Showing papers on "Convergence (routing) published in 1983"

A New Convergence Proof for the Multigrid Method Including the V-Cycle

Abstract: For a positive definite finite element equation we describe a multigrid iteration and prove convergence under natural assumptions on the discretization and the elliptic problem. Hitherto existing convergence proofs require a sufficiently large number of smoothing iterations and exclude the “V-cycle”. The presented proof applies to procedures with any number of smoothing iterations and to the V-cycle.

Second-order convergence analysis of stochastic adaptive linear filtering

Abstract: The convergence of an adaptive filtering vector is studied, when it is governed by the mean-square-error gradient algorithm with constant step size. We consider the mean-square deviation between the optimal filter and the actual one during the steady state. This quantity is known to be essentially proportional to the step size of the algorithm. However, previous analyses were either heuristic, or based upon the assumption that successive observations were independent, which is far from being realistic. Actually, in most applications, two successive observation vectors share a large number of components and thus they are strongly correlated. In this work, we deal with the case of correlated observations and prove that the mean-square deviation is actually of the same order (or less) than the step size of the algorithm. This result is proved without any boundedness or barrier assumption for the algorithm, as it has been done previously in the literature to ensure the nondivergence. Our assumptions are reduced to the finite strong-memory assumption and the finite-moments assumption for the observation. They are satisfied in a very wide class of practical applications.

Self-consistency iterations in electronic-structure calculations

Abstract: The convergence of self-consistency iterations in electronic-structure calculations based on density-functional theory is examined by the linearization of the self-consistency equations around the exact solution. In particular, we study the convergence of the usual procedure employing a mixture of the input and output of the last iteration. We show that this procedure converges for a suitably chosen mixture. However, the convergence is necessarily slow in certain cases. These problems are connected either with large charge oscillations or with the onset of magnetism. We discuss physical situations where such problems occur. Moreover, we propose some improved iteration schemes which are illustrated in calculations for $3d$ impurities in Cu.

Guaranteed convergence in ground state multiconfigurational self‐consistent field calculations

Abstract: We show how an optimization constraint algorithm of Fletcher that guarantees convergence to the lowest state of a given symmetry may be practically implemented in a multiconfigurational self‐consistent field (MCSCF) calculation. Other MCSCF procedures in current use have not been proven mathematically to guarantee convergence. Calculations on the ground states of N2 and CO show that rapid and efficient convergence is obtained with the Fletcher restricted step size algorithm.

Convergence criteria for iterative restoration methods

Abstract: While many iterative signal restoration methods have been shown to converge in the mathematical sense, a practical criterion is needed to indicate when the numerical algorithm has converged. The use of the residual signal in defining such a criterion is investigated. The advantage of this criterion over the criterion of examining successive iterations is demonstrated.

Convergence of functions: equi-semicontinuity

Abstract: The ever increasing complexity of the systems to be modeled and analyzed, taxes the existing mathematical and numerical techniques far beyond our present day capabilities. By their intrinsic nature, some problems are so difficult to solve that at best we may hope to find a solution to an approximation of the original problem. Stochastic optimization problems, except in a few special cases, are typical examples of this class. This however raises the question of what is a valid "approximate" to the original problem. The design of the approximation , must be such that (i) the solution to the approximate provides approximate solutions to the original problem and (ii) a refinement of the approximation yields a better approximate solution. The classical techniques for approximating functions are of little use in this setting. In fact very simple examples show that classical approximation techniques dramatically fail in meeting the objectives laid out above. What is needed, at least at a theoretical level, is to design the approximates to the original problem in such a way that they satisfy an epi-convergence criterion. The convergence of the functions defining the problem is to be replaced by the convergence of the sets defined by these functions. That type of convergence has many properties but for our purpose the main one is that it implies the convergence of the (optimal) solutions. This article is devoted to the relationship between the epi-convergence and the classical notion of pointwise-convergence. A strong semicontinuity condition is introduced and it is shown to be the link between these two types of convergences. It provides a number of useful criteria which can be used in the design of approximates to difficult problems.

