽  
First Principles ◽  
Generalized Langevin Equation ◽  
Biophysical Modeling ◽  
Open Problems ◽  
The Relationship ◽  
Introductory Review

An introductory, pedagogical review of the generalized Langevin equation (GLE) within the classical regime is presented. It is intended to be accessible to biophysicists with an interest in molecular dynamics (MD). Section 1 presents why the equation may be of interest within biophysical modeling. A detailed elementary first principles derivation of the (multidimensional) Kac–Zwanzig model is presented. The literature is reviewed with a focus on biophysical applications and representation by Markovian stochastic differential equations. The relationship with the Mori–Zwanzig formalism is discussed. The framework of model reduction and approximation is emphasized. Some open problems are identified.

scholarly journals Teaching Mathematical Modeling of Cellular Systems with the VCell MathModel

2021 ◽  
Author(s):  
Jasia King ◽  
Kerbaï Saïd Eroumé ◽  
Roman Truckenmüller ◽  
Stefan Giselbrecht ◽  
Ann Cowan ◽  
...  
Keyword(s):  
Mathematical Modeling ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Cellular Systems ◽  
Teaching Tool ◽  
Interactive Teaching ◽  
Advantages And Disadvantages ◽  
Hands On ◽  
Powerful Approach ◽  
Session Plan

ABSTRACT Mathematical biology has emerged as a powerful approach to describe and understand biological systems. Here, we introduce an interactive teaching tool with a practical hands-on skill session plan to introduce students to the various components of a mathematical model with 4 different mathematical approaches (i.e., ordinary differential equations, partial differential equations, stochastic differential equations, and spatial stochastic differential equations) and their advantages and disadvantages. As such, we provide a didactic summary for instructors and students interested in using VCell MathModels for mathematical modeling; this work is also valuable for mathematics-savvy users who would like to exploit fully the capabilities of the VCell software.

scholarly journals Existence Theorems for Solutions to Set-Valued Stochastic Differential Equations and Applications

2012 ◽  
Vol 1 ◽  
pp. 1-15
Author(s):  
Vu Ho ◽  
Van Hoa Ngo
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Wiener Process ◽  
Langevin Equation ◽  
Stock Prices ◽  
Hausdorff Space ◽  
Existence Theorems ◽  
Uniqueness Of Solution ◽  
Hukuhara Derivative ◽  
Interval Valued

In this paper, a class of new stochastic differential equations on semilinear Hausdorff space under Hukuhara derivative, called set-valued stochastic differential equations (SSDEs) driven by a Wiener process. Moreover, some corresponding properties of SSDEs are discussed such as existence, uniqueness of solution. Finaly, we give some applications to models of interval-valued stochastic differential equations such as stock prices model and the Langevin equation.

Stochastic differential equations: Langevin, Fokker–Planck (FP) equations

2021 ◽  
pp. 831-856
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Langevin Equation ◽  
Diffusion Processes ◽  
Random Noise ◽  
Path Integrals ◽  
Brst Symmetry ◽  
Random Motion ◽  
Definition Of ◽  
Fokker Planck

This chapter is devoted to the study of Langevin equations, first order in time differential equations, which depend on a random noise, and which belong to a class of stochastic differential equations that describe diffusion processes, or random motion. From a Langevin equation, a Fokker–Planck (FP) equation for the probability distribution of the solutions, at given time, of the Langevin equation can be derived. It is shown that observables averaged over the noise can also be calculated from path integrals, whose integrands define automatically positive measures. The path integrals involve dynamic actions that have automatically a Becchi–Rouet–Stora–Tyutin (BRST) symmetry and, when the driving force derives from a potential, exhibit the simplest form of supersymmetry. In some cases, like Brownian motion on Riemannian manifolds, difficulties appear in the precise definition of stochastic equations, quite similar to the quantization problem encountered in quantum mechanics (QM). Time discretization provides one possible solution to the problem.

scholarly journals Error estimates of the backward Euler–Maruyama method for multi-valued stochastic differential equations

2021 ◽  
Author(s):  
Monika Eisenmann ◽  
Mihály Kovács ◽  
Raphael Kruse ◽  
Stig Larsson
Keyword(s):  
Differential Equations ◽  
Error Analysis ◽  
Stochastic Differential Equations ◽  
Root Mean Square ◽  
Error Estimates ◽  
Langevin Equation ◽  
Gradient Flow ◽  
Mean Square ◽  
Backward Euler ◽  
Appl Math

AbstractIn this paper we derive error estimates of the backward Euler–Maruyama method applied to multi-valued stochastic differential equations. An important example of such an equation is a stochastic gradient flow whose associated potential is not continuously differentiable but assumed to be convex. We show that the backward Euler–Maruyama method is well-defined and convergent of order at least 1/4 with respect to the root-mean-square norm. Our error analysis relies on techniques for deterministic problems developed in Nochetto et al. (Commun Pure Appl Math 53(5):525–589, 2000). We verify that our setting applies to an overdamped Langevin equation with a discontinuous gradient and to a spatially semi-discrete approximation of the stochastic p-Laplace equation.

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