Parametric resonance in the problem of a heavy rigid body rolling along a straight line on a plane

The problem of the stability of a heavy rigid body, bounded by the surface of an ellipsoid and with a cavity in the form of a coaxial ellipsoid, rolling along a straight line on a horizontal rough plane is investigated. It is shown that in the case of a body that is close to being dynamically symmetrical, parametric resonance always occurs leading to instability of the rolling. Each ellipsoid has its own “individual” resonance angular velocity. In the general case, regions in which the necessary stability conditions are satisfied can be distinguished in parameter space. The problem of calculating the resonance coefficient corresponding to instability for parametric resonance in a reversible third-order system is solved.     

A general importance sampling algorithm for estimating portfolio loss probabilities in linear factor models

This paper develops a novel importance sampling algorithm for estimating the probability of large portfolio losses in the conditional independence framework. We apply exponential tilts to (i) the distribution of the natural sufficient statistics of the systematic risk factors and (ii) conditional default probabilities, given the simulated values of the systematic risk factors, and select parameter values by minimizing the Kullback–Leibler divergence of the resulting parametric family from the ideal (zero-variance) importance density. Optimal parameter values are shown to satisfy intuitive moment-matching conditions, and the asymptotic behaviour of large portfolios is used to approximate the requisite moments. In a sense we generalize the algorithm of Glasserman and Li (2005) so that it can be applied in a wider variety of models. We show how to implement our algorithm in the $t$ copula model and compare its performance there to the algorithm developed by Chan and Kroese (2010). We find that our algorithm requires substantially less computational time (especially for large portfolios) but is slightly less accurate. Our algorithm can also be used to estimate more general risk measures, such as conditional tail expectations, whereas Chan and Kroese (2010) is specifically designed to estimate loss probabilities.     

Parametric mixed-integer 0–1 linear programming: The general case for a single parameter

Two algorithms for the general case of parametric mixed-integer linear programs (MILPs) are proposed. Parametric MILPs are considered in which a single parameter can simultaneously influence the objective function, the right-hand side and the matrix. The first algorithm is based on branch-and-bound on the integer variables, solving a parametric linear program (LP) at each node. The second algorithm is based on the optimality range of a qualitatively invariant solution, decomposing the parametric optimization problem into a series of regular MILPs, parametric LPs and regular mixed-integer nonlinear programs (MINLPs). The number of subproblems required for a particular instance is equal to the number of critical regions. For the parametric LPs an improvement of the well-known rational simplex algorithm is presented, that requires less consecutive operations on rational functions. Also, an alternative based on predictor–corrector continuation is proposed. Numerical results for a test set are discussed.     

Parametric linear programming and cluster analysis

In the cluster analysis problem one seeks to partition a finite set of objects into disjoint groups (or clusters) such that each group contains relatively similar objects and, relatively dissimilar objects are placed in different groups. For certain classes of the problem or, under certain assumptions, the partitioning exercise can be formulated as a sequence of linear programs (LPs), each with a parametric objective function. Such LPs can be solved using the parametric linear programming procedure developed by Gass and Saaty [(Gass, S., Saaty, T. (1955), Naval Research Logistics Quarterly 2, 39–45)]. In this paper, a parametric linear programming model for solving cluster analysis problems is presented. We show how this model can be used to find optimal solutions for certain variations of the clustering problem or, in other cases, for an approximation of the general clustering problem.     

The effect of transverse crack upon parametric instability of a rotor-bearing system with an asymmetric disk

It is well known that either the asymmetric disk or transverse crack brings parametric inertia (or stiffness) excitation to the rotor-bearing system. When both of them appear in a rotor system, the parametric instability behaviors have not gained sufficient attentions. Thus, the effect of transverse crack upon parametric instability of a rotor-bearing system with an asymmetric disk is studied. First, the finite element equations of motion are established for the asymmetric rotor system. Both the open and breathing transverse cracks are taken into account in the model. Then, the discrete state transition matrix (DSTM) method is introduced for numerically acquiring the instability regions. Based upon these, some computations for a practical asymmetric rotor system with open or breathing transverse crack are conducted, respectively. Variations of the primary and combination instability regions induced by the asymmetric disk with the crack depth are observed, and the effect of the orientation angle between the crack and asymmetric disk on various instability regions are discussed in detail. It is shown that for the asymmetric angle around 0, the existence of transverse (either open or breathing) crack has attenuation effect upon the instability regions. Under certain crack depth, the instability regions could be vanished by the transverse crack. When the asymmetric angle is around π/2, increasing the crack depth would enhance the instability regions.     

