Small Step Size Research Articles

Implicit numerical solutions to neutral-type stochastic systems with superlinearly growing coefficients

In this paper, our main aim is to investigate the stability and strong convergence of an implicit numerical approximations for neutral-type stochastic differential equations with superlinearly growing coefficients. After providing moment boundedness and exponential stability for the exact solutions, we show that the backward Euler–Maruyama numerical method preserves stability and boundedness of moments, and the numerical approximations converge strongly to the true solutions for sufficiently small step size.

An efficient runtime mesh smoothing technique for 3D explicit Lagrangian free‐surface fluid flow simulations

SummaryA fast runtime mesh smoothing algorithm for explicit Lagrangian simulations of 3D weakly compressible viscous fluid flows, implemented in conjunction with the particle finite element method (PFEM), is proposed. The formulation for weakly compressible fluids allows for the use of an explicit time integration scheme. Explicit solvers are appealing for large‐scale engineering problems characterized by fast dynamics and/or a high degree of nonlinearity. However, the conditional stability of these schemes requires the use of small time increments, proportional to the size of the element in the mesh with the worst geometrical quality. The Lagrangian description of the PFEM requires an efficient and robust runtime mesh generator algorithm, such as the Delaunay tessellation, to create new meshes during the analysis, whenever the current ones get too distorted because of the motion of the mesh nodes. When 3D problems are considered, a computationally effective mesh‐improving algorithm is also required because, in 3D, the Delaunay tessellation loses some of its optimality properties holding in 2D so that badly shaped tetrahedra are frequently included in the triangulation, leading to unacceptably small stable time step sizes for the explicit solver. To this purpose, a novel and efficient mesh smoothing technique is here proposed, exploiting an elastic analogy that allows for the use of the same explicit and parallelizable architecture of the fluid solver. This smoothing algorithm has been specifically designed to ensure reasonably large critical time step sizes at an acceptable computational cost. This is particularly appealing for the application of explicit Lagrangian PFEM in large‐scale 3D engineering problems, but it could be conveniently applied also to regularize the mesh and improve the solution of implicit solvers.

Stability and bifurcation analysis of a discrete predator\u2013prey system with modified Holling\u2013Tanner functional response

In this paper, we study a discrete predator–prey system with modified Holling–Tanner functional response. We derive conditions of existence for flip bifurcations and Hopf bifurcations by using the center manifold theorem and bifurcation theory. Numerical simulations including bifurcation diagrams, maximum Lyapunov exponents, and phase portraits not only illustrate the correctness of theoretical analysis, but also exhibit complex dynamical behaviors and biological phenomena. This suggests that the small integral step size can stabilize the system into the locally stable coexistence. However, the large integral step size may destabilize the system producing far richer dynamics. This also implies that when the intrinsic growth rate of prey is high, the model has bifurcation structures somewhat similar to the classic logistic one.

Exponential stability of θ-method for stochastic differential equations in the G-framework

Exponential stability of the stochastic θ -method for stochastic differential equations with G -Brownian motion (called G-SDEs for brevity) is investigated. It is proved that under the global Lipschitz condition, a G-SDE is p th ( p ∈ ( 0 , 1 ) ) moment exponentially stable if and only if the stochastic θ -method with a sufficiently small step size is also p th moment exponentially stable. The stochastic θ -method for a G-SDE is p th ( p ∈ ( 0 , 1 ) ) moment exponentially stable, then it is also quasi surely exponentially stable, and furthermore, the G-SDE is quasi surely exponentially stable. Numerical examples are demonstrated to illustrate the obtained theoretical results.

On obtaining limiting values of a class of infinite products

Difference equations have a wide variety of applications, including fluid flow analysis, wave propagation, circuit theory, the study of traffic patterns, queueing analysis, diffusion theory, and many others. Besides these applications, studies into the analogy between ordinary differential equations (ODEs) and difference equations have been a favourite topic of mathematicians (e.g. see [1] and [2]). These applications and studies bring to light the similar character of the solutions of a difference equation with a fixed step size and a corresponding ODE.Also, an important numerical technique for solving both ordinary and partial differential equations (PDEs) is the method of finite differences [3], whereby a difference equation with a small step size is utilised to obtain a numerical solution of a differential equation. In this paper, elements of both of these ideas will be used to solve some intriguing problems in pure and applied mathematics.

