Euler-Maruyama Research Articles

Root Mean Square Error Estimates for the Projection-Difference Method for the Approximate Solution of a Parabolic Equation with a Periodic Condition for the Solution

Using the projection-difference method, we construct an approximate solution of an abstract linear parabolic equation in a separable Hilbert space with a periodic condition for a solution. We use the Galerkin method for the spatial variables and the implicit Euler discretization for time. We obtain root mean square estimates of the error of approximate solutions that are effective both in time and spatial variables; these estimates imply the convergence of approximate solutions to an exact solution and allow one to find the convergence rate.

A Reliable Computational Scheme for Stochastic Reaction–Diffusion Nonlinear Chemical Model

The main aim of this contribution is to construct a numerical scheme for solving stochastic time-dependent partial differential equations (PDEs). This has the advantage of solving problems with positive solutions. The scheme provides conditions for obtaining positive solutions, which the existing Euler–Maruyama method cannot do. In addition, it is more accurate than the existing stochastic non-standard finite difference (NSFD) method. Theoretically, the suggested scheme is more accurate than the current NSFD method, and its stability and consistency analysis are also shown. The scheme is applied to the linear scalar stochastic time-dependent parabolic equation and the nonlinear auto-catalytic Brusselator model. The deficiency of the NSFD in terms of accuracy is also shown by providing different graphs. Many observable occurrences in the physical world can be traced back to certain chemical concentrations. Examining and understanding the inter-diffusion between chemical concentrations is important, especially when they coincide. The Brusselator model is the gold standard for describing the relationship between chemical concentrations and other variables in chemical systems. A computational code for the proposed model scheme may be made available to readers upon request for convenience.

Positivity-Preserving Numerical Method and Relaxed Control for Stochastic Susceptible-Infected-Vaccinated Epidemic Model with Markov Switching.

The stochastic susceptible-infected-vaccinated (SIV) epidemic model includes a nonlinear term, making it difficult to obtain analytical solutions. Thus, numerical approximation schemes are an important tool for predicting the dynamics of infectious diseases and establishing optimal control strategies. However, the convergence rate of the existing numerical methods [e.g., Euler-Maruyama (EM) and truncated EM scheme] is only 1/2 order of the time step . This article describes the construction of a logarithmic truncated EM scheme that achieves order-1 convergence and ensures positive numerical solutions of the stochastic SIV epidemic model. The existence of an invariant measure is proved for the stochastic SIV epidemic model with Markov switching. In addition, relaxed controls for the stochastic SIV epidemic model are investigated by using the Markov chain approximation method. It is demonstrated that the approximation schemes converge to the optimal strategy as the mesh size goes to zero. Finally, the results of numerical examples are presented to illustrate the theoretical results derived in this article.

Hamiltonian Deep Neural Networks Guaranteeing Nonvanishing Gradients by Design

Deep Neural Networks (DNNs) training can be difficult due to vanishing and exploding gradients during weight optimization through backpropagation. To address this problem, we propose a general class of Hamiltonian DNNs (H-DNNs) that stem from the discretization of continuous-time Hamiltonian systems and include several existing DNN architectures based on ordinary differential equations. Our main result is that a broad set of H-DNNs ensures non-vanishing gradients by design for an arbitrary network depth. This is obtained by proving that, using a semi-implicit Euler discretization scheme, the backward sensitivity matrices involved in gradient computations are symplectic. We also provide an upper-bound to the magnitude of sensitivity matrices and show that exploding gradients can be controlled through regularization. The good performance of H-DNNs is demonstrated on benchmark classification problems, including image classification with the MNIST dataset.

