Global sensitivity analysis using support vector regression

Global sensitivity analysis (GSA) plays an important role in exploring the respective effects of input variables on response variables. In this paper, a new kernel function derived from orthogonal polynomials is proposed for support vector regression (SVR). Based on this new kernel function, the Sobol’ global sensitivity indices can be computed analytically by the coefficients of the surrogate model built by SVR. In order to improve the performance of the SVR model, a kernel function iteration scheme is introduced further. Due to the excellent generalization performance and structural risk minimization principle, the SVR possesses the advantages of solving non-linear prediction problems with small samples. Thus, the proposed method is capable of computing the Sobol’ indices with a relatively limited number of model evaluations. The proposed method is examined by several examples, and the sensitivity analysis results are compared with the sparse polynomial chaos expansion (PCE), high dimensional model representation (HDMR) and Gaussian radial basis (RBF) SVR model. The examined examples show that the proposed method is an efficient approach for GSA of complex models.     

Sensitivity analysis of linear dynamical systems in uncertainty quantification

We consider linear dynamical systems including random parameters for uncertainty quantification. A sensitivity analysis of the stochastic model is applied to the input-output behaviour of the systems. Thus the parameters that contribute most to the variance are detected. Both intrusive and non-intrusive methods based on the polynomial chaos yield the required sensitivity coefficients. We use this approach to analyse a test example from electrical engineering. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Second Order Convergence of the Interpolation Based on $Q^c_1$-Element

In this paper, the second order convergence of the interpolation based on $Q^c_1$-element is derived in the case of $d$=1, 2 and 3. Using the integral average on each element, the new basis functions of tensor product type is builded up and we can easily extend it to the higher dimensional case. Finally, some numerical tests are made to show the analytical results of the interpolation errors.     

Simultaneous state‐time approximation of the chemical master equation using tensor product formats

We apply the novel tensor product formats (tensor train, quantized TT [QTT], and QTT‐Tucker) to the solution of d‐dimensional chemical master equations for gene regulating networks (signaling cascades, toggle switches, and phage‐ λ). For some important cases, for example, signaling cascade models, we prove analytical tensor product representations of the system operator. The quantized tensor representations (QTT, QTT‐Tucker) are employed in both state space and time, and the global state‐time (d + 1)‐dimensional system is solved in the tensor product form by the alternating minimal energy iteration, the ALS‐type algorithm. This approach leads to the logarithmic dependence of the computational complexity on the volume of the state space. We investigate the proposed technique numerically and compare it with the direct chemical master equation solution and some previously known approximate schemes, where possible. We observe that the newer tensor methods demonstrate a good potential in simulation of relevant biological systems. Copyright © 2014 John Wiley & Sons, Ltd.     

Fraeijs de Veubeke variational principle-based cylindrical shell finite element model

In this paper a dimensionally reduced cylindrical shell model based on the dual-mixed variational principle of Fraeijs de Veubeke will be presented. The fundamental variables of this variational principle are the not a priori symmetric stress tensor and the skew-symmetric rotation tensor. The tensor of first-order stress functions is applied to satisfy translational equilibrium. A shell model derived in this way makes the application of the classical kinematical hypotheses unnecessary, and enables us to use unmodified three-dimensional constitutive equations. On the basis of this shell model, a new dual-mixed cylindrical shell finite elements capable of both h- and p-approximation can be derived. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于有限元概念的解析解

基于有限元概念,本文研究了结构响应对于设计变量的严格解析解,以空间桁架为例,得到了位移显式的解析表达式——一个关于截面设计变量的有理函数,并进行了严格的数学论证;进而推出了有益于结构敏度分析、结构优化的若干结论,针对含有位移显式的优化模型,提出了采用广义几何规划解法的建议;最后计算了若干简例.     

Project dynamics and emergent complexity

This paper presents a theoretical analysis of project dynamics and emergent complexity in new product development (NPD) projects subjected to the management concept of concurrent engineering. To provide a comprehensive study, the complexity frameworks, theories and measures that have been developed in organizational theory, systematic engineering design and basic scientific research are reviewed. For the evaluation of emergent complexity in NPD projects, an information-theory quantity—termed “effective measure complexity” (EMC)—is selected from a variety of measures, because it can be derived from first principles and therefore has high construct validity. Furthermore, it can be calculated efficiently from dynamic generative models or purely from historical data, without intervening models. The EMC measures the mutual information between the infinite past and future histories of a stochastic process. According to this principle, it is particularly interesting to evaluate the time-dependent complexity in NPD and to uncover the relevant interactions. To obtain analytical results, a model-driven approach is taken and a vector autoregression (VAR) model of cooperative work is formulated. The formulated VAR model provided the foundation for the calculation of a closed-form solution of the EMC in the original state space. This solution can be used to analyze and optimize complexity based on the model’s independent parameters. Moreover, a transformation into the spectral basis is carried out to obtain more expressive solutions in matrix form. The matrix form allows identification of the surprisingly few essential parameters and calculation of two lower complexity bounds. The essential parameters include the eigenvalues of the work transformation matrix of the VAR model and the correlations between components of performance fluctuations.     

