Lower Bounds to Eigenvalues for Vibrations and Buckling of Structures by Analytical Applications of a-priori Known Qualitative Information

Two methods to obtain lower bounds to eigenvalues are presented for cases which have equivalent minimum variational formulations. One method is an extension and elaboration of a theorem presented by the author in 1972, which affected the transfer of a weight function from one location to another over the physical system considered. The extension relies on information known a-priori about the exact solution of the problem, although the exact solution is not obtained. The other method is akin to the Rayleigh–Ritz method but yields lower bounds. The two methods are applied to various physical examples of vibrations and of buckling with rather good results. The application to other examples is direct and may be performed in a way quite similar to those examples shown.     

Invariant imbedding and the solution of Fredholm integral equations with displacement Kernels— comparative numerical experiments

This paper compares the relative efficiencies of the invariant imbedding method with the traditional solution techniques of successive approximations (Picard method), linear algebraic equations, and Sokolov's method of averaging functional corrections in solving numerically two representatives of a class of Fredholm integral equations. The criterion of efficiency is the amount of computing time necessary to obtain the solution to a specified degree of accuracy. The results of this computational investigation indicate that invariant imbedding has definite numerical advantages; more information was obtained in the same length of time as with the other methods, or even in less time. The conclusion emphasized is that a routine application of invariant imbedding may be expected to be computationally competitive with, if not superior to, a routine application of other methods for the solution of some classes of Fredholm integral equations.     

Multi-start methods for combinatorial optimization

Multi-start methods strategically sample the solution space of an optimization problem. The most successful of these methods have two phases that are alternated for a certain number of global iterations. The first phase generates a solution and the second seeks to improve the outcome. Each global iteration produces a solution that is typically a local optimum, and the best overall solution is the output of the algorithm. The interaction between the two phases creates a balance between search diversification (structural variation) and search intensification (improvement), to yield an effective means for generating high-quality solutions. This survey briefly sketches historical developments that have motivated the field, and then focuses on modern contributions that define the current state-of-the-art. We consider two categories of multi-start methods: memory-based and memoryless procedures. The former are based on identifying and recording specific types of information (attributes) to exploit in future constructions. The latter are based on order statistics of sampling and generate unconnected solutions. An interplay between the features of these two categories provides an inviting area for future exploration.     

Improving the computational efficiency in a global formulation (GLIDE) for interactive multiobjective optimization

In this paper, we present a new general formulation for multiobjective optimization that can accommodate several interactive methods of different types (regarding various types of preference information required from the decision maker). This formulation provides a comfortable implementation framework for a general interactive system and allows the decision maker to conveniently apply several interactive methods in one solution process. In other words, the decision maker can at each iteration of the solution process choose how to give preference information to direct the interactive solution process, and the formulation enables changing the type of preferences, that is, the method used, whenever desired. The first general formulation, GLIDE, included eight interactive methods utilizing four types of preferences. Here we present an improved version where we pay special attention to the computational efficiency (especially significant for large and complex problems), by eliminating some constraints and parameters of the original formulation. To be more specific, we propose two new formulations, depending on whether the multiobjective optimization problem to be considered is differentiable or not. Some computational tests are reported showing improvements in all cases. The generality of the new improved formulations is supported by the fact that they can accommodate six interactive methods more, that is, a total of fourteen interactive methods, just by adjusting parameter values.     

The Optimal Classification Using a Linear Discriminant for Two Point Classes Having Known Mean and Covariance

The current study provides a simple algorithm for finding the optimal ROC curve for a linear discriminant between two point distributions, given only information about the classes' means and covariances. The method makes no assumptions concerning the exact type of distribution and is shown to provide the best possible discrimination for any physically reasonable measure of the classification error. This very general solution is shown to specialise to results obtained in other papers which assumed multi-dimensional Gaussian distributed classes, or minimised the maximum classification error. Some numerical examples are provided which show the improvement in classification of this method over previously used methods.     

