Dynamical level set method for parameter identification of nonlinear parabolic distributed parameter systems

This article considers a dynamical level set method for the identification problem of the nonlinear parabolic distributed parameter system, which is based on the solvability and stability of the direct PDE (partial differential equation) in Sobolev space. The dynamical level set algorithms have been developed for ill-posed problems in Hilbert space. This method can be regarded as a asymptotical regularization method as long as a certain stopping rule is satisfied. Hence, the convergence analysis of the method is established similar to the proof of convergence of asymptotical regularization. The level set converges to a solution as the artificial time evolves to infinity. Furthermore, the proposed level set method is proved to be stable by using Lyapunov stability theorem, which is constructed in my previous article.Numerical tests are discussed to demonstrate the efficacy of the dynamical level set method, which consequently confirm the level set method to be a powerful tool for the identification of the parameter.     

Method of particular solutions for linear,two-point boundary-value problems

The methods commonly employed for solving linear, two-point boundary-value problems require the use of two sets of differential equations: the original set and the derived set. This derived set is the adjoint set if the method of adjoint equations is used, the Green's functions set if the method of Green's functions is used, and the homogeneous set if the method of complementary functions is used.With particular regard to high-speed digital computing operations, this paper explores an alternate method, the method of particular solutions, in which only the original, nonhomogeneous set is used. A general theory is presented for a linear differential system ofnth order. The boundary-value problem is solved by combining linearly several particular solutions of the original, nonhomogeneous set. Both the case of an uncontrolled system and the case of a controlled system are considered.This research, supported by the NASA-Manned Spacecraft Center, Grant No. NGR-44-006-089, is a condensed version of the investigations described in Refs. 1 and 2.     

不确定性理论集对分析在海水水质富营养化评价中的应用

集对分析是处理不确定性问题的新的系统理论方法.利用该理论,建立了进行海水水质富营养化综合评价的新方法.实例研究表明,集对分析评价海水水质的富营养化程度是切实可行的,与人工神经网络方法和支持向量机方法相比,集对分析方法概念清晰,计算简便、快捷、精度高,具有较高的分辨率和较大的实用性,减少了评价过程中的人为主观因素;具有不遗失数据中间信息、评价结果与实际情况更为相符的优点,为海水环境质量综合评价提供了一种简单而适用的评价方法.     

基于多项式组主项解耦消元法的几何定理机器证明

基于多项式组主项解耦消元法 ,将几何定理的假设条件 (多项式组 PS)化为主项只含主变元的三角型多项式组 DTS,可得到定理命题成立的不含变元的非退化条件 ,即充分必要或更接近充分必要的非退化条件 .由于多项式主系数不含变元 ,已不存在 DTS多项式之间的约化问题 ,故方法有普遍意义 .文中例为西姆松定理的机器证明 .     

Visualization of Multivariate Density Estimates With Level Set Trees

This article presents a method for visualization of multivariate functions. The method is based on a tree structure—called the level set tree—built from separated parts of level sets of a function. The method is applied for visualization of estimates of multivarate density functions. With different graphical representations of level set trees we may visualize the number and location of modes, excess masses associated with the modes, and certain shape characteristics of the estimate. Simulation examples are presented where projecting data to two dimensions does not help to reveal the modes of the density, but with the help of level set trees one may detect the modes. I argue that level set trees provide a useful method for exploratory data analysis.     

An approximation algorithm for convex multi-objective programming problems

In multi-objective convex optimization it is necessary to compute an infinite set of nondominated points. We propose a method for approximating the nondominated set of a multi-objective nonlinear programming problem, where the objective functions and the feasible set are convex. This method is an extension of Benson’s outer approximation algorithm for multi-objective linear programming problems. We prove that this method provides a set of weakly ε-nondominated points. For the case that the objectives and constraints are differentiable, we describe an efficient way to carry out the main step of the algorithm, the construction of a hyperplane separating an exterior point from the feasible set in objective space. We provide examples that show that this cannot always be done in the same way in the case of non-differentiable objectives or constraints.     

Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems

The subgradient extragradient method can be considered as an improvement of the extragradient method for variational inequality problems for the class of monotone and Lipschitz continuous mappings. In this paper, we propose two new algorithms as combination between the subgradient extragradient method and Mann-like method for finding a common element of the solution set of a variational inequality and the fixed point set of a demicontractive mapping.     

Stability analysis of probabilistic Boolean networks with switching topology

This paper investigates the set stability of probabilistic Boolean networks (PBNs) with switching topology. To deal with this problem, two novel concepts, set reachability and the largest invariant set family, are defined. By constructing an auxiliary system, the necessary and sufficient conditions for verifying set reachability are given and the calculation method for the largest invariant set family is obtained. Based on these two results, an equivalent condition of set stability is derived, which can be used to determine whether a PBN with switching topology can be stabilized to a given set. In addition, the design method of switching signal is proposed by combining the characteristic of the largest invariant set family, and a numerical example is reported to demonstrate the efficiency of presented approach.     

Equivalent weights for lexicographic multi-objective programs: Characterizations and computations

A recent paper [19] demonstrated the existence of a set of equivalent weights for which the optimal solution set of a preemptive priority multi-objective program is precisely equal to the set of optimal solutions to the resulting nonpreemptive program with the objective function given as a linear weighting of the multiple objectives. This paper addresses two further issues. Firstly, for some important special cases or applications, it is demonstrated that not only is the computation of a set of equivalent weights feasible, but it is also highly desirable. Two algorithms are presented to compute a set of equivalent weights. One method is a direct specialization of the approach adopted in [19], whereas the second approach is an alternative technique. The latter method is shown to yield weights of uniformly smaller values than the former method, while being of the same computational complexity, and is hence preferable. Secondly, as opposed to constructing one vector of equivalent weights, a characterization is provided for the entire set of equivalent weights.     