The Heston stochastic volatility model in Hilbert space

We extend the Heston stochastic volatility model to a Hilbert space framework. The tensor Heston stochastic variance process is defined as a tensor product of a Hilbert-valued Ornstein–Uhlenbeck process with itself. The volatility process is then defined by a Cholesky decomposition of the variance process. We define a Hilbert-valued Ornstein–Uhlenbeck process with Wiener noise perturbed by this stochastic volatility, and compute the characteristic functional and covariance operator of this process. This process is then applied to the modeling of forward curves in energy and commodity markets. Finally, we compute the dynamics of the tensor Heston volatility model when the generator is bounded, and study its projection down to the real line for comparison with the classical Heston dynamics.     

Design and simulation of FIR band pass and band stop filters using gravitational search algorithm

In this paper, a new optimization method named gravitational search algorithm (GSA) is adopted for designing optimal linear phase finite impulse response band pass (BP) and band stop (BS) digital filters. Other various population based evolutionary algorithms like real coded genetic algorithm, conventional particle swarm optimization, differential evolution (DE), bee swarm optimization have also been applied for the sake of comparative study of the same optimal designs. In GSA, particles are considered as objects and their performances are measured by their masses. All these objects attract each other by gravity forces, and these forces produce global movements of all objects towards the objects with heavier masses. GSA guarantees the exploitation step of the algorithm and it is apparently free from premature convergence. Extensive simulation results justify superior optimization capability of GSA over the afore-mentioned optimization techniques for the solution of the multimodal, non-differentiable, highly non-linear, and constrained filter design problems.     

Volterra series approximation for rational nonlinear system

Rational nonlinear systems are widely used to model the phenomena in mechanics, biology, physics and engineering. However, there are no exact analytical solutions for rational nonlinear system. Hence, the approximate analytical solutions are good choices as they can give the estimation of the states for system analysis, controller design and reduction. In this paper, an approximate analytical solution for rational nonlinear system is derived in terms of the solution of a polynomial system by Volterra series theory. The rational nonlinear system is transformed to a singular polynomial system with finite terms by adding some algebraic constraints related to the rational terms. The analytical solution of singular polynomial system is approximated by the summation of the solutions of Volterra singular subsystems. Their analytical solutions are derived by a novel regularization algorithm. The first fourth Volterra subsystems are enough to approximate the analytical solution to guarantee the accuracy. Results of numerical experiments are reported to show the effectiveness of the proposed method.     

A generalized separation for the variance contributions of input variables and their distribution parameters

For the structural system with both the uncertainties of input variables and their distribution parameters, this work investigates the generalized separation approach by transforming the original variable into the auxiliary variable with arbitrary distribution. Based on the variance based sensitivity analysis, the generalized sensitivity measures can be given, which are used to identify the influences of the auxiliary variables and distribution parameters simultaneously. For the different auxiliary variables, the variance contributions are proved to be identical, which illustrates the correctness of the generalized separation approach. Then the relationship of the variance contributions of original variables with those of the auxiliary variables and distribution parameters is investigated. Several examples are employed to demonstrate the rationality of the generalized separation approach.     

Krylov subspace methods to solve a class of tensor equations via the Einstein product

This paper is concerned with some of the well‐known iterative methods in their tensor forms to solve a class of tensor equations via the Einstein product. More precisely, the tensor forms of the Arnoldi and Lanczos processes are derived and the tensor form of the global GMRES method is presented. Meanwhile, the tensor forms of the MINIRES and SYMMLQ methods are also established. The proposed methods use tensor computations with no matricizations involved. Numerical examples are provided to illustrate the efficiency of the proposed methods and testify the conclusions suggested in this paper.     

Iterative adaptive RBF methods for detection of edges in two-dimensional functions

Radial basis functions have gained popularity for many applications including numerical solution of partial differential equations, image processing, and machine learning. For these applications it is useful to have an algorithm which detects edges or sharp gradients and is based on the underlying basis functions. In our previous research, we proposed an iterative adaptive multiquadric radial basis function method for the detection of local jump discontinuities in one-dimensional problems. The iterative edge detection method is based on the observation that the absolute values of the expansion coefficients of multiquadric radial basis function approximation grow exponentially in the presence of a local jump discontinuity with fixed shape parameters but grow only linearly with vanishing shape parameters. The different growth rate allows us to accurately detect edges in the radial basis function approximation. In this work, we extend the one-dimensional iterative edge detection method to two-dimensional problems. We consider two approaches: the dimension-by-dimension technique and the global extension approach. In both cases, we use a rescaling method to avoid ill-conditioning of the interpolation matrix. The global extension approach is less efficient than the dimension-by-dimension approach, but is applicable to truly scattered two-dimensional points, whereas the dimension-by-dimension approach requires tensor product grids. Numerical examples using both approaches demonstrate that the two-dimensional iterative adaptive radial basis function method yields accurate results.     