Cube Arithmetic: Improving Euler Method for Ordinary Differential Equation using Cube Mean

The Euler method is a first-order numerical procedure for solving Ordinary Differential Equation (ODEs) problems. It is an effective and easy method to solve initial value problems. Although Euler provides simple procedure for solving ODEs, there have been issues such as complexity, time of processing and accuracy that compelled the use of other, more complex, methods. Improvements to the Euler method have attracted much attention resulting in numerous modified Euler methods. This paper proposes Cube Arithmetic, a modified Euler method with improved accuracy. The efficiency of Cube Arithmetic was compared with Euler Arithmetic and tested using SCILAB against exact solutions. Results indicate that not only Cube Arithmetic provided solutions that are similar to exact solutions at small step size, but also at higher step size, hence producing more accurate results.

Robust Visual Tracking via Hierarchical Convolutional Features.

Visual tracking is challenging as target objects often undergo significant appearance changes caused by deformation, abrupt motion, background clutter and occlusion. In this paper, we propose to exploit the rich hierarchical features of deep convolutional neural networks to improve the accuracy and robustness of visual tracking. Deep neural networks trained on object recognition datasets consist of multiple convolutional layers. These layers encode target appearance with different levels of abstraction. For example, the outputs of the last convolutional layers encode the semantic information of targets and such representations are invariant to significant appearance variations. However, their spatial resolutions are too coarse to precisely localize the target. In contrast, features from earlier convolutional layers provide more precise localization but are less invariant to appearance changes. We interpret the hierarchical features of convolutional layers as a nonlinear counterpart of an image pyramid representation and explicitly exploit these multiple levels of abstraction to represent target objects. Specifically, we learn adaptive correlation filters on the outputs from each convolutional layer to encode the target appearance. We infer the maximum response of each layer to locate targets in a coarse-to-fine manner. To further handle the issues with scale estimation and re-detecting target objects from tracking failures caused by heavy occlusion or out-of-the-view movement, we conservatively learn another correlation filter, that maintains a long-term memory of target appearance, as a discriminative classifier. We apply the classifier to two types of object proposals: (1) proposals with a small step size and tightly around the estimated location for scale estimation; and (2) proposals with large step size and across the whole image for target re-detection. Extensive experimental results on large-scale benchmark datasets show that the proposed algorithm performs favorably against the state-of-the-art tracking methods.

Reduction of the Phase Noise Level of Heterodynes of Modern Spectrum Analyzers Using Fundamental Frequency Images of Direct Digital Synthesizers

A hybrid frequency synthesizer ins developed in which fundamental frequency images of a direct digital synthesizer are used to improve noise characteristics. It is shown that this synthesizer can generate a wide range of output frequencies with a small step size adjustment. Mathematical modeling of the noise characteristics of the synthesizer and comparison with the noise characteristics of heterodynes of modern spectrum analyzers are carried out.

On the zero-stability of multistep methods on smooth nonuniform grids

In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950’s, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. In practical computations, step size control can be implemented using smooth (small) step size changes. The resulting grid {t_n}_{n=0}^N can be modeled as the image of an equidistant grid under a smooth deformation map, i.e., t_n = {varPhi }(tau _n), where tau _n = n/N and the map {varPhi } is monotonically increasing with {varPhi }(0)=0 and {varPhi }(1)=1. The model is justified for any fixed order method operating in its asymptotic regime when applied to smooth problems, since the step size is then determined by the (smooth) principal error function which determines {varPhi }, and a tolerance requirement which determines N. Given any strongly stable multistep method, there is an N^* such that the method is zero stable for N>N^*, provided that {varPhi }in C^2[0,1]. Thus zero stability holds on all nonuniform grids such that adjacent step sizes satisfy h_n/h_{n-1} = 1 + {mathrm {O}}(N^{-1}) as Nrightarrow infty . The results are exemplified for BDF-type methods.

A Linear Algorithm for Optimization Over Directed Graphs With Geometric Convergence

In this letter, we study distributed optimization, where a network of agents, abstracted as a directed graph, collaborates to minimize the average of locally-known convex functions. Most of the existing approaches over directed graphs are based on push-sum (type) techniques, which use an independent algorithm to asymptotically learn either the left or right eigenvector of the underlying weight matrices. This strategy causes additional computation, communication, and nonlinearity in the algorithm. In contrast, we propose a linear algorithm based on an inexact gradient method and a gradient estimation technique. Under the assumptions that each local function is strongly-convex with Lipschitz-continuous gradients, we show that the proposed algorithm geometrically converges to the global minimizer with a sufficiently small step-size. We present simulations to illustrate the theoretical findings.

