Convergence of a canonical expansion for normal fields

In the present study we consider a normal separable stochastic continuous field, and we prove the convergence of a Karhunen series with probability 1 for all parameter values. This leads in particular, to the nonrandomness of points of the discontinuity and values of the discontinuity. A criterion is presented for the convergence of the canonical expansion in a uniform norm.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 565–572, October, 1973.The author thanks S. A. Molchanov for guidance with the work and Yu. K. Belayev for help and valuable instruction.     

Partitions and the grand canonical ensemble

Two disconnected remarks about partitions. First, a pedagogical remark connecting pure mathematics with statistical physics. The grand canonical ensemble of statistical mechanics is applied to the counting of partitions. This picture borrowed from physics gives a simple approximation to the exact calculation of the partition function by Hardy and Ramanujan. Second, an exact formula is guessed for the function N _S (m,n) defined in a recent paper by Andrews, Garvan, and Liang. The formula was subsequently proved by Garvan. We hope that it may lead to a better understanding of the beautiful new congruence properties of partitions discovered by Andrews.     

Convergence rates for Kaczmarz-type algorithms

By making use of the normal and skew-Hermitian splitting (NSS) method as the inner solver for the modified Newton method, we establish a class of modified Newton-NSS method for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. Under proper conditions, the local convergence theorem is proved. Furthermore, the successive-overrelaxation (SOR) technique has been proved quite successfully in accelerating the convergence rate of the NSS or the Hermitian and skew-Hermitian splitting (HSS) iteration method, so we employ the SOR method in the NSS iteration, and we get a new method, which is called modified Newton SNSS method. Numerical results are given to examine its feasibility and effectiveness.     

Outer approximation algorithms for canonical DC problems

The paper discusses a general framework for outer approximation type algorithms for the canonical DC optimization problem. The algorithms rely on a polar reformulation of the problem and exploit an approximated oracle in order to check global optimality. Consequently, approximate optimality conditions are introduced and bounds on the quality of the approximate global optimal solution are obtained. A thorough analysis of properties which guarantee convergence is carried out; two families of conditions are introduced which lead to design six implementable algorithms, whose convergence can be proved within a unified framework.     

Convergence of algorithms for finding eigenvectors

1. IntroductionMethods for finding eigenvectors and eigenvalues of a matrix have importallt applications in colltrol theory, pattern recognition, signal processing and many other fields. Withthese applications the computational methods themselves have been developing rapidly. See[1--4] for some developments.For a square matriX of order n, when n is sufficiently large, it is very difficult to findits eigenvalues directly and one usually uses methods of iteration. Esseniiajly, we can evensac th…     

Convergence bounds for nonlinear programming algorithms

Bounds on convergence are given for a general class of nonlinear programming algorithms. Methods in this class generate at each interation both constraint multipliers and approximate solutions such that, under certain specified assumptions, accumulation points of the multiplier and solution sequences satisfy the Fritz John or the Kuhn—Tucker optimality conditions. Under stronger assumptions, convergence bounds are derived for the sequences of approximate solution, multiplier and objective function values. The theory is applied to an interior—exterior penalty function algorithm modified to allow for inexact subproblem solutions. An entirely new convergence bound in terms of the square root of the penalty controlling parameter is given for this algorithm.     

Convergence theorems for inertial KM-type algorithms

This paper deals with the convergence analysis of a general fixed point method which unifies KM-type (Krasnoselskii–Mann) iteration and inertial type extrapolation. This strategy is intended to speed up the convergence of algorithms in signal processing and image reconstruction that can be formulated as KM iterations. The convergence theorems established in this new setting improve known ones and some applications are given regarding convex feasibility problems, subgradient methods, fixed point problems and monotone inclusions.     

Information theoretic interpretation of Forte's entropy for the grand canonical ensemble

Summary Forte's characterization of the entropy of a grand canonical ensemble in statistical mechanics has a natural interpretation in information theory. This entropy is also shown to be finite under the natural assumption that the average message length be finite. In the proof, an analogue of a well-known inequality of Shannon is obtained.

---

Riassunto La caratterizzazione di Forte dell'entropia di un insieme gran canonico in meccanica statistica viene interpretata in teoria dell'informazione. Si mostra anche che questa entropia è finita nell'ipotesi naturale che la lunghezza media del messaggio sia finita. Nella dimostrazione si ottiene una disuguaglianza analoga a quella ben nota di Shannon.

---

---

The author is a member of G.N.A.F.A. of the Italian C.N.R.

---

An erratum to this article is available at .     

One-particle distribution density in the grand canonical ensemble

The grand canonical ensemble of single-component systems of particles in a region is considered. It is shown that if the distribution density of one particle tends to a finite limit as the diameter of tends to infinity then under fairly general conditions this limit is represented by the function that, as is shown in an earlier paper of the author, is, under certain conditions, the analytic continuation of the Mayer expansion representing the density as a function of the activity.All-Russia Correspondence Institute of the Food Industry. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No.1, pp. 44–58, July, 1994.     

Convergence of some algorithms for convex minimization

We present a simple and unified technique to establish convergence of various minimization methods. These contain the (conceptual) proximal point method, as well as implementable forms such as bundle algorithms, including the classical subgradient relaxation algorithm with divergent series.An important research work of Phil Wolfe's concerned convex minimization. This paper is dedicated to him, on the occasion of his 65th birthday, in appreciation of his creative and pioneering work.     

Convergence of algorithms for perturbed optimization problems

Infinite-dimensional optimization problems occur in various applications such as optimal control problems and parameter identification problems. If these problems are solved numerically the methods require a discretization which can be viewed as a perturbation of the data of the optimization problem. In this case the expected convergence behavior of the numerical method used to solve the problem does not only depend on the discretized problem but also on the original one. Algorithms which are analyzed include the gradient projection method, conditional gradient method, Newton's method and quasi-Newton methods for unconstrained and constrained problems with simple constraints.     

Method of collective variables with references system for the grand canonical ensemble

L'vov Branch of Statistical Physics, Institute of Theoretical Physics, Ukrainian SSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 79, No. 2, pp. 282–296, May, 1989.     

Convergence behavior of decomposition algorithms for linear programs

This paper discusses plausible explanations of the somewhat folkloric, ‘tailing off’ convergence behavior of the Dantzig-Wolfe decomposition algorithm for linear programs. Is is argued that such beahvior may be used to numerical inaccuracy. Procedures to identify and mitigate such difficulties are outlined.     

Convergence proof of minimization algorithms for nonconvex functions

In this paper, we prove a theorem of convergence to a point for descent minimization methods. When the objective function is differentiable, the convergence point is a stationary point. The theorem, however, is applicable also to nondifferentiable functions. This theorem is then applied to prove convergence of some nongradient algorithms.     

Convergence of interval-type algorithms for generalized fractional programming

The purpose of this paper is to analyze the convergence of interval-type algorithms for solving the generalized fractional program. They are characterized by an interval [LB _k , UB _k ] including*, and the length of the interval is reduced at each iteration. A closer analysis of the bounds LB _k and UB _k allows to modify slightly the best known interval-type algorithm NEWMODM accordingly to prove its convergence and derive convergence rates similar to those for a Dinkelbach-type algorithm MAXMODM under the same conditions. Numerical results in the linear case indicate that the modifications to get convergence results are not obtained at the expense of the numerical efficiency since the modified version BFII is as efficient as NEWMODM and more efficient than MAXMODM.This research was supported by NSERC (Grant A8312) and FCAR (Grant 0899).