Stochastic Diffusion Process Research Articles

A hybrid yang transform adomian decomposition method for solving time-fractional nonlinear partial differential equation

Nonlinear time-fractional partial differential equations (NTFPDEs) play a great role in the mathematical modeling of real-world phenomena like traffic models, the design of earthquakes, fractional stochastic systems, diffusion processes, and control processing. Solving such problems is reasonably challenging, and the nonlinear part and fractional operator make them more problematic. Thus, developing suitable numerical methods is an active area of research. In this paper, we develop a new numerical method called Yang transform Adomian decomposition method (YTADM) by mixing the Yang transform and the Adomian decomposition method for solving NTFPDEs. The derivative of the problem is considered in sense of Caputo fractional order. The stability and convergence of the developed method are discussed in the Banach space sense. The effectiveness, validity, and practicability of the method are demonstrated by solving four examples of NTFPEs. The findings suggest that the proposed method gives a better solution than other compared numerical methods. Additionally, the proposed scheme achieves an accurate solution with a few numbers of iteration, and thus the method is suitable for handling a wide class of NTFPDEs arising in the application of nonlinear phenomena.

Generalized autocorrelation function in the family of deterministic and stochastic anomalous diffusion processes

We investigate the observables of the one-dimensional model for anomalous transport in semiconductor devices where diffusion arises from scattering at dislocations at fixed random positions, known as the Lévy-Lorentz gas. To gain insight into the microscopic properties of such a stochastically complex system, deterministic dynamics known as the slicer map and fly-and-die dynamics are used. We analytically derive the generalized position autocorrelation function of these dynamics and study the special case, the 3-point position correlation function. For this, we derive single-parameter-dependent scaling and compare it with the numerically estimated 3-point position autocorrelation of the Lévy-Lorentz gas, for which the analytical expression is still an open question. Here we obtained a remarkable agreement between them, irrespective of any functional relationship with time. Moreover, we demonstrate that the position moments and the position autocorrelations of these systems scale in the same fashion, provided the times are large enough and far enough apart. Other observables, such as velocity moments and correlations, are reported to distinguish the systems. Published by the American Physical Society 2024

A comparison of methods for estimating parameters of the stochastic Lomax process: through simulation study

In this paper, we study the problem of parameter estimation of the stochastic Lomax diffusion process, this process was introduced in [A. Nafidi, I. Makroz, and R. Gutiérrez Sánchez, A stochastic lomax diffusion process: Statistical inference and application, Mathematics, 2021][14], and the authors suggested the method of simulated annealing to find the maximum likelihood estimators of this process. In this work, we propose alternative methods for finding the maximum likelihood estimators, namely Genetic algorithm and Nelder-Mead, we also investigate the use of Markov Chain Monte Carlo method to determine the model parameters. Finally, an example of application through the simulation of paths for the process is suggested. Then, a comparison is made between the application of three algorithms based on their accuracy and time of execution.

Multivariate stochastic Vasicek diffusion process: computational estimation and application to the analysis of $$CO_2$$ and $$N_2O$$ concentrations

Multivariate stochastic Vasicek diffusion process: computational estimation and application to the analysis of $$CO_2$$ and $$N_2O$$ concentrations

Learning Koopman eigenfunctions of stochastic diffusions with optimal importance sampling and ISOKANN

The dominant eigenfunctions of the Koopman operator characterize the metastabilities and slow-timescale dynamics of stochastic diffusion processes. In the context of molecular dynamics and Markov state modeling, they allow for a description of the location and frequencies of rare transitions, which are hard to obtain by direct simulation alone. In this article, we reformulate the eigenproblem in terms of the ISOKANN framework, an iterative algorithm that learns the eigenfunctions by alternating between short burst simulations and a mixture of machine learning and classical numerics, which naturally leads to a proof of convergence. We furthermore show how the intermediate iterates can be used to reduce the sampling variance by importance sampling and optimal control (enhanced sampling), as well as to select locations for further training (adaptive sampling). We demonstrate the usage of our proposed method in experiments, increasing the approximation accuracy by several orders of magnitude.

