Validation and error estimation of computational models

This paper develops a Bayesian methodology for assessing the confidence in model prediction by comparing the model output with experimental data when both are stochastic. The prior distribution of the response is first computed, which is then updated based on experimental observation using Bayesian analysis to compute a validation metric. A model error estimation methodology is then developed to include model form error, discretization error, stochastic analysis error (UQ error), input data error and output measurement error. Sensitivity of the validation metric to various error components and model parameters is discussed. A numerical example is presented to illustrate the proposed methodology.     

Novel Computing for the Delay Differential Two-Prey and One-Predator System

The aim of these investigations is to find the numerical performances of the delay differential two-prey and one-predator system. The delay differential models are very significant and always difficult to solve the dynamical kind of ecological nonlinear two-prey and one-predator system. Therefore, a stochastic numerical paradigm based artificial neural network (ANN) along with the Levenberg-Marquardt backpropagation (L-MB) neural networks (NNs), i.e., L-MBNNs is proposed to solve the dynamical two-prey and one-predator model. Three different cases based on the dynamical two-prey and one-predator system have been discussed to check the correctness of the L-MBNNs. The statistic measures of these outcomes of the dynamical two-prey and one-predator model are chosen as 13% for testing, 12% for authorization and 75% for training. The exactness of the proposed results of L-MBNNs approach for solving the dynamical two-prey and one-predator model is observed with the comparison of the Runge-Kutta method with absolute error ranges between 10⁻⁰⁵ to 10⁻⁰⁷. To check the validation, constancy, validity, exactness, competence of the L-MBNNs, the obtained state transitions (STs), regression actions, correlation presentations, MSE and error histograms (EHs) are also provided.     

Error estimation and model adaptation for a stochastic‐deterministic coupling method based on the Arlequin framework

The paper deals with the issue of accuracy for multiscale methods applied to solve stochastic problems. It more precisely focuses on the control of a coupling, performed using the Arlequin framework, between a deterministic continuum model and a stochastic continuum one. By using residual‐type estimates and adjoint‐based techniques, a strategy for goal‐oriented error estimation is presented for this coupling and contributions of various error sources (modeling, space discretization, and Monte Carlo approximation) are assessed. Furthermore, an adaptive strategy is proposed to enhance the quality of outputs of interest obtained by the coupled stochastic‐deterministic model. Performance of the proposed approach is illustrated on 1D and 2D numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

Stochastic Investigations for the Fractional Vector-Host Diseased Based Saturated Function of Treatment Model

The goal of this research is to introduce the simulation studies of the vector-host disease nonlinear system (VHDNS) along with the numerical treatment of artificial neural networks (ANNs) techniques supported by Levenberg-Marquardt backpropagation (LMQBP), known as ANNs-LMQBP. This mechanism is physically appropriate, where the number of infected people is increasing along with the limited health services. Furthermore, the biological effects have fading memories and exhibit transition behavior. Initially, the model is developed by considering the two and three categories for the humans and the vector species. The VHDNS is constructed with five classes, susceptible humans , infected humans , recovered humans , infected vectors , and susceptible vector based system of the fractional-order nonlinear ordinary differential equations. To solve the number of variations of the VHDNS, the numerical simulations are performed using the stochastic ANNs-LMQBP. The achieved numerical solutions for solving the VHDNS using the stochastic ANNs-LMQBP have been described for training, verifying, and testing data to decrease the mean square error (MSE). An extensive analysis is provided using the correlation studies, MSE, error histograms (EHs), state transitions (STs), and regression to observe the accuracy, efficiency, expertise, and aptitude of the computing ANNs-LMQBP.     

时效性商品最优定价模型及其粒子群优化解

对时效性商品的定价问题进行了研究.基于一种负二项分布的离散需求函数,并在利润最大化原则下,建立了时效商品最优定价模型.由于该模型涉及多个随机变量的概率分布,常规函数极值算法难以获得问题解析解,引入粒子群优化算法,对模型进行演化求解,并给出算例分析.结果表明:利用粒子群算法,可以快速有效得到不同库存量情况下应采取的最优定价.最后提出需要进一步解决的若干问题.     

