Euler-Maruyama Research Articles

Discrete-Time Adaptive Control for Three-Phase PWM Rectifier.

This paper proposes a dual-loop discrete-time adaptive control (DDAC) method for three-phase PWM rectifiers, which considers inductance-parameter-mismatched and DC load disturbances. A discrete-time model of the three-phase PWM rectifier is established using the forward Euler discretization method, and a dual-loop discrete-time feedback linearization control (DDFLC) is given. Based on the DDFLC, the DDAC is designed. Firstly, an adaptive inductance disturbance observer (AIDO) based on the gradient descent method is proposed in the current control loop. The AIDO is used to estimate lump disturbances caused by mismatched inductance parameters and then compensate for these disturbances in the current controller, ensuring its strong robustness to inductance parameters. Secondly, a load parameter adaptive law (LPAL) based on the discrete-time Lyapunov theory is proposed for the voltage control loop. The LPAL estimates the DC load parameter in real time and subsequently adjusts it in the voltage controller, achieving DC load adaptability. Finally, simulation and experimental results show that the DDAC exhibits better steady and dynamic performances, less current harmonic content than the DDFLC and the dual-loop discrete-time PI control (DDPIC), and a stronger robustness to inductance parameters and DC load disturbances.

Discretization of prescribed-time observers in the presence of noises and perturbations

An implicit Euler discretization scheme of the prescribed-time converging observer from Holloway and Krstic, (2019) is proposed, which preserves all main properties of the continuous-time counterpart, and it is investigated how to apply recursively this discretization on any interval of time. In addition, it is demonstrated that the estimation error is robustly stable with respect to the measurement noise with a linear gain (the property that does not hold in the continuous-time setting). The efficiency of the suggested discrete-time observer is illustrated through numeric experiments.

The improvement of the truncated Euler-Maruyama method for non-Lipschitz stochastic differential equations

The improvement of the truncated Euler-Maruyama method for non-Lipschitz stochastic differential equations

Antithetic multilevel particle filters

Abstract In this paper we consider the filtering of partially observed multidimensional diffusion processes that are observed regularly at discrete times. This is a challenging problem which requires the use of advanced numerical schemes based upon time-discretization of the diffusion process and then the application of particle filters. Perhaps the state-of-the-art method for moderate-dimensional problems is the multilevel particle filter of Jasra et al. (SIAM J. Numer. Anal.55 (2017), 3068–3096). This is a method that combines multilevel Monte Carlo and particle filters. The approach in that article is based intrinsically upon an Euler discretization method. We develop a new particle filter based upon the antithetic truncated Milstein scheme of Giles and Szpruch (Ann. Appl. Prob.24 (2014), 1585–1620). We show empirically for a class of diffusion problems that, for $\epsilon>0$ given, the cost to produce a mean squared error (MSE) of $\mathcal{O}(\epsilon^2)$ in the estimation of the filter is $\mathcal{O}(\epsilon^{-2}\log(\epsilon)^2)$ . In the case of multidimensional diffusions with non-constant diffusion coefficient, the method of Jasra et al. (2017) requires a cost of $\mathcal{O}(\epsilon^{-2.5})$ to achieve the same MSE.

Superconvergence error analysis of linearized semi-implicit bilinear-constant SAV finite element method for the time-dependent Navier–Stokes equations

In this paper, based on the scalar auxiliary variable (SAV) approach, the superconvergence error analysis is investigated for the time-dependent Navier–Stokes equations. In which, an equivalent system of the Navier–Stokes equations with three variables and a fully-discrete scheme is developed with semi-implicit Euler discretization for the temporal direction and low-order bilinear-constant finite element discretization for the spatial direction, respectively. With the help of the high-precision estimations of the bilinear-constant finite element pair on the rectangular meshes, the superclose error estimates for velocity in H1-norm and pressure in L2-norm are obtained by treating the trilinear term carefully and skillfully. The global superconvergence results are also derived in terms of a simple and efficient interpolation post-processing technique. Finally, some numerical results are provided to demonstrate the correctness of the theoretical analysis.