Some remarks on the links between the VIR and Gompertz diffusion models and the links between the CIR and Rayleigh diffusion models

The main purpose of this work is to establish new links between the stochastic Cox-Ingersoll-Ross (CIR) model and the stochastic Rayleigh diffusion process (SRDP) and the links between the Vasicek Interest Rate (VIR) process and the stochastic Gompertz diffusion process (SGDP). These links focus on elementary stochastic calculus and Itô’s calculus. Firstly, we prove that the square root of the CIR model is a SRDP. Secondly, we prove that the square of the SRDP is a CIR model. Thirdly, we prove that the exponential of the VIR model is a SGDP. Finally, we prove that the logarithm of the SGDP is a VIR model. New computations of the probability transition density function (PTDF) and the trend functions of the processes have quite simple formulations.

A stochastic log-logistic diffusion process: Statistical computational aspects and application to real data

This article introduces a new stochastic diffusion process based on the theory of diffusion processes whose mean function is proportional to the log-logistic growth curve. The main characteristics of the process are analyzed, including the transition probability density function, the mean functions and in particular, the auto-correlation function between two times of the process. The parameters of the process are estimated by maximum likelihood method using discrete sampling. The simulated annealing algorithm is applied after bounding the parametric space by a strategy procedure to solve the likelihood equations. The behavior of the diffusion process here derived is finally applied to study an example for the growth data of a microorganism culture.

Turbulent Diffusion of the Particles within Pulsar Wind Nebulae

A turbulent diffusion model is presented to account for the energy and spatial diffusion of the particles within pulsar wind nebulae (PWNe), with the assumption that the energy injected from the pulsar into the nebula is split between the turbulence waves and electrons/positrons. In the model, the mutual interactions between the turbulence waves and particles are taken into account, representing the damping effect of the turbulence waves and the stochastic acceleration and spatial diffusion of the particles, respectively. The evolutions of the turbulence waves and particles are described with the coupled kinetic equations, in which the Kolmogorov- and Kraichnan-type turbulence are, respectively, considered. The model is applied to the Crab Nebula and shows that the spectral energy distribution of the PWN can be naturally explained. Our modeling results indicate that, for the Crab Nebula, the stochastic acceleration and spatial diffusion processes play a role in modifying the electron spectrum at the low energies of E e ≲ 1 TeV. The damping process seems more effective for modulating the turbulent spectrum in the Kraichnan-type turbulence, resulting in the nonlinear variations of the current energy and spatial diffusion coefficients with energy generated in the wave–particle systems. In the Kolmogorov-type turbulence, the diffusion coefficients are more consistent with the quasi-linear distributions, due to the energy cascade dominating over the damping effect.

Stochastic models of the dynamics of the maximum and optimal profit of a manufacturing enterprise introducing technological innovations

The published article presents new stochastic models that describe the dynamics of profit formation in manufacturing enterprises using innovative technologies. The dependence of the enterprise's profit not only on the level of production (transformation) costs, but also on the level of non-production (transactional) costs resulting from the search and processing of economic information, financing of negotiation procedures, conclusion of contracts with partners, protection of property rights and payment for opportunistic behavior has been studied. employees and management of the company. It has been established that in order to find the optimal values of profit, it is necessary to maximize not only the profit function, but also the target transactional utility function that redistributes the profit of the enterprise, both in the interests of management and for the implementation of socially oriented programs. It is shown that the presence of transaction costs makes it unattainable for an enterprise to obtain the maximum possible value of profit, instead of which it is necessary to limit itself to its lower optimal value. For indicators of innovative transformations in the production of an enterprise that affect the increase in output and reduce costs, stochastic differential equations are established, random solutions of which describe the stochastic diffusion process of introducing technological innovations. Algorithms for the numerical solution of the stochastic differential equations of the model are constructed using the EulerMaruyama method, according to which each of their implementations is a stochastic trajectory of random functions of indicators of the dynamics of the development of manufacturing enterprises. For the mathematical expectations of the considered random functions, the corresponding differential equations are obtained. Numerical analysis of the developed model showed that taking into account the external random perturbing factor in the stochastic model leads to significant deviations from the deterministic model of the profit formation dynamics of manufacturing enterprises.

Stochastic Pareto diffusion process : Statistical analysis and computational issues. Simulation and Application

Abstract We propose a novel diffusion process having a mean function equal to the Pareto probability density function up to a constant of proportionality. We examine the probabilistic properties of the proposed model. Then, referring to the problem of statistical inference, we describe the approach employed to tackle the issue of obtaining parameter estimates by maximizing the likelihood function based on discrete sampling. This estimation reduces to solving a set of complex equations, that is accomplished using the simulated annealing algorithm. A simulation study is also given to validate the methodology presented. Finally, using a real-world example of the Moroccan child mortality rate, we obtain the fits and forecasts by employing the suggested stochastic process and nonlinear regression model.