An Intelligence Computational Approach for the Fractional 4D Chaotic Financial Model

The main purpose of the study is to present a numerical approach to investigate the numerical performances of the fractional 4-D chaotic financial system using a stochastic procedure. The stochastic procedures mainly depend on the combination of the artificial neural network (ANNs) along with the Levenberg-Marquardt Backpropagation (LMB) i.e., ANNs-LMB technique. The fractional-order term is defined in the Caputo sense and three cases are solved using the proposed technique for different values of the fractional order α. The values of the fractional order derivatives to solve the fractional 4-D chaotic financial system are used between 0 and 1. The data proportion is applied as 73%, 15%, and 12% for training, testing, and certification to solve the chaotic fractional system. The acquired results are verified through the comparison of the reference solution, which indicates the proposed technique is efficient and robust. The 4-D chaotic model is numerically solved by using the ANNs-LMB technique to reduce the mean square error (MSE). To authenticate the exactness, and consistency of the technique, the obtained performances are plotted in the figures of correlation measures, error histograms, and regressions. From these figures, it can be witnessed that the provided technique is effective for solving such models to give some new insight into the physical behavior of the model.     

An RKPM-based formulation of the generalized probability density evolution equation for stochastic dynamic systems

In the stochastic dynamic analysis, the probability density evolution method (PDEM) provides an optional way to capture the complete probability distribution of the stochastic response of general nonlinear systems. In the PDEM, the key point is to solve the generalized probability density evolution equation (GDEE), which governs the evolution of the joint probability density function (PDF) of the response and the randomness. In this paper, a new numerical method based on the reproducing kernel particle method (RKPM) is proposed. The GDEE can be approximated through the RKPM. By some particles in the response domain, the instantaneous PDF and its partial derivative with respect to response are smoothly expressed. Then, the approximated GDEE can be discretized directly at the collocation points in the response domain. At the same time, discretization in the time domain is achieved by the difference scheme. Therefore, the RKPM-based formulation to obtain the numerical solution of GDEE is formed. The implementation procedure of the proposed method is given in detail. The accuracy and efficiency of this method are illustrated with some numerical examples. Some details of parameter analysis are also discussed.     

单自由度随机滞回系统的振动响应分析

在Bouc—Wen滞回模型的基础上，根据求解随机非线性结构动力学的二阶矩法，推导出求解由滞回环本身的随机性而引起的、单自由度随机滞回系统的响应的有效数值方法，得到了系统响应的均值和标准差。并采用Monte—Carlo数值模拟法对系统的响应进行模拟分析，其计算结果与数值模拟结果相基本吻合。从而解决了由滞回环本身的随机性引起的非线性振动系统的随机响应问题。     

谱表示法模拟风场的误差分析

研究了原型谱表示法模拟的非各态历经性多变量风场的统计矩的时域估计值和目标值之间误差的概率描述。基于原型谱表示法的模拟公式，以三变量风场为例，导出了模拟结果的均值、相关函数、功率谱密度函数和根方差等四项统计特征的单样本时域估计表达式，它们是随机变量或随机过程。运用概率论的计算方法，推导出了上述随机变量或过程的前二阶矩的解析表达式，得到了模拟风场的统计特征时域估计的偏度误差和随机误差。将三变量过程的结果加以推广，给出了误差计算的通式。通过算例中统计误差值和理论误差值的对比，验证解析解的正确性。探讨了可能的降低随机误差的方法。求得的误差闭合解将有利于结合误差传播理论进行可靠性分析。     

Optimization of multi-product batch plant design under uncertainty with environmental considerations

This paper presents a framework for the design and optimization of multi-product batch processes under uncertainty with environmental considerations. The uncertainties and environmental impacts are discussed. The profit and environmental impacts are considered as bi-objectives for batch plant design. The problem, thus, is formulated as a multi-objective stochastic programming problem. It can be converted into a single-objective two-stage stochastic linear programming problem using the weighted aggregation method. To solve the two-stage stochastic programming, we introduce both Monte Carlo sampling for the entire domain of the distribution function and the feasibility cut method based on dual theory in Benders’ decomposition. The detailed algorithm for problem-solving is presented. A numerical example is presented to illustrate the proposed framework.     

Intelligent Networks for Chaotic Fractional-Order Nonlinear Financial Model

The purpose of this paper is to present a numerical approach based on the artificial neural networks (ANNs) for solving a novel fractional chaotic financial model that represents the effect of memory and chaos in the presented system. The method is constructed with the combination of the ANNs along with the Levenberg-Marquardt backpropagation (LMB), named the ANNs-LMB. This technique is tested for solving the novel problem for three cases of the fractional-order values and the obtained results are compared with the reference solution. Fifteen numbers neurons have been used to solve the fractional-order chaotic financial model. The selection of the data to solve the fractional-order chaotic financial model are selected as 75% for training, 10% for testing, and 15% for certification. The results indicate that the presented approximate solutions fit exactly with the reference solution and the method is effective and precise. The obtained results are testified to reduce the mean square error (MSE) for solving the fractional model and verified through the various measures including correlation, MSE, regression histogram of the errors, and state transition (ST).     