Randomized methods based on new Monte Carlo schemes for control and optimization

We address randomized methods for control and optimization based on generating points uniformly distributed in a set. For control systems this sets are either stability domain in the space of feedback controllers, or quadratic stability domain, or robust stability domain, or level set for a performance specification. By generating random points in the prescribed set one can optimize some additional performance index. To implement such approach we exploit two modern Monte Carlo schemes for generating points which are approximately uniformly distributed in a given convex set. Both methods use boundary oracle to find an intersection of a ray and the set. The first method is Hit-and-Run, the second is sometimes called Shake-and-Bake. We estimate the rate of convergence for such methods and demonstrate the link with the center of gravity method. Numerical simulation results look very promising.     

多维数据群点主轴旋转的预测模型研究

数据群点的主轴表示该数据群点分布变异最大的若干方向，它是反映数据群点分布构造的主要特征之一。本论文提出一种对高维空间里数据群点主轴旋转运动的预测建模方法，并将其用于中国城市经济发展预测分析，可用于推测城市群体经济发展的主要特征方向     

多项式方程组的主项解耦消元法

本文提出多项式组符号求解的主项解耦 (主项只含主元 )消元法 :视多项式为变元不同幂积的线性组合 ,以主项解耦三角型多项式组 DTS为引导 ,用逐项伪除求余式 ,将多项式组 PS化为与其同解的 DTS.内容涉及 :消元算法、DTS的存在性与结构特性、零点集结构公式等 .亦对 Grobner基法、吴文俊消元法与本文方法之间的相互联系、区别以及特点进行了比较 .研究表明主项解耦消元法适用于一般多项式组且效率较高     

A comparison of two methods for determining the weights of belonging to fuzzy sets

Saaty has solved a basic problem in fuzzy set theory using an eigenvector method to determine the weights of belonging of each member to the set. In this paper, a weighted least-square method is utilized to obtain the weights. This method has the advantage that it involves the solution of a set of simultaneous linear algebraic equations and is thus conceptually easier to understand than the eigenvector method. Examples are given for estimating the relative wealth of nations and the relative amount of foreign trade of nations. Numerical solutions are obtained using both the eigenvector method and the weighted least-square method, and the results are compared.Partial support of this research was provided by NIH Grant No. GM-23732.     

基于TOPSIS思想的技术创新方案的Vague集评价方法及其应用

为了得到理想的技术创新选择方案,论文从技术创新的实际出发,从实力、潜力和外部环境三个角度选择了一组反映技术创新项目实际情况的评价指标。并借助Vague集理论,通过TOPSIS中正负理想解的思想定义了任意创新方案的评价指标值,运用Vague集理论中距离的概念构建了新的评价方法,进而得到了满意的评价结果。该方法一方面体现了Vague集理论描述不确定性问题的优点,另一方面借助TOPSIS思想满足了方案评价的基本准则。这一思想和方法的提出不仅为Vague集理论研究提供了新的应用领域,也为技术创新的研究提供了新的思路和方法。     

Dynamization of order decomposable set problems

Order decomposable set problems are introduced as a general class of problems concerning sets of multidimensional objects. We give a method of structuring sets such that the answer to an order decomposable set problem can be maintained with low worst-case time bounds, while objects are inserted and deleted in the set. Examples include the maintenance of the two-dimensional Voronoi diagram of a set of points, and of the convex hull of a three-dimensional point set in linear time. We show that there is a strong connection between the general dynamization method given and the well-known technique of divide-and-conquer used for solving many problems concerning static sets of objects. Although the upper bounds obtained are low in order of magnitude, the results do not necessarily imply the existence of fast feasible update routines. Hence the results merely assess theoretical bounds for the various set problems considered.     

Extragradient method for solving quasivariational inequalities

We study methods for solving a class of the quasivariational inequalities in Hilbert space when the changeable set is described by translation of a fixed, closed and convex set. We consider one variant of the gradient-type projection method and an extragradient method. The possibilities of the choice of parameters of the gradient projection method in this case are wider than in the general case of a changeable set. The extragradient method on each iteration makes one trial step along the gradient, and the value of the gradient at the obtained point is used at the first point as the iteration direction. In the paper, we establish sufficient conditions for the convergence of the proposed methods and derive a new estimate of the rates of the convergence. The main result of this paper is contained in the convergence analysis of the extragradient method.     

Multiple goal non-cooperative conflicts under uncertainty: A fuzzy set approach

This paper investigates strategy selection for a participant in a two-party non-cooperative conflict which involves both uncertainty and multiple goals. Uncertainty arises from the players not knowing the utility functions. Multiple objectives appear as the result of the payoff being a vector of prizes and the players attempt to attain various goals for each prize separately. The main objective is to present a fuzzy set/fuzzy programming solution concept to the conflict situation. In doing so, we compare a Bayesian player to one that employs fuzzy set techniques. We point out some of the advantages of the fuzzy set method. The necessary computations in the fuzzy set method are explained in detail through an example.     

模糊集对分析用于脱硫除尘器的综合评价

提出了锅炉脱硫除尘器综合评价的模糊集对分析方法.评述了方法的基本原理和计算步骤,通过实例介绍了方法的应用.评价实例结果表明模糊集对分析用于脱硫除尘器综合评价是可行的.与密切值法相比,所得评价结果相同.该法具有计算简便和实用的特点.