Tobler's law and wavefront patterns in the spatial spread of COVID-19 across Europe during the Delta and Omicron waves.

Epidemic wavefront models predict the spread of medieval pandemics such as the plague well. Our aim was to explore whether they contribute to understanding the spread of COVID-19, the first truly global pandemic of the 21st century with its fast and frequent international travel links. We analysed the spatial spread of reaching a threshold of very high incidence of new daily infections of the virus across European countries in the autumn of 2021 in which the Delta variant was dominant, as well as an even higher threshold of incidence in the subsequent spread of infections across the same set of countries during the winter of 2021/2022 when the Omicron variant of the virus became dominant. We found patterns that are consistent with wavefront models for both periods of the pandemic in Europe. Modern means of transportation strongly accelerated the spread of the virus and typically generated diffusion patterns along bidirectional constrained mobility networks in addition to stochastic diffusion processes. However, since the majority of mobility, including mobility across international borders, is over short distances, wavefront patterns in the spread of a pandemic are still to be expected.

Perspective estimation of light emitting diode reliability measures based on multiply accelerated long run stress testing backed up by stochastic diffusion process

Light emitting diodes (LEDs) are currently a key element of many systems. They provide main, informational, safety, support and other functions. Their design, size and low-cost production make them increasingly popular. The relatively long technical life of electronic components is not always easy to prove and properly estimate, and the correlations between luminance, luminous flux intensity and other electrical characteristics of an LED (e.g. voltage) are not usually available. To ensure that their operation is correct and safe, LEDs must have a suitable level of reliability; LEDs are examined in service to detect their degradation and the deterioration of their technical condition. This article is devoted to estimating and demonstrating some LED reliability measures, such as the time during which the LED operation is failure-free and at the required luminance level. Existing research is typically based on short testing periods, one stress factor and more traditional models for analysing recorded data. In contrast, our approach is based on multiply accelerated reliability testing with very long experiment runs supported by degradation modelling using a stochastic diffusion process. Multiple acceleration is provided by intensifying environmental climatic effects (heat exposure) and simultaneously increasing the stress of operating conditions (increased current load). The parameters for degradation models based on stochastic diffusion processes are estimated for the studied LEDs by a specific form of the maximum likelihood estimation (MLE) method, including parameter confidence intervals. Based on the results, a novel correlation between luminance level, luminous flux intensity and voltage magnitude is also determined. The aim is to estimate and demonstrate key LED reliability measures.

Influence Maximization With Visual Analytics.

In social networks, individuals' decisions are strongly influenced by recommendations from their friends, acquaintances, and favorite renowned personalities. The popularity of online social networking platforms makes them the prime venues to advertise products and promote opinions. The Influence Maximization (IM) problem entails selecting a seed set of users that maximizes the influence spread, i.e., the expected number of users positively influenced by a stochastic diffusion process triggered by the seeds. Engineering and analyzing IM algorithms remains a difficult and demanding task due to the NP-hardness of the problem and the stochastic nature of the diffusion processes. Despite several heuristics being introduced, they often fail in providing enough information on how the network topology affects the diffusion process, precious insights that could help researchers improve their seed set selection. In this paper, we present VAIM, a visual analytics system that supports users in analyzing, evaluating, and comparing information diffusion processes determined by different IM algorithms. Furthermore, VAIM provides useful insights that the analyst can use to modify the seed set of an IM algorithm, so to improve its influence spread. We assess our system by: (i) a qualitative evaluation based on a guided experiment with two domain experts on two different data sets; (ii) a quantitative estimation of the value of the proposed visualization through the ICE-T methodology by Wall et al. (IEEE TVCG - 2018). The twofold assessment indicates that VAIM effectively supports our target users in the visual analysis of the performance of IM algorithms.

A γ-power stochastic Lundqvist-Korf diffusion process: Computational aspects and simulation

Abstract In this paper, we introduce a new family of stochastic Lundqvist-Korf diffusion process, defined from a g-power of the Lundqvist-Korf diffusion process. First, we determine the probabilistic characteristics of the process, such as its analytic expression, the transition probability density function from the corresponding It ˆo stochastic differential equation and obtain the conditional and non-conditional mean functions. We then study the statistical inference in this process. The parameters of this process are estimated by using the maximum likelihood estimation method with discrete sampling, thus we obtain a nonlinear equation, which is achieved via the simulated annealing algorithm. Finally, the results of the paper are applied to simulated data.

