Showing papers on "Convergence (routing) published in 1978"

Stochastic Approximation Methods for Constrained and Unconstrained Systems

Abstract: I. Introduction.- 1.1. General Remarks.- 1.2. The Robbins-Monro Process.- 1.3. A "Continuous" Process Version of Section 2.- 1.4. Regulation of a Dynamical System a simple example.- 1.5. Function Minimization: The Kiefer-Wolfowitz Procedure.- 1.6. Constrained Problems.- 1.7. An Economics Example.- II. Convergence w.p.1 for Unconstrained Systems.- 2.1. Preliminaries and Motivation.- 2.2. The Robbins-Monro and Kiefer-Wolfowitz Algorithms: Conditions and Discussion.- 2.3. Convergence Proofs for RM and KW-like Procedures.- 2.3.1. A Basic RM-like Procedure.- 2.3.2. One Dimensional RM and Accelerated RM Procedures.- 2.3.3. A Continuous Parameter RM Procedure.- 2.3.4. The Basic Kiefer-Wolfowitz Procedure.- 2.3.5. Random Directions KW Methods.- 2.4. A General Robbins-Monro Process: "Exogenous Noise".- 2.4.1. The Case of Bounded h(*,*).- 2.4.2. Unbounded h(*,*): Exogenous Noise.- 2.5. A General RM Process State Dependent Noise.- 2.5.1. Extensions and Localizations of Theorem 2.5.2.- 2.6. Some Applications.- 2.7. Mensov-Rademacher Estimates.- III. Weak Convergence of Probability Measures.- IV. Weak Convergence for Unconstrained Systems.- 4.1. Conditions and General Discussion.- 4.2. The Robbins-Monro and Kiefer-Wolfowitz Procedures.- 4.2.1. The Basic Robbins-Monro Procedure.- 4.2.2. The One-Dimensional Robbins-Monro Procedure.- 4.2.3. The Kiefer-Wolfowitz Procedure.- 4.2.4. A Case Where the Limit Satisfies a Generalized ODE.- 4.2.5. A Continuous Parameter KW Procedure.- 4.3. A General Robbins-Monro Process: Exogenous Noise.- 4.4. A General RM Process: State Dependent Noise.- 4.5. The Identification Problem.- 4.6. A Counter-Example to Tightness.- 4.7. Boundedness of {Xn} and Tightness of {Xn(*)}.- V. Convergence w.p.1 For Constrained Systems.- 5.1. A Penalty-Multiplier Algorithm for Equality Constraints.- 5.1.1. A Basic RM-like Algorithm, Conditions and Discussion.- 5.1.2. The Noise Condition, Discussion and Generalization.- 5.1.3. Boundedness of {Xn}.- 5.1.4. Proof of the Main Theorem.- 5.1.5. Constrained Function Minimization and Other Extensions.- 5.2. A Lagrangian Method for Inequality Constraints.- 5.2.1. The Algorithm and Conditions.- 5.2.2. The Convergence Theorem 18.- 5.2.3. A Non-Convergent but Useful Algorithm.- 5.2.4. An Application to the Identification Problem.- 5.3. A Projection Algorithm.- 5.4. A Penalty-Multiplier Method for Inequality Constraints.- VI. Weak Convergence: Constrained Systems.- 6.1. A Multiplier Type Algorithm for Equality Constraints.- 6.1.1. Boundedness of {Xn}.- 6.1.2. The Noise Condition, Discussion.- 6.1.3. The Convergence Theorem.- 6.2. The Lagrangian Method.- 6.3. A Projection Algorithm.- 6.4. A Penalty-Multiplier Algorithm for Inequality Constraints.- VII. Rates of Convergence.- 7.1. The Problem Formulation.- 7.2. Conditions and Discussions.- 7.3. Rates of Convergence for Case 1, the KW Algorithm.- 7.4. Discussion of Rates of Convergence for Two KW Algorithms.

Quasi‐steady‐state approximations in air pollution modeling: Comparison of two numerical schemes for oxidant prediction

Abstract: A numerical method to solve the set of differential equations which describes the chemical development in polluted air is presented. The photochemical lifetimes, continuously monitored for all compounds, determine how the integrations are performed at all times. Components with lifetimes less than 10% of the time step, which is taken as 30 sec, are assumed to be in photochemical equilibrium, while compounds with photochemical lifetimes greater than 100 times the time step are computed according to Euler's method. All other components are calculated from the exponential solution of the continuity equation. The computational accuracy may be improved by iteration on components assumed to be determined by the instant values of other components. The convergence of the iteration is speeded up by ordering the short-lived compounds in a hierarchical sequence. Since computational errors connected with QSSA methods are difficult to assess, comparison with an automatic scheme is necessary. Our method has been compared with Gear's method for a range of model mixtures of hydrocarbons, nitrogen oxides, and air, thought to cover most conceivable situations of atmospheric pollution. The agreement with Gear's method is within a few percent for all components all the time, in most cases even within 1%. No accumulation of deviations occur during long-term integrations (e.g., 48 hr with day and night shiftings), and differences which appear during periods with strong concentration gradients (e.g., after sunrise) vanish when the activity has culminated. The method presented here is considerably more efficient than Gear's method, with respect to both computer time and storage.