Weakly convergent stochastic simulation of electron collisions in plasmas

Collisions between charged particles can be described and solved using Monte Carlo methods in the framework of stochastic differential equations (SDEs). In this paper, we start from an SDE including the extended Lorentz collision operator, which can recover the collisions between a sampling electron and background ions and electrons. On this basis, we construct a second order weakly convergent algorithm (WCA2) to simulate collisional effects of electrons in plasmas. Superseding the Weiner process by a three-point distribution, WCA2 possesses high weakly convergent accuracy as well as low computational costs. The definition and properties of weak convergence are discussed in detail. The weakly convergent order of WCA2 is verified both theoretically and numerically. Through two trial moment functions, we carefully analyze the numerical solutions of the SDE using rigorous statistical tests in the sense of weak convergence. The criteria and practical operations of finding the benchmark solution of SDEs are introduced at length. In order to illustrate the power of WCA2, we apply it to simulate the backward runaways in plasmas, which is a dramatic physical phenomenon. By comparison with the Euler-Maruyama method and the Cadjan-Ivanov method, the advantage and efficiency of WCA2 is exhibited. The backward runaway probability and its dependence on initial conditions are accurately studied using WCA2.

A robust and contact resolving Riemann solver for the two-dimensional ideal magnetohydrodynamics equations

This paper presents a new cell-centered numerical method for ideal magnetohydrodynamics (MHD) that can be used in Lagrangian or Eulerian discretization. A two-dimensional Riemann solver based on the HLLC-type MHD method is established, which can be viewed as an extension from the HLLC-2D in hydrodynamics. The main feature of the algorithm is to introduce a nodal contact velocity and ensure the compatibility between edge fluxes and the nodal flow intrinsically. It transforms naturally from Lagrangian setting to the Eulerian setting in terms of grid nodal velocity, and gains benefits of the Lagrangian nature of the scheme. In the Lagrangian approach, the finite volume scheme itself can keep the magnetic field divergence-free strictly, while in the Eulerian case, a special constrained transport (CT) algorithm is constructed from the discontinuous fluxes on cell interfaces to ensure solenoidal nature again. Numerical tests are presented to demonstrate the performance of this new solver and compare the difference between the Lagrangian and Eulerian methods.

A Numerical Approach of Handling Fractional Stochastic Differential Equations

This work proposes a new numerical approach for dealing with fractional stochastic differential equations. In particular, a novel three-point fractional formula for approximating the Riemann–Liouville integrator is established, and then it is applied to generate approximate solutions for fractional stochastic differential equations. Such a formula is derived with the use of the generalized Taylor theorem coupled with a recent definition of the definite fractional integral. Our approach is compared with the approximate solution generated by the Euler–Maruyama method and the exact solution for the purpose of verifying our findings.

A positivity preserving Lamperti transformed Euler–Maruyama method for solving the stochastic Lotka–Volterra competition model

A new positivity preserving numerical scheme is presented for a class of d-dimensional stochastic Lotka–Volterra competitive models, which are characterized by super-linear coefficients and positive solutions. The scheme, dubbed the Lamperti transformed Euler–Maruyama method, approximates the exact solution by integrating a Lamperti-type transformation with an explicit Euler–Maruyama method that has the benefit of being explicit and straightforward to implement. Even though the coefficients of the transformed models grow exponentially and do not satisfy the general monotonicity condition, based on the exponential integrability of the solution, it is proved that the proposed numerical method is of 12-order strong convergence. In particular, when matrix A of the model is a diagonal matrix, the first-order strong convergence is also obtained. Without any step size constraints, the method can preserve long-time dynamical properties such as extinction and pth moment exponential asymptotic stability. Numerical examples are given to support our theoretical conclusions.

A mathematical model for transmission dynamics of COVID-19 infection.

In this paper, a mathematical model of COVID-19 has been proposed to study the transmission dynamics of infection by taking into account the role of symptomatic and asymptomatic infected individuals. The model has also considered the effect of non-pharmaceutical interventions (NPIs) in controlling the spread of virus. The basic reproduction number ( ) has been computed and the analysis shows that for , the disease-free state becomes globally stable. The conditions of existence and stability for two other equilibrium states have been obtained. Transcritical bifurcation occurs when basic reproduction number is one (i.e. ). It is found that when asymptomatic cases get increased, infection will persist in the population. However, when symptomatic cases get increased as compared to asymptomatic ones, the endemic state will become unstable and infection may eradicate from the population. Increasing NPIs decrease the basic reproduction number and hence, the epidemic can be controlled. As the COVID-19 transmission is subject to environmental fluctuations, the effect of white noise has been considered in the deterministic model. The stochastic differential equation model has been solved numerically by using the Euler-Maruyama method. The stochastic model gives large fluctuations around the respective deterministic solutions. The model has been fitted by using the COVID-19 data of three waves of India. A good match is obtained between the actual data and the predicted trajectories of the model in all three waves of COVID-19. The findings of this model may assist policymakers and healthcare professionals in implementing the most effective measures to prevent the transmission of COVID-19 in different settings.

