A method for determining the equilibrium states of dynamic systems

A heuristic method is presented for determining the equilibrium states of motion of dynamic systems, in particular, spacecraft. The method can also be applied to the solution of sets of linear or nonlinear algebraic equations. A positive-semidefinite functional is formed to convert the problem to that of finding those minimum points where the functional vanishes. The process is initiated within a selecteddomain of interest by random search; convergence to a minimum is obtained by a modified Davidon's deflected gradient technique. To render this approach feasible in the presence of constraints, the functional is modified to include penalty terms which cause the functional to approach infinity at the constraint boundaries. Close approximations to solutions near the constraint boundaries are found by applying Carroll's approach in successively reducing the weighting factors of the penalty terms. After finding a minimum, the local domain around this point is eliminated by adding to the functional an interior constraint term, representing the surface under a hypersphere centered at the minimum point. The domain of consideration now becomes the subdomain formed by subtracting the space contained within this hypersphere from the previous domain of interest. Minima are now sought within the remaining space, as before.This paper is derived from research performed by the author while employed by TRW Systems Group, Redondo Beach, California.The author acknowledges the helpful suggestions of Dr. G. Bekey, University of Southern California, and those of Mr. E. A. Quast, Dr. M. P. Scher, and Dr. R. J. Wiley, Dynamics Department, TRW Systems Group, Redondo Beach, California.     

Computation of equilibrium measures

We present a new way of computing equilibrium measures numerically, based on the Riemann–Hilbert formulation. For equilibrium measures whose support is a single interval, the simple algorithm consists of a Newton–Raphson iteration where each step only involves fast cosine transforms. The approach is then generalized for multiple intervals.     

Topological dynamic consistency of non-standard finite difference schemes for dynamical systems

This work expands the mathematical theory which connects continuous dynamical systems and the discrete dynamical systems obtained from the associated numerical schemes. The problem is considered within the setting of Topological Dynamics. The topological dynamic consistency of a family of DDSs and the associated continuous system is defined as topological equivalence between the evolution operator of the continuous system and the set of maps defining the respective DDSs, for all positive time-step sizes. The one-dimensional theory is developed and a few important representative examples are studied in detail. It is found that the design of non-standard topologically dynamically consistent schemes requires some care.     

Multiscale variety in complex systems

The Law of Requisite Variety is a mathematical theorem relating the number of control states of a system to the number of variations in control that is necessary for effective response. The Law of Requisite Variety does not consider the components of a system and how they must act together to respond effectively. Here we consider the additional requirement of scale of response and the effect of coordinated versus uncoordinated response as a key attribute of complex systems. The components of a system perform a task, with a number of such components needed to act in concert to perform subtasks. We apply the resulting generalization—a Multiscale Law of Requisite Variety—to understanding effective function of complex biological and social systems. This allows us to formalize an understanding of the limitations of hierarchical control structures and the inadequacy of central control and planning in the solution of many complex social problems and the functioning of complex social organizations, e.g., the military, healthcare, and education systems. © 2004 Wiley Periodicals, Inc. Complexity 9: 37–45, 2004     

Smooth equilibrium measures and approximation

A necessary and sufficient condition is given for approximation with weighted expressions of the form wnPn, where w is a given continuous weight function and Pn are polynomials of degree . The condition is that the extremal measure that solves an associated equilibrium problem is smooth (asymptotically optimal doubling). As corollaries we get all previous (positive and negative) results for approximation, as well as the solution of a problem of T. Bloom and M. Branker. A connection to level curves of homogeneous polynomials of two variables is also explored.     

Convergence analysis for finite family of relatively quasi nonexpansive mappings and systems of equilibrium problems

In this paper, a new iterative scheme by hybrid method is constructed. Strong convergence of the scheme to a common element of the set of common fixed points of finite family of relatively quasi-nonexpansive mappings and set of common solutions of a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space is proved using the properties of generalized f-projection operator. Our results extend important recent results.     

Multiscale KAM theorem for Hamiltonian systems

In this paper, we study the persistence of invariant tori in nearly integrable multiscale Hamiltonian systems with highorder degeneracy in the integrable part. Such Hamiltonian systems arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem connects closely to the stability of the systems. We introduce multiscale nondegenerate condition and multiscale Diophantine condition, comparable to the usual Diophantine condition. Using quasilinear KAM method, we prove a multiscale KAM theorem.     