A Fast Load Flow Method Retaining Nonlinearity

Abstract: One of the most recognized load flow methods at present is the N-R(Newton-Raphson) method [1] with the sparsity techniques and suboptimal ordering [2]. This paper presents a new AC load flow method which is several to more than ten times faster than the N-R method and has the same memory requirement, mathematical complexity and accuracy as the N-R method. As an example, the convergence time for 118 bus system was only one eighth of the N-R method.

On spectral approximation. Part 1. The problem of convergence

Abstract: — One studies the problem of the numerical approximation of the spectrum of noncompact operators in Banach spaces. Special results are derived for the self adjoint case. An example is presented.

Stochastic Convergence of Weighted Sums of Random Elements in Linear Spaces

Abstract: General introduction.- Mathematical preliminaries.- Random elements in linear spaces.- Laws of large numbers, uncorrelation, and convergence of weighted sums of random variables.- Laws of large numbers in normed linear spaces.- Convergence of weighted sums in normed linear spaces.- Randomly weighted sums.- Laws of large numbers in D[0,1].- Possible applications.

On Newton-Iterative Methods for the Solution of Systems of Nonlinear Equations

Abstract: In this paper we consider the local rates of convergence of Newton-iterative methods for the solution of systems of nonlinear equations. We show that under certain conditions on the inner, linear i...

Strong Convergence of a Stochastic Approximation Algorithm

Abstract: Convergence with probability one of a recursive stochastic approximation algorithm is considered. Some extensions of previous results for the Robbins-Monro and the Kiefer-Wolfowitz procedures are given. An inportant feature of the approach taken here is that the convergence analysis can be directly extended to more complex algorithms.

The Convergence of General Step-Length Algorithms for Regular Optimum Design Criteria

Abstract: For a regular optimality criterion function $\Phi$, a sequence of design measures $\{\xi_n\}$ is generated using the iteration $\xi_{n+1} = (1 - \alpha_n)\xi_n + \alpha_n\xi_n$, where $\xi_n$ is chosen to minimize $ abla \Phi(M(\xi_n), M(\xi))$ over all $\xi$ and $\{\alpha_n\}$ is a prescribed sequence of numbers from (0, 1). This is called a general step-length algorithm for $\Phi$. Typical conditions on $\{\alpha_n\}$ are $\alpha_n \rightarrow 0$ and $\Sigma_n\alpha_n = \infty$. In this paper, a dichotomous behavior of $\{\xi_n\}$ is proved under the above conditions on $\{\alpha_n\}$ for $\Phi$ satisfying some mild regularity conditions. Sufficient conditions for convergence to optimal designs are also established. This can be applied to show that the $\{\xi_n\}$ as constructed above do converge to an optimal design for most of the trace-related and determinant-related design criteria.

Numerical solutions of coupled Burgers' equation

Abstract: Two new algorithms based on cubic spline function technique are proposed for solving Burgers' equation in one space variable and coupled Burgers' equation in two space variables. The algorithms have been analysed for their stability and convergence. Two test examples have been solved for illustrating the merits of the proposed numerical method. The method can be extended for solving non-linear problems arising in mechanics and other areas.

A Quasi‐Bayes Sequential Procedure for Mixtures

Abstract: SUMMARY Coherent Bayes sequential learning and classification procedures are often useless in practice because of ever-increasing computational requirements. On the other hand, computationally feasible procedures may not resemble the coherent solution, nor guarantee consistent learning and classification. In this paper, a particular form of classification problem is considered and a "quasi-Bayes" approximate solution requiring minimal computation is motivated and defined. Convergence properties are established and a numerical illustration provided.

Orthogonal Filters for Model Error Compensation in the Control of Nonrigid Spacecraft

Abstract: This paper seeks to improve the convergence properties of state estimators as used in spacecraft controller designs, when the linearized models upon which the estimators are based are subject to parameter errors, truncated modes, and neglected disturbances. Instead of choosing mode shapes (which are orthogonal in space) multiplied by time varying coefficients (a conventional approach for modeling elastic modes) to represent the truncated modes, the ''model error vector" discussed herein is approximated, over short "observation windows," T units long, by functions which are orthogonal over the time interval T, where the coefficients of the orthogonal functions are automatically updated via the use of real-time measurements from the system. The device which updates the coefficients of the orthogonal functions is called an orthogonal filter and takes on the form of a state estimator for the synthetic modes of a "model error system" which generates the orthogonal functions. The method is illustrated for a 14th-order model of a flexible spacecraft, resulting in 2nd-, 3rd-, and 4th-order controllers.

Considerations for efficient wire/surface modeling

Abstract: The most significant aspects of a moment method surface patch/wire formulation are speed, accuracy, convergence, and versatility. Techniques for improving these parameters are discussed and applied to a solution based on the piecewise sinusoidal reaction formulation.