The truncated Euler-Maruyama method for highly nonlinear stochastic differential equations with multiple time delays

The truncated Euler-Maruyama method for highly nonlinear stochastic differential equations with multiple time delays

The truncated Euler–Maruyama method of one-dimensional stochastic differential equations involving the local time at point zero

Abstract Recently, Mao developed a new explicit method, called the truncated Euler–Maruyama method for nonlinear SDEs, and established the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition. The key aim of this paper is to establish the rate of strong convergence of the truncated Euler–Maruyama method for one-dimensional stochastic differential equations involving that the local time at point zero under the drift coefficient satisfies a one-sided Lipschitz condition and plus some additional conditions.

Physics-informed recurrent neural networks and hyper-parameter optimization for dynamic process systems

Many of the processes in chemical engineering applications are of dynamic nature. Mechanistic modeling of these processes is challenging due to the complexity and uncertainty. On the other hand, recurrent neural networks are useful to be utilized to model dynamic processes by using the available data. Although these networks can capture the complexities, they might contribute to overfitting and require high-quality and adequate data. In this study, two different physics-informed training approaches are investigated. The first approach is using a multi-objective loss function in the training including the discretized form of the differential equation. The second approach is using a hybrid recurrent neural network cell with embedded physics-informed and data-driven nodes performing Euler discretization. Physics-informed neural networks can improve test performance even though decrease in training performance might be observed. Finally, smaller and more robust architecture are obtained using hyper-parameter optimization when physics-informed training is performed.

A fast Euler–Maruyama method for Riemann–Liouville stochastic fractional nonlinear differential equations

In this paper, based on the sum-of-exponentials approximation, a fast Euler–Maruyama (EM) method is constructed to solve a kind of multi-term Riemann–Liouville stochastic fractional differential equations. Then the strong convergence order min{1−αm,0.5} of the proposed EM method is proved with Riemann–Liouville fractional derivatives’ orders satisfying 0<α1<α2<⋯<αm<1. Finally, two numerical examples are given to support the theoretical results and show the powerful computational performance of the fast EM method.

Dynamics of an HIV/AIDS transmission model with protection awareness and fluctuations

We establish a stochastic HIV/AIDS model involving the susceptible with protection awareness within a total population. The mechanism of dynamic behaviors of the stochastic HIV/AIDS model with protection awareness are investigated in order to control the spread of HIV/AIDS. We firstly show that the stochastic model admits a global positive solution with any initial positive values. By constructing Lyapunov functions, the ergodic stationary distribution under the condition R0s>1, and the extinction under the condition R0e<1 for the stochastic model are further obtained respectively. Moreover, by using positive preserving truncated Euler–Maruyama method (PPTEM), the related numerical simulations are performed, which demonstrate the quantitative properties of persistence and extinction of the solution. Precisely, the increasing of protection efficiency of the susceptible with protection awareness reduces the scale of the infected individuals with AIDS; and the continuous antiretroviral therapy (ART) also benefits the control of the scale of the infected individuals with HIV/AIDS; the media reports with more instructions and details such as taking post-exposure prophylaxis (PEP) within 72 h to help the individuals avoid the infection to meet the 90%–90%–90% plan of WHO.

Lower bounds, elliptic reconstruction and a posteriori error control of parabolic problems

Abstract A popular approach for proving a posteriori error bounds in various norms for evolution problems with partial differential equations uses reconstruction operators to recover conforming objects from the approximate solutions. So far, lower bounds in reconstruction-based a posteriori error estimators have been proven only for time-discrete schemes for parabolic problems; the proof of lower bounds for fully discrete schemes in reconstruction-based a posteriori error estimators has eluded. In this work, we provide a complete framework addressing this issue for energy-type norms. We consider Backward Euler discretizations and time-discontinuous Galerkin schemes, combined with dynamically changing conforming finite element methods in space, approximating linear parabolic problems. The results presented include sharp upper and lower a posteriori error bounds. Localized versions of the lower bounds are also considered.

