Stochastic Runge-Kutta Research Articles

Stability and convergence analysis of stochastic Runge–Kutta and balanced stochastic Runge–Kutta methods for solving stochastic differential equations

Stability and convergence analysis of stochastic Runge–Kutta and balanced stochastic Runge–Kutta methods for solving stochastic differential equations

An exponential stochastic Runge–Kutta type method of order up to 1.5 for SPDEs of Nemytskii-type

Abstract For the approximation of solutions for stochastic partial differential equations, numerical methods that obtain a high order of convergence and at the same time involve reasonable computational cost are of particular interest. We therefore propose a new numerical method of exponential stochastic Runge–Kutta type that allows for convergence with a temporal order of up to $\frac{3}/{2}$ and that can be combined with several spatial discretizations. The developed family of derivative-free schemes is tailored to stochastic partial differential equations of Nemytskii-type, i.e., with pointwise multiplicative noise operators. We prove the strong convergence of these schemes in the root mean-square sense and present some numerical examples that reveal the theoretical results.

A study on generalized balanced split drift stochastic Runge- Kutta methods for stochastic differential equations

To reduce computational complexity, the balanced numerical approximations of the general split drift stochastic Runge-Kutta methods are analyzed. The primary reasons for considering the numerical approximations of these balanced split stochastic Runge-Kutta methods are their improved stability characteristics and lower mean square error compared to other methods. By balancing the drift and diffusion components, the splitting techniques outperform the mean square error over longer time increments. For Ito multi-dimensional stochastic differential equations, we propose a novel family of balanced universal split stochastic Runge-Kutta procedures. The Kronecker product concept is utilized to derive the mean-square stability conditions. We conduct numerical tests to evaluate these methods against an existing weak order 2 split drift method. Ultimately, a specific numerical example validates the theoretical outcomes of the balanced general split stochastic Runge-Kutta procedures.

Gaussian and Lévy noises excited delayed tumor growth model: first-passage behavior and stochastic resonance

In this paper, we analyze the dynamical behavior of a delayed tumor growth model under the joint effect of Gaussian white noise and Lévy noise by studying the mean first passage time (MFPT) and stochastic resonance (SR). Firstly, the tumor growth model under the joint effect of Gaussian white noise, Lévy noise and time delay is introduced. Then, the Lévy noise sequence is simulated by Janicki-Weron algorithm, and the MFPT and signal-to-noise ratio(SNR) of the system are simulated by using fourth-order stochastic Runge–Kutta algorithm. The effects of noise parameters, time delay and periodic signal parameters on MFPT, SR are discussed in detail, respectively. In addition, we find the phenomenon of noise enhanced stability. The results of the study can help to select the optimal regulatory parameters in the tumor growth model and promote the treatment of tumors.

Stochastic Runge-Kutta Methods for Preserving Maximum Bound Principle of Semilinear Parabolic Equations. Part I: Gaussian Quadrature Rule

Stochastic Runge-Kutta Methods for Preserving Maximum Bound Principle of Semilinear Parabolic Equations. Part I: Gaussian Quadrature Rule

Stochastic Runge–Kutta for numerical treatment of dengue epidemic model with Brownian uncertainty

Stochastic Runge–Kutta for numerical treatment of dengue epidemic model with Brownian uncertainty

Exotic B-Series and S-Series: Algebraic Structures and Order Conditions for Invariant Measure Sampling

AbstractB-Series and generalizations are a powerful tool for the analysis of numerical integrators. An extension named exotic aromatic B-Series was introduced to study the order conditions for sampling the invariant measure of ergodic SDEs. Introducing a new symmetry normalization coefficient, we analyze the algebraic structures related to exotic B-Series and S-Series. Precisely, we prove the relationship between the Grossman–Larson algebras over exotic and grafted forests and the corresponding duals to the Connes–Kreimer coalgebras and use it to study the natural composition laws on exotic S-Series. Applying this algebraic framework to the derivation of order conditions for a class of stochastic Runge–Kutta methods, we present a multiplicative property that ensures some order conditions to be satisfied automatically.

Role of asymmetry and external noise in the development and synchronization of oscillations in the analog Hopfield neural networks with time delay.