基于分片响应面的钢筋混凝土梁变形随机模拟

将响应面与蒙特卡洛方法结合,可降低随机模拟的计算代价。但当钢筋混凝土梁所受荷载接近开裂荷载时,梁体的变形响应面形状将出现不光滑的现象。采用传统的单一响应面方法来描述结构响应将与实际产生明显的偏差。采用改进的分片响应面技术结合蒙特卡洛方法,建立了钢筋混凝土梁变形随机分析模型。以钢筋混凝土试验梁作为算例,讨论了分片参数的合理取值,并验证了方法的正确性。     

Fractional Order Nonlinear Bone Remodeling Dynamics Using the Supervised Neural Network

This study aims to solve the nonlinear fractional-order mathematical model (FOMM) by using the normal and dysregulated bone remodeling of the myeloma bone disease (MBD). For the more precise performance of the model, fractional-order derivatives have been used to solve the disease model numerically. The FOMM is preliminarily designed to focus on the critical interactions between bone resorption or osteoclasts (OC) and bone formation or osteoblasts (OB). The connections of OC and OB are represented by a nonlinear differential system based on the cellular components, which depict stable fluctuation in the usual bone case and unstable fluctuation through the MBD. Untreated myeloma causes by increasing the OC and reducing the osteoblasts, resulting in net bone waste the tumor growth. The solutions of the FOMM will be provided by using the stochastic framework based on the Levenberg-Marquardt backpropagation (LVMBP) neural networks (NN), i.e., LVMBPNN. The mathematical performances of three variations of the fractional-order derivative based on the nonlinear disease model using the LVMPNN. The static structural performances are 82% for investigation and 9% for both learning and certification. The performances of the LVMBPNN are authenticated by using the results of the Adams-Bashforth-Moulton mechanism. To accomplish the capability, steadiness, accuracy, and ability of the LVMBPNN, the performances of the error histograms (EHs), mean square error (MSE), recurrence, and state transitions (STs) will be provided.     

Stochastic response of nonlinear system in probability domain

A stochastic averaging procedure for obtaining the probability density function (PDF) of the response for a strongly nonlinear single-degree-of-freedom system, subjected to both multiplicative and additive random excitations is presented. The procedure uses random Van Der Pol transformation, Ito’s equation of limiting diffusion process and stochastic averaging technique as outlined by Zhu and others. However, the equations are rederived in generalized form and arranged in such a way that the procedure lends itself to a numerical computational scheme using FFT. The main objective of the modification is to consider highly irregular nonlinear functions which cannot be integrated in closed form and also to solve problems where analytical expressions for probability density function cannot be obtained. The procedure is applied to obtain the PDF of the response of Duffing oscillator subjected to additive and multiplicative random excitations represented by rational power spectral density functions (PSDFs). The results are verified by digital simulation. It is shown that the procedure provides results which compare very well with those obtained from simulation analysis not only for wide-band excitations but also for very narrow-band excitations, which are weak (when normalized with respect to mass of the system.) This paper is dedicated to Prof R N Iyengar of the Indian Institute of Science on the occasion of his formal retirement.     

Enhancing piecewise-defined surrogate response surfaces with adjoints on sets of unstructured samples to solve stochastic inverse problems

Many approaches for solving stochastic inverse problems suffer from both stochastic and deterministic sources of error. The finite number of samples used to construct a solution is a common source of stochastic error. When computational models are expensive to evaluate, surrogate response surfaces are often employed to increase the number of samples available for approximating the solution. This leads to a reduction in finite sampling errors while the deterministic error in the evaluation of each sample is potentially increased. The pointwise accuracy of sampling the surrogate is primarily impacted by two sources of deterministic error: the local order of accuracy in the surrogate and the numerical error from the numerical solution of the model. In this work, we use adjoints to simultaneously give a posteriori error and derivative estimates in order to construct low-order, piecewise-defined surrogates on sets of unstructured samples. Several examples demonstrate the computational gains of this approach in obtaining accurate estimates of probabilities for events in the design space of model input parameters. This lays the groundwork for future studies on goal-oriented adaptive refinement of such surrogates.     