Operator splitting for chemical reaction systems with fast chemistry

Splitting methods are frequently used for large scale chemical reaction systems. The main reason is the reduced computational cost for solving the subproblems in comparison to the time integration of the full problem. However, a splitting error is introduced. The most popular splitting schemes are the first order Lie–Trotter splitting and the second order Strang splitting. However, in case of stiff differential equations the Strang splitting suffers from order reduction and both schemes have order one for stiff differential equations. Therefore, the step size restrictions due to the low order can result in a prohibitive small step size. Hence, splitting schemes with order larger than one are necessary for stiff differential equations. The Richardson extrapolation of the Lie–Trotter splitting is a second order scheme. In this paper we examine differential equations with a fast chemical source term and a slow transport term. Thus, all stiffness of the system is related to the chemical source term. We show that the Richardson extrapolation does not suffer from order reduction in this case. Furthermore, we perform a stability analysis of the extrapolated splitting scheme for a linear test problem. Thereby the operators of the test problem do not commute.

A Discrete Numerical Scheme of Modified Leslie-Gower With Harvesting Model

Recently, exploitation of biological resources and the harvesting of two populations or more are widely practiced, such as fishery or foresty. The simplest way to describe the interaction of two species is by using predator prey model, that is one species feeds on another. The Leslie-Gower predator prey model has been studied in many works. In this paper, we use Euler method to discretisize the modified Leslie-Gower with harvesting model. The model consists of two simultanious predator prey equations. We show numerically that this discrete numerical scheme model is dynamically consistent with its continuous model only for relatively small step-size. By using computer simulation software, we show that equlibrium points can be stable, saddles, and unstable. It is shown that the numerical simulations not only illustrate the results, but also show the rich dynamics behaviors of the discrete system.

Almost sure stability with general decay rate of exact and numerical solutions for stochastic pantograph differential equations

The exponential stability and the polynomial stability have been well studied for the exact and numerical solutions of the stochastic pantograph differential equations (SPDEs), while the result on the general decay stabilities is insufficient for SPDEs. In this paper, the sufficient conditions of the almost sure stability with general decay rate of the exact and numerical solutions for SPDEs have been considered. The stability of two different theta numerical methods, namely the split–step theta method and the stochastic linear theta method, have been discussed respectively. From the conditions, we established for the two theta approximations to reproduce the stability of the exact solution, and we see that the conditions for $\theta \in \left [0,\frac {1}{2}\right ]$ are stronger than those for $\theta \in \left (\frac {1}{2},1\right ]$ . The bound of the Lyapunov exponent of the split-step theta method is little bigger than that of the stochastic linear theta method for sufficiently small step size. To illustrate the theoretical results, we give two examples to examine the almost sure exponential stability.

ADD-OPT: Accelerated Distributed Directed Optimization

In this paper, we consider distributed optimization problems where the goal is to minimize a sum of objective functions over a multi-agent network. We focus on the case when the inter-agent communication is described by a strongly-connected, \emph{directed} graph. The proposed algorithm, ADD-OPT (Accelerated Distributed Directed Optimization), achieves the best known convergence rate for this class of problems,~$O(\mu^{k}),0<\mu<1$, given strongly-convex, objective functions with globally Lipschitz-continuous gradients, where~$k$ is the number of iterations. Moreover, ADD-OPT supports a wider and more realistic range of step-sizes in contrast to existing work. In particular, we show that ADD-OPT converges for arbitrarily small (positive) step-sizes. Simulations further illustrate our results.

Stabilized low‐order finite elements for strongly coupled poromechanical problems

SummaryFinite element solutions of poromechanical problems often exhibit oscillating pore pressures in the limits of low permeability, fast loading rates, coarse meshes, and/or small time step sizes. To suppress completely the pore pressure oscillations, a stabilized finite element scheme with a better performance on monotonicity is proposed for modeling compressible fluid‐saturated porous media. This method, based on the polynomial pressure projection technique, allows the use of linear equal‐order interpolation for both displacement and pore pressure fields, which is more straightforward for both code development and maintenance compared to others. By employing the discrete maximum principle, a proper stabilization parameter is deduced, which is efficient to guarantee the monotonicity and optimal in theory in the 1‐dimensional case. An appealing feature of the method is that the stabilization parameter is evaluated in terms of the properties of porous material only, while no mesh or time step size is involved. Through comparing the numerical simulations with the analytical benchmarks, the efficiency of the proposed stabilization scheme is confirmed.