等距丢失模型下的框架张量积重构方法

框架理论常应用于信号重构.当编码系数在传输过程中发生等距丢失时,基于框架张量积的一些性质,我们可以利用框架张量积对信号进行编码从而降低数据丢失对重构信号的影响.本文由此提出了一种等距丢失模型,并在此模型下,研究了数据等距丢失下的最优对偶框架张量积,得出了对偶框架和正则对偶框架的张量积是最优对偶框架张量积的两个充分必要条件.最后数值实验也说明了：在等距丢失模型下,最优对偶框架张量积比一般对偶框架张量积的信号重构结果更优.     

强Khler-Finsler流形上(p,q)形式的中值Laplace算子(英文)

本文给出了强Khler-Finsler流形上中值Laplace算子的一些性质,如自伴性质,散度形式等。与Khler流形上利用逆变基本张量及其在Finsler流形上的变形作为密度函数定义流形上的逐点内积及整体内积不同,作者利用强Khler-Finsler流形上的逆变密切Khler度量作为密度函数定义了流形上的逐点内积和整体内积,并定义了强Khler-Finsler流形上的Hodge-Laplace算子,它可看作函数情形中值Laplace算子的推广。     

Diffusion of methylene blue in glass fibers – Application of the shrinking core model

Diffusion of mass in a solid cylinder with concentration dependent diffusivity (or temperature-dependent thermal conductivity in case of heat diffusion) does not admit of an analytical solution except in special cases. The ‘shrinking core model’ has been used to develop an approximate analytical solution in certain circumstances. The model, generally useful to describe heterogeneous solid–fluid reactions, is applied to theoretically analyze the adsorption–diffusion phenomena of methylene blue dye in a glass fiber in the present work. Theoretical equations have been derived for the case of diffusivity as an exponential function of concentration. The diffusivity parameters are evaluated by global minimization of the error between the experimental and the theoretical concentration history. Other forms of diffusivity, namely constant diffusivity and diffusivity varying linearly with concentration are found to involve larger errors. A parametric sensitivity analysis of the error has been done. The shrinking core model could satisfactorily interpret the experimental dye concentration profile in the substrate.     

Analysis of general second-order fluid flow in double cylinder rheometer

The fractional calculus approach in the constitutive relationship model of second-order fluid is introduced and the flow characteristics of the viscoelastic fluid in double cylinder rheometer are studied. First, the analytical solution of which the derivative order is 1/2 is derived with the analytical solution and the reliability of Laplace numerical inversion based on Crump algorithm for the problem is verified, then the characteristics of second-order fluid flow in the rheometer by using Crump method is analyzed. The results indicate that the more obvious the viscoelastic properties of fluid are, the more sensitive the dependence of velocity and stress on fractional derivative order is.     

基于Kriging模型的供应链优化设计

供应链优化的目的之一是确定使得总成本最小的最佳运作水平.供应链系统是复杂的动态系统,由于库存系统的复杂性和供应链本身的不确定性,利用传统优化方法往往需要耗费一定的计算成本和经济成本.而元模型则能以简单的数学表达式较精确地刻画仿真系统的输入输出关系,为研究者分析复杂系统提供了一种分析方法.针对供应链优化问题,给出了一种基于计算机试验设计中的元模型一Kriging模型的供应链优化方法,并通过一个三级供应链问题对所提方法进行了实证研究.研究结果表明了所提方法的有效性和可用性,为供应链优化提供了一种新的研究思路.     

A generalized Borgonovo's importance measure for fuzzy input uncertainty

In this work, the way in which fuzzy uncertainty in a model's output is apportioned to fuzzy uncertainty in model inputs is studied through a sensitivity analysis. Here, an optimization technique is employed to obtain the membership functions of the fuzzy structural response, for which a global sensitivity indicator is introduced to measure the influence of fuzzy input uncertainty on fuzzy output uncertainty. The global sensitivity indicator is the important measure of the fuzzy input uncertainty, which extends Borgonovo's measure. In this study, the mathematical properties of the important measure of the fuzzy input uncertainty are discussed and proved. The results of numerical examples and engineering examples show that the proposed importance measure can effectively describe the effect of fuzzy input uncertainty on fuzzy structural response. When the sensitivity indicator is larger, the basic fuzzy-valued variable becomes more important. The sensitivity indicators of the fuzzy structural response can give an essential importance sequence of all the basic fuzzy-valued variables and identify key contributing fuzzy-valued variables. The sensitivity indicators can provide the availability guidance to reduce the number of basic variables and optimize the fuzzy response model.     

Multi-valued characteristics and Morse decompositions

A rapid growth of molecular and systems biology in recent years challenges mathematicians to develop robust modeling and analytical tools for this area.We combine a theory of monotone input-output systems with a classical theory of Morse decompositions in the context of ordinary differential equations models of biochemical reactions. We show that a multi-valued input-output characteristic can be used to define non-trivial Morse decompositions which provide information about a global structure of the attractor. The previous work on input-output characteristics is shown to apply locally to individual Morse sets and is seamlessly incorporated into our global theory.We apply our tools to a model of cell cycle maintenance. We show that changing the strength of the negative feedback loop can lead to cessation of cell cycle in two different ways: it can either lead to globally attracting equilibrium or to a pair of equilibria that attract almost all solutions.