Load Flow Convergence in the Vicinity of a Voltage Stability Limit

Abstract: Because load flow problems are expressed as sets of nonlinear simultaneous equations, they have no unique solutions. In this paper a region where a set of initial values converges to a stable load flow solution under specified conditions, is investigated theoretically when the Newton-Raphson method is applied to a set of nodal power equations expressed in either polar or rectangular coordinates. The results are tested in load flow calculations on a 28-node power system and the convergence characteristics for the two types of coordinates are compared.

Recursive prediction error methods for adaptive estimation

Abstract: Convenient recursive prediction error algorithms for identification and adaptive state estimation are proposed, and the convergence of these algorithms to achieve off-line prediction error minimization solutions is studied. To set the recursive prediction error algorithms in another perspective, specializations are derived from significant simplifications to a class of extended Kalman filters. The latter are designed for linear state space models with the unknown parameters augmenting the state vector and in such a way as to yield good convergence properties. Also, specializations to approximate maximum likelihood recursions, Kalman filters with adaptive gains, and connections to the extended least squares algorithms are noted.

Convergence of an Adaptive Filter Algorithm

Abstract: A state-space representation of a dynamical, stochastic system is given. A corresponding model, parametrized in a particular way, is considered and an algorithm for the estimation of its parameters is analysed. The class of estimation algorithms thus considered contains general output error methods and model reference methods applied to stochastic systems. It also contains adaptive filtering schemes and, e.g. the extended least squares method. It is shown that if a certain transfer function associated with the true system is positive real, then the estimation algorithm converges with probability 1 to a value that gives a correct input-output model.

A Computational Test for Convergence of Iterative Methods for Nonlinear Systems

Abstract: A simple computational test for existence of a solution to a nonlinear system of equations and convergence of iterative methods is given for n-cubes. The test is eventually satisfied by any convergent Newton-type sequence.

Strong Convergence of Semigroups of Nonlinear Contractions in Hilbert Space.

Abstract: : The present paper deals with the strong convergence of trajectories S(t)x of a strongly continuous semigroup of contractions S(t), as t approaches infinity A general sufficient condition for such convergence to occur is introduced and some examples in which the condition is satisfied are provided Strengthening the general convergence condition, sufficient conditions for certain rates of convergence of S(t)x to its limit are exhibited In particular a sufficient condition for a trajectory to reach equilibrium in finite time is given The convergence as t approaches infinity of solutions of certain nonautonomous equations and a discrete version of all the previous results are briefly discussed (Author)

Convergence acceleration on a general class of power series

Abstract: In this paper we present several efficient methods for evaluating functions defined by power series expansions. Simple computer codes for two rapid algorithms are given in a companion paper. The convergence rates of the proposed computational schemes are investigated theoretically and the results are illustrated by numerical examples.

A New Proof of Global Convergence for the Tridiagonal $QL$ Algorithm

Abstract: By exploiting the relation of the $QL$ algorithm to inverse iteration we obtain a proof of global convergence which is more conceptual and less computational than previous analyses. The proof uses a new, but simple, error estimate for the first step of inverse iteration.

The convergence of variable metric methods for non-linearly constrained optimization calculations

Abstract: Variable metric methods for unconstrained optimization calculations can be extended to the constrained case by regarding the positive definite matrix that is revised on each iteration as an approximation to the second derivative matrix of the Lagrangian function. Linear approximations to the constraints are used. Han (1976) has analyzed the convergence of these methods in the case when the true second derivative matrix of the Lagrangian function is positive definite at the solution. However, this matrix sometimes has negative eigenvalues so we analyze the rate of convergence in this case. We find that it is still superlinear. Therefore we may continue to use positive definite second derivative approximations and there is no need to introduce any penalty terms. The given theory helps to explain the excellent numerical results that are obtained by a recent algorithm (Powell, 1977).

A combined Remes-differential correction algorithm for rational approximation

Abstract: : In this paper a hybrid Remes-differential correction algorithm for computing best uniform rational approximants on a compact subset of the real line is developed. This algorithm differs from the classical multiple exchange Remes algorithm in two crucial aspects. First of all, the solving of a nonlinear system to find a best approximation on a given reference set in each iteration of the Remes algorithm is replaced with the differential correction algorithm to compute the desired best approximation on the reference set. Secondly, the exchange procedure itself has been modified to eliminate the possibility of cycling that can occur in the usual exchange procedure. This second modification is necessary to guarantee the convergence of this algorithm on a finite set without the usual normal and sufficiently dense assumptions that exist in other studies. (Author)

Bounding Sets in Function Spaces with Applications to Nonlinear Operator Equations

Abstract: We can compute with bounding sets of numbers, vectors, or functions using the techniques of interval mathematics. The techniques can be used to computationally verify sufficient conditions for existence, uniqueness, and convergence and to construct upper and lower bounds on solutions of nonlinear operator equations. Rounding errors are taken into account. We can compute with set-valued functions and operators on them. The techniques are also useful in search procedures for finding safe starting regions for iterative methods and for constructing natural stopping criteria.