A distributed-order fractional stochastic differential equation driven by Lévy noise: Existence, uniqueness, and a fast EM scheme.

In this paper, we consider a distributed-order fractional stochastic differential equation driven by Lévy noise. We, first, prove the existence and uniqueness of the solution. A Euler-Maruyama (EM) scheme is constructed for the equation, and its strong convergence order is shown to be min{1-α∗,0.5}, where α∗ depends upon the weight function. Besides, we present a fast EM method and also the error analysis of the fast scheme. In addition, several numerical experiments are carried out to substantiate the mathematical analysis.

Fixed-Time Algorithms for Time-Varying Convex Optimization

To resolve the time-varying convex optimization problems with the cost function, the constraints or both being time dependent, in this brief we investigate a novel type of fixed-time algorithms. First, with the unconstrained time-varying optimization problem considered, a general framework algorithm is developed for tracking its optimal trajectory within fixed time, which contains the gradient flow-based scheme and Newton-type method as its special cases. Then, considering the equality constraint being involved in the time-varying optimization problem, we design another algorithm with fixed-time convergence, which includes Newton-type scheme as its special case. To verify that the given approach achieves fixed-time convergence, the simulation result is given with first-order Euler discretization.

Non-Overlapping Schwarz Waveform-Relaxation for Nonlinear Advection-Diffusion Equations

Nonlinear advection-diffusion equations often arise in the modeling of transport processes. We propose for these equations a non-overlapping domain decomposition algorithm of Schwarz waveform-relaxation (SWR) type. It relies on nonlinear zeroth-order (or Robin) transmission conditions between the sub-domains that ensure the continuity of the converged solution and of its normal flux across the interface. We prove existence of unique iterative solutions and the convergence of the algorithm. We then present a numerical discretization for solving the SWR problems using a forward Euler discretization in time and a finite volume method in space, including a local Newton iteration for solving the nonlinear transmission conditions. Our discrete algorithm is asymptotic preserving, i.e., robust in the vanishing viscosity limit. Finally, we present numerical results that confirm the theoretical findings, in particular the convergence of the algorithm. Moreover, we show that the SWR algorithm can be successfully applied to two-phase flow problems in porous media as paradigms for evolution equations with strongly nonlinear advective and diffusive fluxes.

Strong approximation of non-autonomous time-changed McKean–Vlasov stochastic differential equations

In this paper, we investigate the time-changed McKean–Vlasov stochastic differential equations (MV-SDEs) via interacting particle systems, where the MV-SDEs contain two drift terms, one driven by the random time change Et and the other driven by a regular, non-random time variable t. Strong convergence and convergence rate in the finite time of the Euler–Maruyama (EM) method on the particle system is discussed. Numerical example illustrates that “particle corruption” will not occur on the whole particle system and show the consistency with convergence result.

Discrete-time adaptive predictive sliding mode trajectory tracking control for dynamic positioning ship with input magnitude and rate saturations

This paper investigates the discrete-time adaptive predictive sliding mode trajectory tracking control problem based on a reference sliding mode trajectory for the dynamic positioning ship with model uncertainty, environmental disturbances and input magnitude and rate saturations. Initially, a two-step delay discrete-time disturbance observer is designed to estimate the model uncertainty and environmental disturbances, and the input constraints are processed by quadratic programming. Then, an adaptive expected heading is proposed to ensure faster trajectory tracking and more stable heading changes simultaneously. Furthermore, using a double closed-loop control strategy, a new discrete-time virtual velocity controller is designed in the outer loop based on Euler discretization to convert trajectory tracking into velocity tracking. In the inner loop, a discrete-time adaptive predictive sliding mode velocity controller is designed based on a new adaptive reference sliding mode trajectory and an integral terminal sliding mode surface with disturbance estimation error. The proposed strategy can adaptively adjust the sliding mode bandwidth and convergence speed, which not only ensures the system’s rapid convergence but also reduces the tracking error and weakens the chattering. Finally, the stability analysis proves that the closed-loop system is uniformly bounded stable. Comparative simulation results verify the effectiveness of the proposed control scheme.