On the convergence of numerical results in modal analysis

Abstract: Previous studies of the relative convergence (RC) phenomenon in modal analysis showed the need to introduce the edge condition by using an explicit asymptotic behavior of modal coefficients. We do not agree that the edge condition is the only means of uniquely defining a solution. It is demonstrated that the RC can be avoided by including only the bounded feature of modal coefficients. The analytical computations are given in solving the bifurcated parallel plate waveguide. To solve the general problem of convergence of numerical results, a-proof is given of this convergence in modal analysis. The required condition is the use of a "well-conditioned" linear system. With this demonstration, the use of the condition number as purely numerical criterion is justified to ensure the convergence of results.

Convergence of Numerical Solutions of Step-Type Waveguide Discontinuity Problems by Modal Analysis

Abstract: Convergence of modal analysis solutions of step-type waveguide discontinuity problems is studied. The convergence rate depends on the ratio between the number of modal terms retained in different regions. Guidelines for accurate and efficient computations are indicated.

Solution of underdetermined nonlinear equations by stationary iteration methods

Abstract: Nonlinear stationary fixed point iterations inR n are considered. The Perron-Ostrowski theorem [23] guarantees convergence if the iteration functionG possesses an isolated fixed pointu. In this paper a sufficient condition for convergence is given ifG possesses a manifold of fixed points. As an application, convergence of a nonlinear extension of the method of Kaczmarz is proved. This method is applicable to underdetermined equations; it is appropriate for the numerical treatment of large and possibly ill-conditioned problems with a sparse, nonsquare Jacobian matrix. A practical example of this type (nonlinear image reconstruction in ultrasound tomography) is included.

Order of Convergence of One-Step Methods for Volterra Integral Equations of the Second Kind

Abstract: Nice proofs of convergence and asymptotic expansions are known for one-step methods for ordinary differential equations. It is shown that these proofs can be generalized in a natural way to “extended” one-step methods for Volterra integral equations of the second kind. Furthermore, the convergence of “mixed” one-step methods is investigated. For both types general Volterra–Runge–Kutta methods are considered as examples.

Approximations for chance-constrained programming problems

Abstract: This paper proposes an approximation approach to the solution of chance-constrained stochastic programming problems. The results rely in a fundamental way on the theory of convergence of sequences ...

A Cartesian Coordinate Algorithm for Power System State Estimation

Abstract: This paper presents a new steady-state estimator based on the Cartesian coordinate formulation of nodal and line flow equations and minimization of weighted least squares (WLS) of the residuals. The fact that the rectangular coordinate version of network Performance equations is completely expressible in a Taylor series and contains terms up to the second order derivatives only, results in a fast exact second order state (FESOS) estimator. In this estimator, the Jacobian and information matrices are constant, and hence need to be computed once only. The size of the mathematical model for the new estimator is the same as that of the widely used fast decoupled state (FDS) estimator, and hence characterized by comparable computational requirements (storage and time per iteration). Digital simulation results are presented on several sample power systems (well-conditioned/ill-conditioned) under normal as well as unusual operating modes to illustrate the range of application of the method vis-a-vis the FDS estimator. It is found that the exactness of the FESOS algorithm provides an accurate solution during all modes of system operation, and assures convergence to the right solution in spite of network ill-conditioning. In particular, the convergence behaviour and accuracy of solution of the FESOS estimator approach those of the FDS estimator for lightly loaded and well- conditioned power systems during normal modes of operation, but are vastly superior during unusual modes of system operation or in relation to ill-conditioned networks.