Formal Solutions for Polarized Radiative Transfer. III. Stiffness and Instability

Abstract Efficient numerical approximation of the polarized radiative transfer equation is challenging because this system of ordinary differential equations exhibits stiff behavior, which potentially results in numerical instability. This negatively impacts the accuracy of formal solvers, and small step-sizes are often necessary to retrieve physical solutions. This work presents stability analyses of formal solvers for the radiative transfer equation of polarized light, identifies instability issues, and suggests practical remedies. In particular, the assumptions and the limitations of the stability analysis of Runge–Kutta methods play a crucial role. On this basis, a suitable and pragmatic formal solver is outlined and tested. An insightful comparison to the scalar radiative transfer equation is also presented.

On improving the numerical convergence of highly nonlinear elasticity problems

Finite elasticity problems commonly include material and geometric nonlinearities and are solved using various numerical methods. However, for highly nonlinear problems, achieving convergence is relatively difficult and requires small load step sizes. In this work, we present a new method to transform the discretized governing equations so that the transformed problem has significantly reduced nonlinearity and, therefore, Newton solvers exhibit improved convergence properties. We study exponential-type nonlinearity in soft tissues and geometric nonlinearity in compression, and propose novel formulations for the two problems. We test the new formulations in several numerical examples and show significant reduction in iterations required for convergence, especially at large load steps. Notably, the proposed formulation is capable of yielding convergent solution even when 10–100 times larger load steps are applied. The proposed framework is generic and can be applied to other types of nonlinearities as well.

Stability equivalence between the neutral delayed stochastic differential equations and the Euler–Maruyama numerical scheme

In this paper, the equivalence theorem for the mean square exponential stability between the neutral delayed stochastic differential equations (NDSDEs) and the Euler–Maruyama numerical scheme is investigated via the continuous time Euler–Maruyama solutions. Firstly, with some preliminaries on basic notations and assumptions, we establish the approximation degree of the numerical scheme to the underlying NDSDE under the global Lipschitz condition for the dynamics and contractive mapping condition for the neutral operator of the equation, which guarantee the existence and uniqueness of the global solution. Then we show that the underlying NDSDE is exponentially stable in mean square if and only if, for some sufficiently small stepsize, the Euler–Maruyama numerical scheme is exponentially stable in mean square. With such a theoretical result, the mean square exponential stability of NDSDEs can be affirmed just by the simulation approach in practice. Finally, a constructive example is proposed to verify the theoretical result by simulation. Relatively, some analysis around the present topic will be given by remarks and some challenging problems for further works will be proposed in the conclusion section.

Node-Specific Diffusion LMS-Based Distributed Detection Over Adaptive Networks

Diffusion adaptation techniques have shown great promise in addressing the problem of node-specific distributed estimation where the nodes in the network are interested in different, possibly overlapping, sets of parameters. In this work, node-specific distributed detection, which has remained largely unexamined, is considered. In particular, the problem is formulated as one of binary hypothesis testing at each node for each of its parameters of interest. A distributed, online solution for this problem is sought based on diffusion adaptation techniques. In this setting, a signal to be detected by one node constitutes interference that may compromise the ability of the other nodes to detect their signals of interest reliably. Under mild assumptions on the data and network, it is shown that, for sufficiently small adaptation step-sizes, interference can be kept in check. Local detectors are developed where the test-statistics and thresholds adapt to changing conditions in real time. The distributed algorithm is analyzed; and its detection performance characterized and illustrated through numerical simulations.

Computationally Efficient Non-linear Electro-hydraulic Actuation System Model for Real-time Simulation

An electro-hydraulic actuation system with critically lapped servovalve and pressurized return line is exposited. The dynamic equations governing the behavior of various components are explained; using these nonlinear equations, a simulation model is developed in Simulink® software for the purpose of controller design and verification. Model requires very small integration step size as the equations involved are stiff. A relationship between the various pressures in the system is derived, and a simplified method for the nonlinear modeling of electro hydraulic actuating system is developed using these relationships. The simplified model is computationally efficient as the integration step-size is large compared to the original model. The computationally efficient model finds application in real-time hardware-in-loop simulation (HILS).