Parametric continuation of the solitary traveling pulse solution in the reaction-diffusion system using the Newton-Krylov method

The matrix-free Newton-Krylov method that uses the GMRES algorithm (an iterative algorithm for solving systems of linear algebraic equations) is used for the parametric continuation of the solitary traveling pulse solution in a three-component reaction-diffusion system. Using the results of integration on a short time interval, we replace the original system of nonlinear algebraic equations by another system that has more convenient (from the viewpoint of the spectral properties of the GMRES algorithm) Jacobi matrix. The proposed parametric continuation proved to be efficient for large-scale problems, and it made it possible to thoroughly examine the dependence of localized solutions on a parameter of the model.     

Modelling the process of incoming problem reports on released software products

For big software developing companies, it is important to know the amount of problems of a new software product that are expected to be reported in a period after the date of release, on a weekly basis. For each of a number of past releases, weekly data are present on the number of such reports. Based on the type of data that is present, we construct a stochastic model for the weekly number of problems to be reported. The (non‐parametric) maximum likelihood estimator for the crucial model parameter, the intensity of an inhomogeneous Poisson process, is defined. Moreover, the expectation maximization algorithm is described, which can be used to compute this estimate. The method is illustrated using simulated data. Copyright © 2004 John Wiley & Sons, Ltd.     

Asymptotic Analysis of a Multiple Frequency Estimation Method

A recently proposed method of multiple frequency estimation for mixed-spectrum time series is analyzed. The so-called PF method is a procedure that combines the autoregressive (AR) representation of superimposed sinusoids with the idea of parametric filtering. The gist of the method is to parametrize a linear filter in accord with a certain parametrization property, as suggested by the particular form of the bias encountered by Prony′s least-squares estimator for the AR model. It is shown that for any parametric filter with this property, the least-squares estimator obtained from the filtered data is almost surely contractive as a function of the filter parameter and has a unique multivariate fixed-point in the vicinity of the true AR parameter. The fixed-point, known as the PF estimator, is shown to be stronly consistent for estimating the AR model, and the chronic bias of Prony′s estimator is thus eliminated. The almost sure convergence of an iterative algorithm that calculates the fixed-point and the asymptotic normality of the PF estimator are also established. The all-pole filter is considered as an example and application of the developed theory.     

On one method of finding the equilibrium state

An algorithm for solving a linear “production-exchange” model is described. The model reduces to a parametric model with strictly concave utility functions. Some properties of the parametric model are studied. Since the excessive demand map satisfies the discovered preference condition, it is possible to apply the second form of Chebyshev centers. For a sufficiently small parameter, the algorithm converges to the equilibrium state of the initial linear model. Bibliography:11 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 80, 1996, pp. 59–67.     

A branch-and-bound algorithm for 0–1 parametric mixed integer programming

A branch-and-bound algorithm to solve 0–1 parametric mixed integer linear programming problems has been developed. The present algorithm is an extension of the branch-and-bound algorithm for parametric analysis on pure integer programming. The characteristic of the present method is that optimal solutions for all values of the parameter can be obtained.     

On Solving Parametric Polynomial Systems

Border polynomial and discriminant variety are two important notions related to parametric polynomial system solving, in particular, for partitioning the parameter space into regions where the solutions of the system depend continuously on the parameter values. In this paper, we study the relations between those notions in the case of parametric triangular systems. We also investigate the properties and computation of the non-properness locus of the canonical projection restricted at a parametric regular chain or at its saturated ideal.     