Numerical treatment of stochastic and deterministic alcohol drinker dynamics with Euler–Maruyama method

Alcohol abuse is a substantial cause of various health and societal issues, as well as a significant factor in global disease. Once alcohol is consumed in the gastrointestinal tract, it undergoes metabolism in the liver and lungs. In this investigation, the nonlinear deterministic and stochastic differential frameworks are analyzed numerically to predict the dynamic evolution of the virus in the drinker alcohol model. The framework for apprehending drinking patterns is categorized into three distinct groups: the susceptible population, risk drinkers, and moderate drinkers. The approximate solution for each population group is determined by exhaustively creating scenarios that vary the probability ratio of infection in susceptible individuals who do not consume alcohol, the increasing rate of alcohol consumption, the rate at which individuals transition from acute to chronic drinking categories, the rate at which new non-drinking consumers are attracted, the death rate of the population, the ratio affecting the rate of sociability in heavy drinkers, and the overall population rate. The Euler–Maruyama approach for the stochastic framework and the Adams method for the deterministic framework are utilized, respectively, to determine the solutions of the alcohol drinker model. This study compares deterministic and stochastic frameworks to underscore their distinct characteristics and efficiency, achieved through comprehensive simulations and in-depth analysis of the numerical outcomes.

Multi-stage Euler–Maruyama methods for backward stochastic differential equations driven by continuous-time Markov chains

Multi-stage Euler–Maruyama methods for backward stochastic differential equations driven by continuous-time Markov chains

Brownian motion based multi-objective particle swarm optimization methodology and application in binary classification

The particle swam optimization (PSO) method has been widely applied in evolutionary computation and related industrial areas. However, the optimization performance of traditional PSO is usually influenced by particle updating displacement and velocity, which is randomly distributed and makes the solution solving process inefficient. To deal with this challenge, a Brownian particle motion and Euler-Maruyama (EM) methodology-based model is proposed to improve the performance of the multi-objective PSO. The EM based PSO principles and algorithm framework are introduced by theoretical analysis and mathematical modeling, and then the optimization performance of four comparing PSO models is tested, while the feasibility of the new PSO4 is verified by typical benchmark problems. Afterwards, the characterizations of the EM based PSO method are investigated from the perspective of particle motion mechanism analysis, binary classification application and various optimization performance comparison. The result manifests that the new PSO4 features higher accuracy and less running time among 24 related optimization methods for 4 typical standard datasets of Ionosphere, Breast, Diabetes and Wine. Finally, the multi-objective optimization and classification application potential of the Brownian motion and EM based PSO4 is demonstrated, and the future work regarding the new PSO model is discussed.

Some stabilities of stochastic differential equations with delay in the G-framework and Euler–Maruyama method

This paper discusses some stabilities of stochastic differential equations with delay in the G-framework (G-SDDEs, in short) and Euler-Maruyama method. We construct a weaker condition instead of using the Lyapunov functional method to obtain the p-th moment exponential stability of the G-SDDE. We prove that the Euler-Maruyama method can reproduce the p-th moment exponential stability of the G-SDDE under some step size restrictions. We also prove that the stability of the discrete Euler-Maruyama method is equivalent to that of the continuous Euler-Maruyama method. Furthermore, we introduce some sufficient conditions under which the p-th moment exponential stability of the G-SDDE can imply its quasi sure exponential stability. Finally, we give two numerical examples to confirm the theoretical results.

A variational PDNet network using a learning reaction–diffusion equation

Due to their high performance in modeling and forecasting a large amount of real-world complex phenomena, deep convolutional neural networks have received a great deal of attention over the past ten years. It has been extensively utilized in several recent vision problems, such as image denoising and deblurring. By incorporating recent developments in learning-based nonlinear diffusion models and highlighting the connection between partial differential equations (PDEs) and networks, especially residual networks, we propose a flexible learning reaction–diffusion system with spatially computed convolution kernels. This trained PDE is employed to prevent the low-level features from deteriorating, which is needed for tasks involving denoising. In fact, we introduce an improved network architecture for a denoising network entitled VPDNet through an optimal control problem. More precisely, the network architecture is governed by an explicit Euler discretization of a reaction–diffusion system. We give the main algorithm to identify the kernel parameters using the Alternating Direction Method of Multipliers (ADMM) combined with a Primal–Dual algorithm. Finally, numerical simulations are provided with appropriate comparisons to demonstrate the efficiency of the elaborated network.

