Generalized stochastic finite element method in elastic stability problems

The main issue of this paper is the stability analysis of elastic systems with random parameters using the Generalized Stochastic Finite Element Method. The Taylor expansion with random coefficients of nth order is used to express all random functions and to determine up to fourth order probabilistic moments of the critical force or critical pressure. The response function method assists to determine higher order partial derivatives of the structural response instead of the Direct Differentiation Method employed widely before. This approach is examined on the classical Euler problem, 2D and 3D steel frames as well as in addition to the cylindrical shell with some geometrical parameters defined as the Gaussian variables. The comparison of the GSFEM versus the Monte-Carlo simulation on the Euler problem proves the probabilistic convergence of this new technique.     

Multi-level Monte Carlo methods with the truncated Euler–Maruyama scheme for stochastic differential equations

The truncated Euler–Maruyama method is employed together with the Multi-level Monte Carlo method to approximate expectations of some functions of solutions to stochastic differential equations (SDEs). The convergence rate and the computational cost of the approximations are proved, when the coefficients of SDEs satisfy the local Lipschitz and Khasminskii-type conditions. Numerical examples are provided to demonstrate the theoretical results.     

Euler index of uncertain random graph: concepts and properties

We are usually in the state of indeterminacy. Uncertainty and randomness are two basic types of indeterminacy. In many cases, uncertainty and randomness exist simultaneously in a complex system. This paper considers the Euler tour of an uncertain random graph, in which some edges exist with some degrees in uncertain measure and others exist with some degrees in probability measure. In order to show how likely an uncertain random graph is Eulerian, an Euler index of an uncertain random graph is proposed first. Then a method to calculate the Euler index of an uncertain random graph is given. In addition, some properties of the Euler index are discussed.     

Convergence and stability of the semi-implicit Euler method for linear stochastic delay integro-differential equations

This paper deals with the convergence and stability of the semi-implicit Euler method for linear stochastic delay integro-differential equations. It is proved that the semi-implicit Euler method is convergent with strong order p=0.5. The condition under which the method is asymptotic mean square stable is determined and numerical experiments are presented.     

A note on convergence of semi-implicit Euler methods for stochastic pantograph equations

In the literature [1] [Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equation, J. Math. Anal. Appl. 325 (2007) 1142–1159], Fan and Liu investigated the existence and uniqueness of the solution for stochastic pantograph equation and proved the convergence of the semi-implicit Euler methods under the Lipschitz condition and the linear growth condition. Unfortunately, the main result of convergence derived by the conditions is somewhat restrictive for the purpose of practical application, because there are many stochastic pantograph equations that only satisfy the local Lipschitz condition. In this note we improve the corresponding results in the above-mentioned reference.     

连续时间 Hopfield网络模型数值实现分析

讨论使用Euler方法和梯形方法在数值求解连续时间的Hopfield网络模型时,离散时间步长的选择和迭代停止条件问题.利用凸函数的定义研究了能量函数下降的条件,根据凸函数的性质分析它的共轭函数减去二次函数之差仍为凸函数的条件.分析连续时间Hopfield网络模型的收敛性证明,提出了一个广义的连续时间Hopfield网络模型.对于常用的Euler方法和梯形方法数值求数值实现连续时间Hopfield网络,讨论了离散时间步长的选择.由于梯形方法为隐式方法,分析了它的迭代求算法的停止条件.根据连续时间Hopfield网络的特点,提出改进的迭代算法,并对其进行了分析.数值实验的结果表明,较大的离散时间步长不仅加速了数值实现,而且有利于提高优化性能.     

Euler method for solving hybrid fuzzy differential equation

In this paper, we study the numerical method for solving hybrid fuzzy differential using Euler method under generalized Hukuhara differentiability. To this end, we determine the Euler method for both cases of H-differentiability. Also, the convergence of the proposed method is studied and the characteristic theorem is given for both cases. Finally, some numerical examples are given to illustrate the efficiency of the proposed method under generalized Hukuhara differentiability instead of suing Hukuhara differentiability.     

Qualitative properties of stochastic iterative processes under random structural perturbations

In this work, under different modes of stochastic convergence, several convergence and stability results for stochastic iterative processes are developed. Difference inequalities and a comparison method in the context of Lyapunov-like functions are utilized. The presented method does not demand the knowledge of the probability distributions of solution processes. By decomposing random perturbations in nonlinear iterative processes into internal and external random perturbations, effects of these stochastic disturbances on the convergence and the stability of the the iterative processes are investigated. In fact, it is shown that the convergence and stability analysis is robust under random structural perturbations. The presented conditions are easy to verify, algebraically simple, and computationally attractive. The results provide new tests for distributed iterative processes in decentralized external regulation, adaptation, parameter estimation and the numerical analysis schemes.     

A continuous Hopfield network equilibrium points algorithm

The continuous Hopfield network (CHN) is a classical neural network model. It can be used to solve some classification and optimization problems in the sense that the equilibrium points of a differential equation system associated to the CHN is the solution to those problems. The Euler method is the most widespread algorithm to obtain these CHN equilibrium points, since it is the simplest and quickest method to simulate complex differential equation systems. However, this method is highly sensitive with respect to initial conditions and it requires a lot of CPU time for medium or greater size CHN instances. In order to avoid these shortcomings, a new algorithm which obtains one equilibrium point for the CHN is introduced in this paper. It is a variable time-step method with the property that the convergence time is shortened; moreover, its robustness with respect to initial conditions will be proven and some computational experiences will be shown in order to compare it with the Euler method.     