A parameterized hessian quadratic programming problem

We present a general active set algorithm for the solution of a convex quadratic programming problem having a parametrized Hessian matrix. The parametric Hessian matrix is a positive semidefinite Hessian matrix plus a real parameter multiplying a symmetric matrix of rank one or two. The algorithm solves the problem for all parameter values in the open interval upon which the parametric Hessian is positive semidefinite. The algorithm is general in that any of several existing quadratic programming algorithms can be extended in a straightforward manner for the solution of the parametric Hessian problem.This research was supported by the Natural Sciences and Engineering Research Council under Grant No. A8189 and under a Postgraduate Scholarship, by an Ontario Graduate Scholarship, and by the University of Windsor Research Board under Grant No. 9432.     

A parameterized hessian quadratic programming problem

We present a general active set algorithm for the solution of a convex quadratic programming problem having a parametrized Hessian matrix. The parametric Hessian matrix is a positive semidefinite Hessian matrix plus a real parameter multiplying a symmetric matrix of rank one or two. The algorithm solves the problem for all parameter values in the open interval upon which the parametric Hessian is positive semidefinite. The algorithm is general in that any of several existing quadratic programming algorithms can be extended in a straightforward manner for the solution of the parametric Hessian problem. This research was supported by the Natural Sciences and Engineering Research Council under Grant No. A8189 and under a Postgraduate Scholarship, by an Ontario Graduate Scholarship, and by the University of Windsor Research Board under Grant No. 9432.     

Nonparametric conditional autoregressive expectile model via neural network with applications to estimating financial risk

The parametric conditional autoregressive expectiles (CARE) models have been developed to estimate expectiles, which can be used to assess value at risk and expected shortfall. The challenge lies in parametric CARE modeling is the specification of a parametric form. To avoid any model misspecification, we propose a nonparametric CARE model via neural network. The nonparametric CARE model can be estimated by a classical gradient based nonlinear optimization algorithm, and the consistency of nonparametric conditional expectile estimators is established. We then apply the nonparametric CARE model to estimating value at risk and expected shortfall of six stock indices. Empirical results for the new model is competitive with those classical models and parametric CARE models. Copyright © 2016 John Wiley & Sons, Ltd.     

非线性最优化一个超线收敛的可行下降算法

本文讨论非线性等式和不等式约束最优化的求解方法。首先将原问题扩充成一个只含不等式约束的参数规划，对于充分大的参数，扩充问题与原问题是等价的。然手建立具有以下特点的一个新算法。１）算法对扩充问题而言是可行下降的，参数只须自动调整有限次；２）每次迭代仅需解一个二次规划；３）在适当的假设下，算法超线性收敛于原问题的最优解。     

Stability analysis of a single-walled carbon nanotube conveying pulsating and viscous fluid with nonlocal effect

For a single-walled carbon nanotube (CNT) conveying fluid, the internal flow is considered to be pulsating and viscous, and the resulting instability and parametric resonance of the CNT are investigated by the method of averaging. The partial differential equation of motion based on the nonlocal elasticity theory is discretized by the Galerkin method and the averaging equations for the first two modes are derived. The stability regions in frequency–amplitude plane are obtained and the influences of nonlocal effect, viscosity and some system parameters on the stability of CNT are discussed in detail. The results show that decrease of nonlocal parameter and increase of viscous parameter both increase the fundamental frequency and critical flow velocity; the dynamic stability of CNT can be enhanced due to nonlocal effect; the contributions of the fluid viscosity on the stability of CNT depend on flow velocity; the axial tensile force can only influence natural frequencies of the system however the viscoelastic characteristic of materials can enhance the dynamic stability of CNT. The conclusions drawn in this paper are thought to be helpful for the vibration analysis and structural design of nanofluidic devices.     

Clusters, outliers, and regression: fixed point clusters

Fixed point clustering is a new stochastic approach to cluster analysis. The definition of a single fixed point cluster (FPC) is based on a simple parametric model, but there is no parametric assumption for the whole dataset as opposed to mixture modeling and other approaches. An FPC is defined as a data subset that is exactly the set of non-outliers with respect to its own parameter estimators. This paper concentrates upon the theoretical foundation of FPC analysis as a method for clusterwise linear regression, i.e., the single clusters are modeled as linear regressions with normal errors. In this setup, fixed point clustering is based on an iteratively reweighted estimation with zero weight for all outliers. FPCs are non-hierarchical, but they may overlap and include each other. A specification of the number of clusters is not needed. Consistency results are given for certain mixture models of interest in cluster analysis. Convergence of a fixed point algorithm is shown. Application to a real dataset shows that fixed point clustering can highlight some other interesting features of datasets compared to maximum likelihood methods in the presence of deviations from the usual assumptions of model based cluster analysis.     