Multiscale finite element for problems with highly oscillatory coefficients

Summary. In this paper, we study a multiscale finite element method for solving a class of elliptic problems with finite number of well separated scales. The method is designed to efficiently capture the large scale behavior of the solution without resolving all small scale features. This is accomplished by constructing the multiscale finite element base functions that are adaptive to the local property of the differential operator. The construction of the base functions is fully decoupled from element to element; thus the method is perfectly parallel and is naturally adapted to massively parallel computers. We present the convergence analysis of the method along with the results of our numerical experiments. Some generalizations of the multiscale finite element method are also discussed. Received April 17, 1998 / Revised version received March 25, 2000 / Published online June 7, 2001     

Iterative approximation method for finite family of relatively quasi nonexpansive mappings and systems of equilibrium problems

In this paper, a new iterative scheme by hybrid method is constructed. Strong convergence of the scheme to a common element of the set of common fixed points of a finite family of relatively quasi-nonexpansive mappings and set of common solutions to a system of equilibrium problems in a uniformly convex real Banach space which is also uniformly smooth is proved. Our results extend important recent results.     

Multiscale failure Modeling of composites using generalized finite element method

In this work multiscale failure modeling of composites is made using generalized finite element method (GFEM). In this method the global approximation are constructed by combining the local basis with partition of unity functions. The enrichment functions for the GFEM approximation are computed using a proper orthogonal decomposition (POD) technique. The approximation is then used in a two scale Galerkin scheme for failure modeling of composites. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptation on the space of finite signed measures

Given an i.i.d. sample from a probability measure P on ℝ, an estimator is constructed that efficiently estimates P in the bounded-Lipschitz metric for weak convergence of probability measures, and, at the same time, estimates the density of P — if it exists (but without assuming it does) — at the best possible rate of convergence in total variation loss (that is, in L ¹-loss for densities).      

On equilibrium finite elements in three-dimensional case

The space of divergence-free functions with vanishing normal flux on the boundary is approximated by subspaces of finite elements that have the same property. The easiest way of generating basis functions in these subspaces is considered.     

Projected subgradient algorithms on systems of equilibrium problems

The paper proposes two parallel and cyclic algorithms for solving systems of equilibrium problems in Hilbert spaces. The algorithms combine two methods including the diagonal subgradient method and the projection method with parallel or cyclic computations. The obtained results can be considered as improvements over several previously known methods for systems of equilibrium problems in computational steps. The algorithms have also allowed to reduce several assumptions imposed on bifunctions. The strongly convergent theorems are established under suitable conditions.     

Gibbs and equilibrium measures for elliptic functions

Because of its double periodicity, each elliptic function canonically induces a holomorphic dynamical system on a punctured torus. We introduce on this torus a class of summable potentials. With each such potential associated is the corresponding transfer (Perron-Frobenius-Ruelle) operator. The existence and uniquenss of Gibbs states and equilibrium states of these potentials are proved. This is done by a careful analysis of the transfer operator which requires a good control of all inverse branches. As an application a version of Bowens formula for expanding elliptic maps is obtained.The research of the second author was supported in part by the NSF Grant DMS 0400481 and INT 0306004.     

Optimal dynamic equilibrium in insurance policies

Insurance companies sell contracts of various types each of them having a specific probability of return. Insurers may also own, at the same time, several insurance contracts which evolve through time. In this context, expectation and variance of the free reserves appear as functions of the number of customers in different classes as well as their evolution. Assuming that the customer system can be formulated as an open Markov one characterized by free entry, it is interesting to seek the optimal new customer distribution over the different customer classes j, which permits the minimization of the variance of free reserves for a desired average level of free reserves at a given time horizon. It is shown that, under some conditions, the customer system converges to an optimal growth steady state.     

Dual bounds on the equilibrium distribution of a finite Markov chain

Sharp variational expressions are given for the equilibrium state probabilities of a finite Markov chain, along with an iterative scheme that drives the upper and lower bounds inward and converging geometrically.