The analog Hopfield neural network with time delay and random connections has been studied for its similarities in activity to human electroencephalogram and its usefulness in other areas of the applied sciences such as speech recognition, image analysis, and electrocardiogram modeling. Our goal here is to understand the mechanisms that affect the rhythmic activity in the neural network and how the addition of a Gaussian noise contributes to the network behavior. The neural network studied is composed of ten identical neurons. We investigated the excitatory and inhibitory networks with symmetric (square matrix) and asymmetric (triangular matrix) connections. The differential equations that model the network are solved numerically using the stochastic second-order Runge-Kutta method. Without noise, the neural networks with symmetric and asymmetric matrices possessed different synchronization properties: fully connected networks were synchronized both in time and in amplitude, while asymmetric networks were synchronized in time only. Saturation outputs of the excitatory neural networks do not depend on the time delay, whereas saturation oscillation amplitudes of inhibitory networks increase with the time delay until the steady state. The addition of the Gaussian noise is shown to significantly amplify small-amplitude oscillations, dramatically accelerates the rate of amplitude growth to saturation, and changes synchronization properties of the neural network outputs.

Computational analysis of the coronavirus epidemic model involving nonlinear stochastic differential equations

Stochastic methods significantly solve stochastic differential equations such as stochastic equations with a delay, stochastic fractional and fractal equations, stochastic partial differential equations, and many more. The coronavirus is still a threat to humans and puts people in danger. The model is a symmetric and compatible distribution family. In this case, the present model contains seven sub-populations of humans: susceptible, exposed, infected, quarantined, vaccinated, recovered, and dead. Two deterministic to stochastic formation types are studied, namely, transition probabilities and nonparametric perturbations. The positivity and boundedness of the stochastic model are analyzed. The stochastic Euler, stochastic Runge–Kutta, and Euler–Maruyama methods solve the stochastic system. Unfortunately, many issues originate, such as negativity, boundedness, and violation of dynamical consistency. The nonstandard finite difference method is designed in the sense of stochasticity to restore the dynamic properties of the model. In the end, simulations are carried out in contrast to deterministic and stochastic solutions. Overall, our findings shed light on the underlying mechanisms of COVID-19 dynamics and the influence of environmental factors on the spread of the disease, which can help make informed policy decisions and public health interventions.

Strong stochastic Runge-Kutta–Munthe-Kaas methods for nonlinear Itô SDEs on manifolds

In this paper we present numerical methods for the approximation of nonlinear Itô stochastic differential equations on manifolds. For this purpose, we extend Runge-Kutta–Munthe-Kaas (RKMK) schemes for ordinary differential equations on manifolds to the stochastic case and analyse the strong convergence of these schemes. Since these schemes are based on the application of a stochastic Runge-Kutta (SRK) scheme in a corresponding Lie algebra, we address the question under which circumstances the stochastic RKMK method has the same strong order of convergence as the applied SRK scheme. To illustrate our answer to this question and the effectiveness of our schemes, we show some numerical results of applying these methods to a problem with an autonomous underwater vehicle.

A non-standard computational method for stochastic anthrax epidemic model

This study employing a non-standard computational method for a stochastic anthrax epidemic model can enhance accuracy, evaluate control measures, and identify critical factors. The mathematical modeling of an anthrax disease includes the four-compartment of the population as susceptible animals (s), infected animals (i), carcasses animals (c), and grams spores of animals in the environment (a). The continuous model analysis (equilibria, reproduction number, and local stability of equilibria) is studied rigorously. The stochastic model is based on transition probabilities and parametric perturbation techniques. The fundamental properties of the model with standard computational methods such as Euler Maruyama, stochastic Euler, and stochastic Runge Kutta are studied. Unfortunately, these methods are time-dependent and even valid for short-period analysis of the disease. In comparison, the non-standard computational method, like the non-standard finite difference method nonstandard finite difference in the sense of stochastic, is designed for the given model. The non-standard computational method and its dynamical properties (positivity, boundedness, and dynamical consistency) are studied thoroughly. In the end, numerical results of the non-standard computational method with the existing standard computational methods are provided. These benefits contribute to a more comprehensive understanding of anthrax epidemiology and support effective decision-making in public health interventions.