Semi-Iterative Minimum Cross-Entropy Algorithms for Rare-Events,Counting, Combinatorial and Integer Programming

We present a new generic minimum cross-entropy method, called the semi-iterative MinxEnt, or simply SME, for rare-event probability estimation, counting, and approximation of the optimal solutions of a broad class of NP-hard linear integer and combinatorial optimization problems (COP’s). The main idea of our approach is to associate with each original problem an auxiliary single-constrained convex MinxEnt program of a special type, which has a closed-form solution. We prove that the optimal pdf obtained from the solution of such a specially designed MinxEnt program is a zero variance pdf, provided the “temperature” parameter is set to minus infinity. In addition we prove that the parametric pdf based on the product of marginals obtained from the optimal zero variance pdf coincides with the parametric pdf of the standard cross-entropy (CE) method. Thus, originally designed at the end of 1990s as a heuristics for estimation of rare-events and COP’s, CE has strong connection with MinxEnt, and thus, strong mathematical foundation. The crucial difference between the proposed SME method and the standard CE counterparts lies in their simulation-based versions: in the latter we always require to generate (via Monte Carlo) a sequence of tuples including the temperature parameter and the parameter vector in the optimal marginal pdf’s, while in the former we can fix in advance the temperature parameter (to be set to a large negative number) and then generate (via Monte Carlo) a sequence of parameter vectors of the optimal marginal pdf’s alone. In addition, in contrast to CE, neither the elite sample no the rarity parameter is needed in SME. As result, the proposed SME algorithm becomes simpler, faster and at least as accurate as the standard CE. Motivated by the SME method we introduce a new updating rule for the parameter vector in the parametric pdf of the CE program. We show that the CE algorithm based on the new updating rule, called the combined CE (CCE), is at least as fast and accurate as its standard CE and SME counterparts. We also found numerically that the variance minimization (VM)-based algorithms are the most robust ones. We, finally, demonstrate numerically that the proposed algorithms, and in particular the CCE one, allows accurate estimation of counting quantities up to the order of hundred of decision variables and hundreds of constraints. This research was supported by the Israel Science Foundation (grant No 191-565).     

Adaptive partitioning techniques for ordinary differential equations

Many stiff systems of ordinary differential equations (ODEs) modeling practical problems can be partitioned into loosely coupled subsystems. In this paper the objective of the partitioning is to permit the numerical integration of one time step to be performed as the solution of a sequence of small subproblems. This reduces the computational complexity compared to solving one large system and permits efficient parallel execution under appropriate conditions. The subsystems are integrated using methods based on low order backward differentiation formulas.This paper presents an adaptive partitioning algorithm based on a classical graph algorithm and techniques for the efficient evaluation of the error introduced by the partitioning.The power of the adaptive partitioning algorithm is demonstrated by a real world example, a variable step-size integration algorithm which solves a system of ODEs originating from chemical reaction kinetics. The computational savings are substantial. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65L06, 65Y05     

The theory of parametric oscillations

The correctness of the existing definitions of the parametric oscillations of linear and non-linear systems is discussed. The possibility of an erroneous choice of the mathematical parametric model instead of the autooscillatory model, connected with the existence in such systems of the same periodic solutions, is pointed out. Some non-local properties of parametric oscillations in Hamiltonian systems are established. It is shown, in particular, that stability regions are convex with respect to the frequency of the parametric excitation (i.e., all the points between the boundaries of neighbouring instability regions correspond to stable solutions). At the critical frequencies of parametric resonance the well-known Rayleigh and Zhuravlev theorems on the behaviour of the frequencies of natural oscillations when the stiffness and inertia changes are generalized. Some additional assertions on the limits of the first instability region for the Hill vector equations are established.