Approximation of the Heat Fluxes Dynamics in the North Atlantic by Stochastic Diffusion Process

Approximation of the Heat Fluxes Dynamics in the North Atlantic by Stochastic Diffusion Process

Determination of the solution of a stochastic parabolic equation by the terminal value

This paper studies the inverse problem of determination the history for a stochastic diffusion process, by means of the value at the final time T. By establishing a new Carleman estimate, the conditional stability of the problem is proven. Based on the idea of Tikhonov method, a regularized solution is proposed. The analysis of the existence and uniqueness of the regularized solution, and proof for error estimate under an a priori assumption are present. Numerical verification of the regularization, including numerical algorithm and examples are also illustrated.

Efficient information diffusion in time-varying graphs through deep reinforcement learning

Network seeding for efficient information diffusion over time-varying graphs (TVGs) is a challenging task with many real-world applications. There are several ways to model this spatio-temporal influence maximization problem, but the ultimate goal is to determine the best moment for a node to start the diffusion process. In this context, we propose Spatio-Temporal Influence Maximization (STIM), a model trained with Reinforcement Learning and Graph Embedding over a set of artificial TVGs that is capable of learning the temporal behavior and connectivity pattern of each node, allowing it to predict the best moment to start a diffusion through the TVG. We focus on the scenario where some nodes in the TVG present periodic connectivity patterns, an aspect that received little attention in previous approaches. We also develop a special set of artificial TVGs used for training that simulate a stochastic diffusion process in TVGs, showing that the STIM network can learn an efficient policy even over a non-deterministic environment. After trained, STIM can be used in TVGs of any size, since the number of parameters of the model is independent to the size of the TVG being processed. STIM is also evaluated in two real-world TVGs, where it also manages to efficiently propagate information through the nodes. Finally, we also show that the STIM model has a time complexity of O(|E|). STIM is also highly versatile, where one can change the goal of the model by simply changing the adopted reward function.

Product innovation, diffusion and endogenous growth

We develop a model of Schumpeterian growth featuring a stochastic diffusion process where the rate of commercial success of product innovations is endogenously determined by advertising intensity. We consider both informative advertising, which young technological leaders use to increase the probability of diffusion, and defensive advertising, which incumbents use to prevent the diffusion of competing products. Economic growth depends positively on the arrival rate of product innovations and the diffusion rate of innovations into the mainstream market. We show that R&D subsidies shift relative investment incentives towards innovation and away from diffusion. This creates an inverted U-shaped relationship between R&D subsidies and both economic growth and welfare as innovations arrive more frequently, but fewer commercialize successfully. We find that advertising subsidies increase diffusion, growth, and welfare when advertising is purely informative. In the presence of defensive advertising, advertising subsidies lead to socially wasteful increases in resources devoted to advertising without large increases in diffusion, reducing growth and welfare.

Modeling of Adverse Weather Events in an Integrated Reliability and Power Quality Assessment of Distribution System Using Stochastic Diffusion Process

This paper presents the modeling of adverse weather events and their impact on the integrated reliability and power quality assessment of power distribution systems. With respect to previous works, stochastic modeling of adverse weather events is integrated into the Monte Carlo Simulation approach and its impacts on reliability and voltage sag indices are highlighted. In this paper, the aleatory and epistemic uncertainty are modeled into sampling of Time to Failure (TTF) using the Stochastic Diffusion Process. The Stochastic Diffusion Process, including the Jump diffusion is used to model the impact of adverse weather events on permanent and temporary failure rates. The proposed method is applied to the modified IEEE 34 node test feeder, and three case studies have been performed to investigate the impact of various uncertainties and adverse weather events. Numerical results for the IEEE 34 node test feeder are presented to quantify adverse weather impacts on both reliability and voltage sag indices.

Nonparametric inference for diffusion processes in systems with smooth evolution

Dynamics of complex systems can often be successfully modeled as a stochastic diffusion process, even if the real dynamics are not strictly diffusive. We show that for such systems current methods for nonparametric estimation of the drift and diffusion terms may lead to results that are inconsistent with the probability distribution of the system. We present a novel estimation technique that for the two systems studied, turbulent flow and molecular motion in gas, produces drift and diffusion consistent with the observed probability density functions. The presented method is applicable to systems with smooth real dynamics.