DEVELOPMENT OF ALGORITHMS AND SOFTWARE FOR MODELING CONTROLLED DYMAMIC SYSTEMS USING SYMBOLIC COMPUTATIONS AND STOCHASTIC METHODS

The development of software for synthesizing and analyzing models of controlled systems taking into account their deterministic and stochastic description is an important direction of research. Results of the development of software for modeling dynamic systems the behavior of which can be described by onestep processes are presented. Models of population dynamics are considered as an example. The software uses a deterministic description of the model at its input to obtain a corresponding stochastic model in symbolic form and also analyze the model in detail (calculate trajectories in the deterministic and stochastic cases, find control functions, and visualize the results). An important aspect of the development is the use of computer algebra for analyzing the model and synthesizing controls. Methods and algorithms based on deterministic and stochastic Runge–Kutta methods, stability and control theory, methods for designing self-consistent stochastic models, numerical optimization algorithms, and artificial intelligence are implemented. The software was developed using high-level programming languages Python and Julia. As the basic tools, high-performance libraries for vector–matrix computations, symbolic computation libraries, libraries for the numerical solution of ordinary differential equations, and libraries of global optimization algorithms are used.

Fractional discrete Temimi–Ansari method with singular and nonsingular operators: applications to electrical circuits

The goal of this article is to present a recently developed numerical approach for solving fractional stochastic differential equations with a singular Caputo kernel and a nonsingular Caputo–Fabrizio and Atangana–Baleanu (ABC) kernel. The proposed method is based on the discrete Temimi–Ansari method, which is combined with three different numerical schemes that are appropriate for the new fractional derivative operators. The proposed technique is used to investigate the effects of Gaussian white-noise and Gaussian colored-noise perturbations on the potential source and resistance in fractional stochastic electrical circuits. The proposed method’s robustness and efficiency were demonstrated by comparing its results to those of the stochastic Runge–Kutta method (SRK). The valuable point in this article is that the resulting numerical scheme is able to combine two powerful methods that can be extended into more complex stochastic models. The comparison of different fractional derivatives using Mathematica 12 software has been obtained and the simulation results demonstrate the merit of the contributed method.

Weak convergence of balanced stochastic Runge–Kutta methods for stochastic differential equations

In this paper, weak convergence of balanced stochastic one-step methods and especially balanced stochastic Runge–Kutta (SRK) methods for Itô multidimensional stochastic differential equations is analyzed. Generalizing a corresponding result obtained by H. Schurz for the standard Euler method, it is shown that under certain conditions, balanced one-step methods preserve the weak convergence properties of their underlying methods. As an application, this allows to prove the weak convergence order of the balanced SRK methods presented in earlier work by A. Rathinasamy, P. Nair and D. Ahmadian.

Conservative Continuous-Stage Stochastic Runge–Kutta Methods for Stochastic Differential Equations

In this paper, we develop a new class of conservative continuous-stage stochastic Runge–Kutta methods for solving stochastic differential equations with a conserved quantity. The order conditions of the continuous-stage stochastic Runge–Kutta methods are given based on the theory of stochastic B-series and multicolored rooted tree. Sufficient conditions for the continuous-stage stochastic Runge–Kutta methods preserving the conserved quantity of stochastic differential equations are derived in terms of the coefficients. Conservative continuous-stage stochastic Runge–Kutta methods of mean square convergence order 1 for general stochastic differential equations, as well as conservative continuous-stage stochastic Runge–Kutta methods of high order for single integrand stochastic differential equations, are constructed. Numerical experiments are performed to verify the conservative property and the accuracy of the proposed methods in the longtime simulation.

Adaptive Step Size Stochastic Runge-Kutta Method of Order 1.5(1.0) for Stochastic Differential Equations (SDEs)

The stiff stochastic differential equations (SDEs) involve the solution with sharp turning points that permit us to use a very small step size to comprehend its behavior. Since the step size must be set up to be as small as possible, the implementation of the fixed step size method will result in high computational cost. Therefore, the application of variable step size method is needed where in the implementation of variable step size methods, the step size used can be considered more flexible. This paper devotes to the development of an embedded stochastic Runge-Kutta (SRK) pair method for SDEs. The proposed method is an adaptive step size SRK method. The method is constructed by embedding a SRK method of 1.0 order into a SRK method of 1.5 order of convergence. The technique of embedding is applicable for adaptive step size implementation, henceforth an estimate error at each step can be obtained. Numerical experiments are performed to demonstrate the efficiency of the method. The results show that the solution for adaptive step size SRK method of order 1.5(1.0) gives the smallest global error compared to the global error for fix step size SRK4, Euler and Milstein methods. Hence, this method is reliable in approximating the solution of SDEs.