Discrete kinetic analysis of a general reaction–diffusion model constructed by Euler discretization and coupled map lattices

In this paper, we provide a framework for the Turing instability in a general two-dimensional discrete reaction–diffusion model that utilizes Euler discretization and coupled map lattices. We obtained explicit criterions for the normal forms of Neimark–Sacker bifurcation and flip bifurcation in the absence of diffusion. We derive the general conditions that govern the emergence of pure Turing instability, Neimark–Sacker–Turing instability, Flip–Turing instability, spatially homogeneous stable states, spatially homogeneous periodic oscillation states, and spatially homogeneous period-doubling oscillatory states under spatially inhomogeneous conditions. As an example, we employ a ratio-dependent predation model and verify the derived general conditions through a series of numerical simulations. Additionally, we apply the calculation of Maximum Lyapunov Exponent(MLE) to simulate and demonstrate the path from flip bifurcation to chaos.

Parameter identification algorithm for ship manoeuvrability and wave peak model based multi-innovation stochastic gradient algorithm use data filtering technique

This paper addresses the issue of identifying ship motion parameters and wave peak frequency. Utilising the Euler discretisation principle, we establish a discrete-time auto-regressive moving average model with exogenous input (ARMAX) for the ship-wave system. Furthermore, we develop a filtering-based stochastic gradient algorithm for the system by applying filtering techniques and auxiliary model identification idea. A filtering-based multi-innovation stochastic gradient algorithm, utilizing the multi-innovation identification theory, was developed to enhance the convergence rate and accuracy of parameter identification. This approach was found to be more effective than the filtering-based stochastic gradient algorithm. Simulation results validate the efficacy of the proposed algorithm in parameter identification.

Euler State Networks: Non-dissipative Reservoir Computing

Inspired by the numerical solution of ordinary differential equations, in this paper, we propose a novel Reservoir Computing (RC) model, called the Euler State Network (EuSN). The presented approach makes use of forward Euler discretization and antisymmetric recurrent matrices to design reservoir dynamics that are both stable and non-dissipative by construction.Our mathematical analysis shows that the resulting model is biased towards a unitary effective spectral radius and zero local Lyapunov exponents, intrinsically operating near the edge of stability. Experiments on long-term memory tasks show the clear superiority of the proposed approach over standard RC models in problems requiring effective propagation of input information over multiple time steps. Furthermore, results on time-series classification benchmarks indicate that EuSN can match (or even exceed) the accuracy of trainable Recurrent Neural Networks, while retaining the training efficiency of the RC family, resulting in up to ≈464-fold savings in computation time and ≈1750-fold savings in energy consumption. At the same time, our results on time-series modeling tasks show competitive results against standard RC when the architecture is complemented by direct input-readout connections.

Predictive trajectory tracking control for the USV in networked environments with communication constraints

The trajectory tracking control problem of under-actuated unmanned surface vessels (USVs) suffering from communication constraints (network delay, packet loss, packet disorder) as well as external environment disturbance is investigated in this paper. To solve the aforementioned trajectory control problem, a nonlinear networked predictive control strategy based on the discrete sliding mode framework is proposed. Initially, using the forward Euler discretization method, a novel discrete virtual velocity control law is designed to transform trajectory tracking control into virtual velocity tracking control. Then, a sliding mode controller is suggested to suppress external disturbance of the USV. In addition, a networked sliding mode predictive control (NSMPC) scheme is proffered to guarantee active compensation of the communication constraints and stability of the trajectory tracking system simultaneously. Afterward, a theoretical analysis is given to show that the closed-loop system is uniformly bounded stable after the introduction of the digital network. Finally, extensive comparative simulation examples are provided to demonstrate the effectiveness of the proposed control strategy.

Strong convergence of Euler–Maruyama schemes for doubly perturbed McKean–Vlasov stochastic differential equations

In this paper, we develop strong convergence of the Euler–Maruyama (EM) scheme for approximating the doubly perturbed McKean–Vlasov stochastic differential equations. In contrast to the existing work, a novel feature is that we use more general conditions for parameters α and β. To obtain the desired approximation, this paper also proves the existence and uniqueness of strong solution for this class of McKean–Vlasov SDEs. Combining with the results of propagation of chaos, the overall convergence rate is obtained for the EM scheme. Finally, two numerical examples are provided to demonstrate our results.