算子方程近似解直接方法的优化和信息复杂性研究进展

算子方程近似解直接方法的优化和信息复杂性是80年代发展起来的连续复杂性理论的两个主要方面,是计算机科学和数学的交叉研究领域．本文拟就这两个方面的研究进展做一简要介绍．     

Inverse problems of reconstructing parameters of the Navier-Stokes system

An inverse problem of reconstructing parameters not known a priori of the dynamical system described by the boundary-value problem for the Navier-Stokes system is considered. The reconstruction is based on one piece of admissible information or another about the motion of the dynamical system (solution of the corresponding boundary-value problem). In particular, one of the problems considered is the inverse problem consisting of reconstruction of the a priori unknown right-hand side of the Navier-Stokes system. The right-hand side characterizes the density of exterior mass forces acting on the system. This problem, as well as many other similar problems, is ill-posed. Two methods are proposed for its solution: the statistical method and the dynamical method. These methods use different initial information. In solving the problem by using the statistical method, initial information for the solution is the results of approximate measurement (in one sense or another) of the motion of the dynamical system on a given interval of time. Here, the reconstruction is performed after the corresponding interval of time. For solution of the problem by this method, the concepts and constructions of open-loop control theory are used. In solving the problem by using the dynamical method, initial information for its solution is the results of approximate (in one sense or another) measurements of the current states of the system, which are dynamically obtained by the observer. Here, the reconstruction is dynamically performed during the process. For solution of the problem by the dynamical method, the concepts and constructions of the dynamical regularization method based on positional control theory are used. Also, the author considers various modifications and regularizations of the methods for solution of problems proposed that are based on one piece of a priori information or another about the desired solution and solvability conditions of the problem. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 26, Nonlinear Dynamics, 2005.     

Generalization of Exp-function and other standard function methods

Exp-function and other standard function methods for solving nonlinear differential equations are generalized in this paper. An analytical criterion determining if a solution can be expressed in a form comprising standard functions is derived. New computational algorithm for automatic identification of the structure of the solution is constructed. The algorithm provides information if the solution can be expressed as a sum of standard functions, a ratio of sums of standard functions, or even a more complex algebraic form involving standard functions. Several examples are used to illustrate the proposed concept.     

Quantitative parametric connections between methods for generating noninferior solutions in multiobjective optimization

In multiobjective optimization, the trade-off information between different objective functions is probably the most important piece of information in a solution process to reach the most preferred solution. Generating methods are the methods in generating noninferior solutions. Different generating methods provide decision makers trade-off information in different forms. Various generating methods are characterized in this paper and the quantitative parametric connections between these generating methods are established. The result in this paper is then used to consolidate trade-off information with different formats associated with different generating methods.     

Unresolved computation and optimal predictions

We present methods for predicting the solution of time‐dependent partial differential equations when that solution is so complex that it cannot be properly resolved numerically, but when prior statistical information can be found. The sparse numerical data are viewed as constraints on the solution, and the gist of our proposal is a set of methods for advancing the constraints in time so that regression methods can be used to reconstruct the mean future. For linear equations we offer general recipes for advancing the constraints; the methods are generalized to certain classes of nonlinear problems, and the conditions under which strongly nonlinear problems and partial statistical information can be handled are briefly discussed. Our methods are related to certain data acquisition schemes in oceanography and meteorology. © John Wiley & Sons, Inc.     

Shooting method for the numerical solution of optimal control problems with bounded state variables

Abtract Various methods have been proposed for the numerical solution of optimal control problems with bounded state variables. In this paper, a new method is put forward and compared with two other methods, one of which makes use of adjoint variables whereas the other does not. Some conclusions are drawn on the usefulness of the three methods involved.     

一类Riccati方程组对称自反解的两种迭代算法

针对源于Markov跳变线性二次控制问题中的一类对偶代数Riccati方程组,分别采用修正共轭梯度算法和正交投影算法作为非精确Newton算法的内迭代方法,建立求其对称自反解的非精确Newton-MCG算法和非精确Newton-OGP算法.两种迭代算法仅要求Riccati方程组存在对称自反解,对系数矩阵等没有附加限定.数值算例表明,两种迭代算法是有效的.     

A novel car-following inertia gray model and its application in forecasting short-term traffic flow

Real-time and accurate short-term traffic flow prediction results can provide real-time and effective information for traffic information systems. Based on classic car-following models, this paper establishes differential equations according to the traffic state and proposes a car-following inertial gray model based on the information difference of the differential and gray system, in combination with the mechanical characteristics of traffic flow data and the characteristics of an inertial model. Furthermore, analytical methods are used to study the parameter estimation and model solution of the new model, and the important properties, such as the original data, inertia coefficient and simulation accuracy, are studied. The effectiveness of the model is verified in two cases. The performance of the model is better than that of six other prediction models, and the structural design of the new model is more reasonable than that of the existing gray models. Moreover, the new model is applied to short-term traffic flow prediction for three urban roads. The results show that the simulation and prediction effects of the model are better than those of other gray models. In terms of the traffic flow state, an optimal match between short-term traffic flow prediction and the new model is achieved.     