Stationary Policies in Dynamic Programming Models Under Compactness Assumptions

Abstract: The present work deals with the usual stationary decision model of dynamic programming The imposed convergence condition on the expected total rewards is so general that both the negative (unbounded) case and the positive (unbounded) case are included However, the gambling model studied by Dubins and Savage is not covered by the present model In addition to the convergence condition, a continuity and compactness condition is imposed The main result states that the supremum of the expected total rewards under all stationary policies is equal to the supremum under all (possibly randomized and non-Markovian) policies

A hybrid algorithm for finding a global minimum

Abstract: One of the most important problems in non-linear programming is to find out the global minimum of a given objective function. In this paper, a new hybrid algorithm which combines the random optimization method of Matyas (1965) and one of the well-known ordinary descent algorithms having an effective convergence property (for example, the Fletcher-Reeves conjugate gradient method, the Davidon-Fletcher-Powell quasi Newton method, etc.) is proposed in order to find out a global minimum in as small a number of steps as possible. Several computational experiments on multimodal objective functions are carried out in order to test the efficiency of the proposed hybrid algorithm. The results obtained imply that the proposed hybrid algorithm is useful for finding out a global minimum in a small number of steps. A theorem that predicts convergence to a global minimum is also given.

A convergence for bivariate functions aimed at the convergence of saddle values

Abstract: Epi/hypo-convergence is introduced from a variational view-point The known topological properties are reviewed and extended Finally, it is shown that the (partial) Legendre-Fenchel transform is biocontinuous with respect to the topology induced by epi/hypoconvergence on the space of convex-concave bivariate functions

An economical technique for forcing convergence in conventional SCF methods

Abstract: We present an economical technique for ensuring convergence of the open‐ and closed‐shell SCF methods. In this technique, the number of operations required is proportional to the square of the number of basis functions and all the employed quantities are present in any conventional SCF procedure. We test their efficacy with several numerical calculations.

A control parametrization algorithm for optimal control problems involving linear systems and linear 1

Abstract: Computational schemes based on control parametrization techniques are known to be very efficient for solving optimal control problems. However, the convergence result is only available for the case in which the dynamic system is linear and without the terminal equality and inequality constraints. This paper is to improve this convergence result by allowing the presence of the linear terminal inequality. For illustration, an example arising in the study of optimally one-sided heating of a metal slab in a furnace is considered.

Regularity and integrability of spherical means

Abstract: The regularity and integrability of spherical means of functions inLp(ℝn),n≥2, are studied. An application is given to convergence of Fourier integrals.

Chapter 1 Augmented Lagrangian Methods in Quadratic Programming

Abstract: Publisher Summary This chapter discusses the augmented Lagrangian methods in quadratic programming The advantage of the augmented Lagrangian is that, because of the presence of the term (q,Bv), the exact Solution of problem can be determined without making r tend to infinity, unlike ordinary penalization methods where this has the effect of causing deterioration in the conditioning of the systems to be solved The conjugate gradient method is especially attractive for solving quadratic problems because theoretically it converges in a finite number of iterations (≤M) and because moreover in the general case it leads to quadratic convergence It would be too lengthy and inappropriate to study here the convergence of this method For M "large" quadratic convergence becomes a greater attraction than convergence in a finite number of iterations

Convergence of a decision-directed adaptive equalizer

Abstract: An adaptive decision directed equalizer, wherein the estimates of the transmitted data are used for the adaptation of the equalizer parameters, is analyzed. In the paper the stability of the limit equation associated with the stochastic adaptation algorithm is proved. The convergence of the stochastic algorithm then follows by the application of the standard weak convergence theory. The approach of this paper is different from the earlier such analysis in the literature. Thus in the paper we work with the discrete probability distribution of the transmitted symbols, as is the case in practice, rather than assuming a continuous distribution of these for the sake of theoretical convenience.

The effect of perturbations on the convergence rates of optimization algorithms

Abstract: The problem of minimizing a functionF over a set Ω is approximated by a sequence of problems whereF and Ω are replaced byF(n) and Ω(n), respectively We show in which manner the convergence rates of the conditional gradient and projected gradient methods are influenced by the approximation In particular, it becomes evident how the convergence theory for infinite dimensional problems such as control problems explains the behavior of finite dimensional implementations