Simulation of random fields on random domains

This paper focuses on the simulation of random fields on random domains. This is an important class of problems in fields such as topology optimization and multiphase material analysis. However, there is still a lack of effective methods to simulate this kind of random fields. To this end, we extend the classical Karhunen–Loève expansion (KLE) to this class of problems, and we denote this extension as stochastic Karhunen–Loève expansion (SKLE). We present three numerical algorithms for solving the stochastic integral equations arising in the SKLE. The first algorithm is an extension of the classical Monte Carlo simulation (MCS), which is used to solve the stochastic integral equation on each sampled domain. However, such approach demands remeshing each sampled domain and solving the corresponding integral equation, which can become computationally very demanding. In the second algorithm, a domain transformation is used to map the random domain into a reference domain, and only one mesh for the reference domain is required. In this way, remeshing different sample realizations of the random domain is avoided and much computational effort is thus saved. MCS is then adopted to solve the corresponding stochastic integral equation. Further, to avoid the computational effort of MCS, the third algorithm proposed in this contribution involves a reduced-order method to solve the stochastic integral equation efficiently. In this third algorithm, stochastic eigenvectors are represented as a sum of products of unknown random variables and deterministic vectors, where the deterministic vectors are efficiently computed by solving deterministic eigenvalue problems. The random variables and stochastic eigenvalues that appear in this third algorithm are calculated by a reduced-order stochastic eigenvalue problem constructed by the obtained deterministic vectors. Based on the obtained stochastic eigenvectors, the target random field is then simulated and reformulated as a classical KLE-like representation. Finally, three numerical examples are presented to demonstrate the performance of the proposed methods.     

结构随机响应分析的一种直接积分法

对结构随机响应分析的数值积分方法进行了深入的研究。首先，直接从控制方程出发计算加速度，其次，给出了位移和速度的一种近似计算公式，最后，推导出响应均方值的离散计算表达式。就两个算例进行了数值仿真，结果表明该方法对于结构的随机响应分析是有效的。     

色噪声与确定性谐波联合激励下Bouc-Wen动力系统响应的统计线性化方法

提出了一种用于求解色噪声和确定性谐波联合作用下单自由度Bouc?Wen系统响应的统计线性化方法.基于系统响应可分解为确定性谐波和零均值随机分量之和的假定,将原滞回运动方程等效地化为两组耦合的且分别以确定性和随机动力响应为未知量的非线性微分方程.利用谐波平衡法求解确定性运动方程,利用统计线性化方法求解色噪声激励下的随机运...     

概率密度演化方程差分格式的计算精度及初值条件改进

由于随机荷载作用下工程结构响应的一阶时间导数往往大于结构响应自身，采用仅满足确定性结构响应分析计算精度的时间步长求解概率密度演化方程(Probability density evolution equation, PDEE)时，总变差减小(Total variation diminishing, TVD)性质差分法计算的结构响应标准差通常无法满足精度要求。该文分别对比了不同差分时间步长下单边差分、Lax-wendroff (L-W)双边差分和TVD差分三种PDEE数值求解方法的计算精度。为提升TVD差分的求解精度，该文提出了差分时间步长的合理选取方法和正态分布型初值条件，并通过数值算例验证了时间步长选取方法和正态分布型初值条件的准确性及普适性。数值算例结果表明:L-W双边差分法的样本离散性最小，误差允许范围内可采用的时间差分步长最大；当仅关注均值和标准差等统计指标时，建议使用L-W双边差分法；利用该文方法可有效降低TVD差分所带来的数值误差；正态分布型初值条件的标准差大小等于空间离散步长时，可以获得最小均值误差。     

Dynamic Response Analysis of Fuzzy Stochastic Truss Structures under Fuzzy Stochastic Excitation

A novel method (Fuzzy factor method) is presented, which is used in the dynamic response analysis of fuzzy stochastic truss structures under fuzzy stochastic step loads. Considering the fuzzy randomness of structural physical parameters, geometric dimensions and the amplitudes of step loads simultaneously, fuzzy stochastic dynamic response of the truss structures is developed using the mode superposition method and fuzzy factor method. The fuzzy numerical characteristics of dynamic response are then obtained by using the random variable’s moment method and the algebra synthesis method. The influences of the fuzzy randomness of structural physical parameters, geometric dimensions and step load on the fuzzy randomness of the dynamic response are demonstrated via an engineering example, and Monte-Carlo method is used to simulate this example, verifying the feasibility and validity of the modeling and method given in this paper.