Stochastic convergence of random search methods to fixed size Pareto front approximations

In this paper we investigate to what extent random search methods, equipped with an archive of bounded size to store a limited amount of solutions and other data, are able to obtain good Pareto front approximations. We propose and analyze two archiving schemes that allow for maintaining a sequence of solution sets of given cardinality that converge with probability one to an ?-Pareto set of a certain quality, under very mild assumptions on the process used to sample new solutions. The first algorithm uses a hierarchical grid to define a family of approximate dominance relations to compare solutions and solution sets. Acceptance of a new solution is based on a potential function that counts the number of occupied boxes (on various levels) and thus maintains a strictly monotonous progress to a limit set that covers the Pareto front with non-overlapping boxes at finest resolution possible. The second algorithm uses an adaptation scheme to modify the current value of ? based on the information gathered during the run. This way it will be possible to achieve convergence to the best (smallest) ? value, and to a corresponding solution set of k solutions that ?-dominate all other solutions, which is probably the best possible result regarding the limit behavior of random search methods or metaheuristics for obtaining Pareto front approximations.     

Unconditionally stable modified methods for the solution of two‐ and three‐dimensional telegraphic equation with Robin boundary conditions

In this article, we discuss modified three level implicit difference methods of order two in time and four in space for the numerical solution of two‐ and three‐dimensional telegraphic equation with Robin boundary conditions. Ghost points are introduced to obtain fourth‐order approximations for boundary conditions. Matrix stability analysis is carried out to prove stability of the method for telegraphic equations in two and three dimensions with Neumann boundary conditions. Numerical experiments are carried out and the results are found to be better when compared with the results obtained by other existing methods.     

Integral-Functional Equation Arising in the Study of an Inverse Problem for a Quasilinear Hyperbolic Equation

A problem with data on the characteristics is considered for a quasilinear hyperbolic equation. The inverse problem of determining two unknown coefficients of the equation from some additional information about the solution is posed. One of the unknown coefficients depends on the independent variable, and the other, on the solution of the equation. Uniqueness theorems are proved for the solution of the inverse problem. The proof is based on the derivation of the integro-functional equation and the analysis of the uniqueness of its solution.     

Multiderivative methods of eighth algebraic order with minimal phase-lag for the numerical solution of the radial Schrödinger equation

Multiderivative methods with minimal phase-lag are introduced in this paper, for the numerical solution of the one-dimensional Schrödinger equation. The methods are called multiderivative since they use derivatives of order two, four or six. Numerical application of the newly introduced method to the resonance problem of the one-dimensional Schrödinger equation shows its efficiency compared with other similar well-known methods of the literature.     

基于结构元理论及信息熵的模糊多属性决策方法

主要目的是讨论当权重完全未知时的模糊多属性决策方法,其中运用了结构元理论和信息熵方法.首先,简单地介绍了这两种方法的基本原理.然后,提出了基于这两种方法的决策模型,给出了模糊权重函数的求解公式.最后,把该方法应用于考核、评估干部的问题.     

Corrected sequential linear programming for sparse minimax optimization

We present a new algorithm for nonlinear minimax optimization which is well suited for large and sparse problems. The method is based on trust regions and sequential linear programming. On each iteration a linear minimax problem is solved for a basic step. If necessary, this is followed by the determination of a minimum norm corrective step based on a first-order Taylor approximation. No Hessian information needs to be stored. Global convergence is proved. This new method has been extensively tested and compared with other methods, including two well known codes for nonlinear programming. The numerical tests indicate that in many cases the new method can find the solution in just as few iterations as methods based on approximate second-order information. The tests also show that for some problems the corrective steps give much faster convergence than for similar methods which do not employ such steps.Research supported partly by The Nordic Council of Ministers, The Icelandic Science Council, The University of Iceland Research Fund and The Danish Science Research Council.     

Query and construction of multi‐attribute social accounts

This paper surveys methods for query of systems of social accounts that are constructed from collections of intersecting data sets. Such query methods are used to ascertain, e.g., the number of people who possess certain attribute characteristics not explicitly provided by an original account or survey. Three approaches to the query problem are considered: (1) a generalized inverse solution suggested by Ijiri (1965); (2) a row reduced echelon matrix formulation devised by Hellerman and Cavallo (1977); and (3) a Boolean matrix operation introduced by Homer (1974). The latter two methods are extended to provide a general approach for using existing multi‐attribute information for the construction of new systems of social accounts.