Exploring stochastic differential equation for analyzing uncertainty in wastewater treatment plant-activated sludge modeling

Abstract The management of wastewater treatment plant (WWTP) and the assessment of uncertainty in its design are crucial from an environmental engineering perspective. One of the key mechanisms in WWTP operation is activated sludge, which is related to the biological oxygen demand (BOD) parameter. In the modeling of BOD, the conventional approach utilizing ordinary differential equations (ODEs) fails to incorporate the stochastic nature of this parameter, leading to a considerable level of uncertainty in the design of WWTP. To address this issue, this study proposes a stochastic model that utilizes stochastic differential equations (SDEs) instead of ODE to simulate BOD activities of microorganisms and wastewater flow rate (Q). The SDEs and integral It̂o are solved using the Euler–Maruyama method for a period of 15 sequential days and the timespan of 2019–2020 for a WWTP in Tabriz City. SDE improves the design of WWTP facilities by identifying uncertainties and enhancing reliability. This, in turn, increases the reliability of the technical structures within the WWTP and improves the performance of its biological system. By considering the randomness of the problem, the proposed method significantly improves the results, with an enhancement of 11.47 and 10.11% for the BOD and Q models, respectively.

Spectral analysis of spin noise in an optically spin-polarized stochastic Bloch equation driven by noisy magnetic fields

We theoretically analyze an optically spin-polarized collection of atoms, which serves as a basis for atomic sensors. Assuming that the intrinsic atomic spin projection noise is negligible, we provide the closed-form autocorrelation function and the power spectral density (PSD) of the solution to a noisy version of an optically pumped Bloch equation, wherein each component of the external magnetic field is subjected to white noise. We conclude that noise in the bias B-field direction does not affect the autocorrelation function, up to first order in white noise covariance amplitudes. Moreover, the noise terms for the remaining two axes make different contributions to the magnetic noise-driven spin PSD; in particular, the contribution corresponding to noises perpendicular to the probing direction dominates at high frequencies. Some results concerning the second (and higher)-order terms are given. In particular, we anticipate a decrease in the effective Larmor frequency despite an increase in the magnetic field magnitude in the case of anisotropic transversal B-field noises. The analytic results are supported by Monte Carlo simulations employing the Euler–Maruyama method. The analytic methodology is applied to the case of a Bell–Bloom magnetometer, which reveals a non-linearity in the PSD of the magnetometer output and also a broadening effect due to magnetic field noise.

A comparative study of discretization techniques for augmented Urysohn type nonlinear functional Volterra integral equations and their convergence analysis

This article considers the nonlinear functional Volterra integral equation, the Urysohn type class of nonlinear integral equations. In an approach based on the Picard iterative method, the solution's existence and uniqueness are demonstrated. The trapezoidal and Euler discretization methods are used for the approximation of a numerical solution, and a nonlinear algebraic system of equations is obtained. The first and second order of convergence for the Euler and the trapezoidal methods respectively to the solution is manifested by employing the Gronwall inequality and its discrete form. A new Gronwall inequality is devised in order to prove the trapezoidal method's convergence. Finally, some numerical examples are provided which attest to the application, effectiveness, and reliability of the methods.

Well-posedness and an Euler-Maruyama method for multi-term caputo tempered fractional stochastic differential equations

In this paper, we first prove the existence, uniqueness and continuity dependence on the initial value of multi-term Caputo tempered fractional stochastic differential equations with initial value. Then, an Euler-Maruyama (EM) method is presented for solving the considered equation. The strong convergence of the presented EM method is strictly investigated with the order to be min{αm−αm−1,αm−0.5} with 0 < α 1 < α 2 < ⋯ < α m−1 < α m < 1, α m > 0.5, and m is a given positive integer. Finally, three numerical examples are provided to support our theoretical findings.

The logarithmic truncated EM method with weaker conditions

In 2014, Neuenkirch and Szpruch established the drift-implicit Euler-Maruyama method for a class of SDEs which take values in a given domain. However, expensive computational cost is required for implementation of an implicit numerical method. A competitive positivity preserving explicit numerical method for SDEs which take values in the positive domain is the logarithmic truncated Euler-Maruyama method. However, assumptions for the logarithmic truncated Euler-Maruyama method used in previous work are restrictive which exclude some important SDE models with specific parameters. The main aim of this paper is to use weaker assumptions to establish strong convergence theory for the logarithmic truncated Euler-Maruyama method.