Stochastic Analysis for the Dynamics of a Poliovirus Epidemic Model

Most developing countries such as Afghanistan, Pakistan, India, Bangladesh, and many more are still fighting against poliovirus. According to the World Health Organization, approximately eighteen million people have been infected with poliovirus in the last two decades. In Asia, still, some countries are suffering from the virus. The stochastic behavior of the poliovirus through the transition probabilities and non-parametric perturbation with fundamental properties are studied. Some basic properties of the deterministic model are studied, equilibria, local stability around the stead states, and reproduction number. Euler Maruyama, stochastic Euler, and stochastic Runge-Kutta study the behavior of complex stochastic differential equations. The main target of this study is to develop a nonstandard computational method that restores dynamical features like positivity, boundedness, and dynamical consistency. Unfortunately, the existing methods failed to fix the actual behavior of the disease. The comparison of the proposed approach with existing methods is investigated.

Stochastic Bayesian Runge-Kutta method for dengue dynamic mapping

Dengue Hemorrhagic Fever (DHF) is still a threat to humanity that cause death and disability due to changes in environmental and socioeconomic conditions, especially in tropical areas. A critical assessment of the models and methods is necessary. The vital role of stochastic processes of infectious disease research in Bayesian statistical models helps provide an explicit framework for understanding disease transmission dynamics between hosts (humans) and vectors (mosquitoes). This research presents a Bayesian stochastic process for the cross-infection SIR-SI model as a differential equation (susceptibility–infection–recovery for the human population; susceptible–infectious for the vector population) for the dynamic transmission of DHF. Given the difficulties of solving the differential equations precisely, we propose a computational model to approach the solution based on the Runge-Kutta family approach, namely the Euler and the four-order Runge-Kutta methods. The methods used for the discretization process of the SIR-SI model for computational process needs. For comparison purposes, we use a monthly DHF dataset of 10 Kendari, Indonesia, districts from 2019 to 2021. Parameter estimation of the Bayesian SIR-SI model based on the Euler and four-order Runge-Kutta method was updated using Markov Chain Monte Carlo (MCMC) Bayesian. The Euler and four-order Runge-Kutta methods have converged at 10,000 iterations with burn-in 80,000. The numerical simulation results show that the four-order Runge-Kutta approach has the slightest deviance, 106.5. Therefore, this approach is the best one. The relative risk analysis shows that the dynamics of DHF cases fluctuate from January to July every year. However, from January to May, there was a high consistency of DHF cases. Two districts with high case consistency were found, namely Kadia and Wua-Wua. Furthermore, because the spread of DHF cases has a spatial effect, the Kadia and Wua-Wua districts need serious attention to suppress the rate of reach of DHF in Kendari City. Intensive observation and intervention in the Kadia and Wua-Wua districts should be carried out in early January to stop the breeding of mosquito larvae. In other neighbourhoods such as Puwatu, Kambu and Kendari Barat, the DHF cases occur temporally as the impacts of DHF cases in the Kadia and Wua-Wua districts. It may facilitate the statisticians to develop further models to adopt a better understanding to control the DHF dynamically. In brief,

Explicit pseudo-symplectic Runge-Kutta methods for stochastic Hamiltonian systems

We give conditions for stochastic Runge-Kutta methods to near preserve quadratic invariants, and we discuss the associated simplified order conditions. For stochastic Hamiltonian systems we propose a systematic approach to construct explicit stochastic Runge-Kutta pseudo-symplectic schemes. Our approach is based on colored trees and B-series. We construct some pseudo-symplectic stochastic Runge-Kutta methods with strong convergence order, and we illustrate numerically the long term performance of the proposed schemes.

Stochastic Capital–Labor Lévy Jump Model with the Precariat Labor Force

In this work, we study a capital–labor model by considering the interaction between the new proposed and the confirmed free jobs, the precariat labor force, and the mature labor force by introducing Brownian motion and Lévy noise. Moreover, we illustrate the well-posedness of the solution. In addition, we establish the conditions of the extinction of both the free jobs and labor force; subsequently, we prove the persistence of only the free jobs, and we also show the conditions of the persistence of both the free jobs and labor force. Finally, we validate our theoretical finding by numerical simulation by building a new stochastic Runge–Kutta method.