Steady-State Computations Using Summation-by-Parts Operators

This paper concerns energy stability on curvilinear grids and its impact on steady-state calulations. We have done computations for the Euler equations using fifth order summation-by-parts block and diagonal norm schemes. By imposing the boundary conditions weakly we obtain a fifth order energy-stable scheme. The calculations indicate the significance of energy stability in order to obtain convergence to steady state. Furthermore, the difference operators are improved such that faster convergence to steady state are obtained.     

Optimum transonic airfoils based on the Euler equations

We solve the problem of determining airfoils that approximate, in a least square sense, a given surface pressure distributions in transonic flight regimes. The flow is modeled by means of the Euler equations and the solution procedure is an adjoint-based minimization algorithm that makes use of the inverse Theodorsen transform in order to parameterize the airfoil. Fast convergence to the optimal solution is obtained by means of the pseudo-time method. Results are obtained using three different pressure distributions for several unperturbed flow conditions. The airfoils obtained have given a trailing edge angle.     

Adaptive Iteration to Steady State of Flow Problems

Runge–Kutta time integration is used to reach the steady state solution of discretized partial differential equations. Continuous and discrete parameters in the method are adapted to the particular problem by minimizing the residual in each step, if this is possible, or the work to reach convergence. Algorithms for parameter optimization are devised and analyzed. Solutions of the linearized Euler equations and the nonlinear Euler and Navier–Stokes equations for compressible flow illustrate the methods.     

A robust layer adapted difference method for singularly perturbed two-parameter parabolic problems

A finite difference method for a time-dependent singularly perturbed convection–diffusion–reaction problem involving two small parameters in one space dimension is considered. We use the classical implicit Euler method for time discretization and upwind scheme on the Shishkin–Bakhvalov mesh for spatial discretization. The method is analysed for convergence and is shown to be uniform with respect to both the perturbation parameters. The use of the Shishkin–Bakhvalov mesh gives first-order convergence unlike the Shishkin mesh where convergence is deteriorated due to the presence of a logarithmic factor. Numerical results are presented to validate the theoretical estimates obtained.     

Convergence Conditions for the Observed Mean Method in Stochastic Programming

The paper analyzes convergence conditions of the method of observed mean under nonstandard conditions, where dependent observations of random parameters are used and probabilistic optimization functions may be discontinuous indicators. For the case of dependent observations, large deviation type theorems for approximate optimal values and solutions are established.     

Convergence and stability of numerical methods with variable step size for stochastic pantograph differential equations

The stochastic pantograph equations (SPEs) are very special stochastic delay differential equations (SDDEs) with unbounded memory. When the numerical methods with a constant step size are applied to the pantograph equations, the most difficult problem is the limited computer memory. In this paper, we construct methods with variable step size to solve SPEs. The analysis is motivated by the example of a mean-square stable linear SPE for which the Euler–Maruyama (EM) method with variable step size fails to reproduce this behaviour for any nonzero timestep. Then we consider the Backward Euler (BE) method with variable step size and develop the fundamental numerical analysis concerning its strong convergence and mean-square linear stability. It is proved that the numerical solutions produced by the BE method with variable step size converge to the exact solution under the local Lipschitz condition and the Bounded condition. Furthermore, the order of convergence p=½ is given under the Lipschitz condition. The result of the mean-square linear stability is given. Some illustrative numerical examples are presented to demonstrate the order of strong convergence and the mean-square linear stability of the BE method.     

Stability in the numerical simulation of stochastic delayed Hopfield neural networks

This paper is concerned with the mean-square stability of the Split-Step Backward Euler method for stochastic delayed Hopfield neural networks. The sufficient conditions to guarantee the mean-square stability of the Split-Step Backward Euler method are given. Moreover, an example of the comparison of our method with the Euler–Maruyama method is used to show the superiority of our method.     

Least squares solutions of bilinear equations

The problem of finding a least squares solution for a system of bilinear equations is investigated. Sufficient conditions to have a unique minimum are given in the case of random inputs. Three methods, the normalized iterative method, the over-parameterization method and the numerical method are presented for solving the problem along with their convergence properties. Simulation examples are provided.     

A Very High-Order Accurate Staggered Finite Volume Scheme for the Stationary Incompressible Navier–Stokes and Euler Equations on Unstructured Meshes

We propose a sixth-order staggered finite volume scheme based on polynomial reconstructions to achieve high accurate numerical solutions for the incompressible Navier–Stokes and Euler equations. The scheme is equipped with a fixed-point algorithm with solution relaxation to speed-up the convergence and reduce the computation time. Numerical tests are provided to assess the effectiveness of the method to achieve up to sixth-order convergence rates. Simulations for the benchmark lid-driven cavity problem are also provided to highlight the benefit of the proposed high-order scheme.     

Accelerated convergence of Jameson's finite-volume Euler scheme using van der Houwen integrators

An efficient method for obtaining converged solutions to the time-dependent Euler equations has recently been proposed by Jameson. The convergence of the method is stabilised by using a four-stage Runge-Kutta scheme. In this paper the convergence is assessed of multistage algorithms which are of Runge-Kutta type, and are stable for larger time steps. This is done by means of numerical experiments using a coarse mesh. A six-stage algorithm is found to be best. It gives answers 15–20% more quickly than the standard method. No overheads in terms of increased storage are involved, and the results are indistinguishable at plotting accuracy from those obtained with the standard method.     

随机连续系统的连续时间最小二乘辨识的数值实现

讨论了随机连续系统的连续时间最小二乘（ＣＴＬＳ）辨识的数值实现及仿真，首先回顾了随机连续系统的ＣＴＬＳ辨识法和理论分析结果，然后基于数值积分技术和求解常微分方程的数值解欧拉法和龙格－库塔法，给出了ＣＴＬＳ法的两种数值实现方法，仿真结果显示出此方法的有效性。