Showing papers on "Rate of convergence published in 1975"
TL;DR: The convergence rate for difference approximations to mixed initial boundary value problems has been shown to be linear in the convergence rate of the difference approximation as mentioned in this paper, which is the best known convergence rate.
Abstract: The convergence rate for difference approximations to mixed initial boundary value problems
386 citations
TL;DR: In this article, the authors proposed an orthogonal collocation method for heat and mass transfer with chemical reaction in a catalyst pellet, which is especially suited to situations corresponding to high Thiele modulus when the solution is confined to a thin boundary region near the catalyst surface.
287 citations
TL;DR: Using numerical integration in the formation of the finite element mass matrix and placing the movable nodes at integration points causes it to become lumped or diagonal (block diagonal) with the optimal rate of energy convergence retained.
149 citations
TL;DR: An algorithm is developed for solving the convex programming problem by constructing a cutting plane through the center of a polyhedral approximation to the optimum, which generates a sequence of primal feasible points whose limit points satisfy the Kuhn—Tucker conditions of the problem.
Abstract: An algorithm is developed for solving the convex programming problem which iteratively proceeds to the optimum by constructing a cutting plane through the center of a polyhedral approximation to the optimum. This generates a sequence of primal feasible points whose limit points satisfy the Kuhn—Tucker conditions of the problem. Additionally, we present a simple, effective rule for dropping prior cuts, an easily calculated bound on the objective function, and a rate of convergence.
137 citations
TL;DR: In this paper, the authors studied the time between the first infection and the last removal in the closed stochastic epidemic and proved limit theorems for the distribution of this time, as the population size becomes large, and then used the limits to provide an approximate distribution.
Abstract: SUMMARY The paper discusses the time between the first infection and the last removal in the closed stochastic epidemic. The method is to prove limit theorems for the distribution of this time, as the population size becomes large, and then to use the limits to provide an approximate distribution: the epidemic is allowed to start either with a large number of immigrant infectives, or with a single case. The accuracy of the approximation in finite populations is illustrated by some examples, and the method of proof also gives a theoretical estimate of the rate of convergence to the limit. The problem is typical of a much wider class of boundary problems, and the method used can be adapted to them without difficulty.
84 citations
TL;DR: The main purpose of this work is to associate a wide class of Lagrangian functions with a nonconvex, inequality and equality constrained optimization problem in such a way that unconstrained stationary points and local saddle points of each Lagrangians are related to Kuhn–Tucker points or local or global solutions of the optimization problem.
Abstract: The main purpose of this work is to associate a wide class of Lagrangian functions with a nonconvex, inequality and equality constrained optimization problem in such a way that unconstrained stationary points and local saddle points of each Lagrangian are related to Kuhn–Tucker points or local or global solutions of the optimization problem. As a consequence of this we are able to obtain duality results and two computational algorithms for solving the optimization problem. One algorithm is a Newton algorithm which has a local superlinear or quadratic rate of convergence. The other method is a locally linearly convergent method for finding stationary points of the Lagrangian and is an extension of the method of multipliers of Hestenes and Powell to inequalities.
79 citations
TL;DR: This paper considers a class of combined primal-dual and penalty methods often called methods of multipliers, and focuses mainly on the rate of convergence of these methods.
Abstract: In this paper we consider a class of combined primal-dual and penalty methods often called methods of multipliers. The analysis focuses mainly on the rate of convergence of these methods. It is sho...
76 citations
TL;DR: In this paper, it is proved that, when a controller (an observation) can be arbitrarily constructed, observability (controllability) of the system is necessary and sufficient for stabilizing the system so that it has an arbitrarily large damping constant.
Abstract: In this paper, we study feedback stabilization of a class of distributed systems governed by partial differential equations of parabolic type and its application to constructing a state estimator for asymptotic state identification. It is proved that, when a controller (an observation) can be arbitrarily constructed, observability (controllability) of the system is necessary and sufficient for stabilizing the system so that it has an arbitrarily large damping constant. As an application of this result, it is shown that a state estimator can be constructed, the output of which approaches asymptotically the real state of the system with an arbitrary convergence rate.
67 citations
TL;DR: Perturbation theory, when applied to charged-particle transport, generates a series solution that requires a double quadrature per term as discussed by the authors, and the continuity of higher-order terms leads to numerical evaluation of the series.
Abstract: Perturbation theory, when applied to charged-particle transport, generates a series solution that requires a double quadrature per term. The continuity of higher-order terms leads to numerical evaluation of the series. The high rate of convergence of the series makes the method a practical tool for charged-particle transport problems. The coupling of the neutron component in the case of proton transport in tissue does not greatly alter the rate of convergence. The method holds promise for a practical high-energy proton transport theory.
63 citations
TL;DR: In this paper, the Dirichlet problem for biharmonic equation in a rectangular region is considered and the method of splitting is used and two classes of finite difference approximations are defined.
Abstract: The Dirichlet problem for biharmonic equation in a rectangular region is considered. The method of splitting is used and two classes of finite difference approximations are defined. Two semi-iterative procedures are considered for obtaining the solution of the resulting coupled system of algebraic equations. It is shown that the rate of convergence of the iterative procedures depends upon the choice of the difference approximation. Estimates for optimum iteration parameters are given and several comparisons are made. An attempt is made to unify the ideas on the splitting technique for solving the first biharmonic boundary value problem.
59 citations
TL;DR: In this article, an iterative procedure using conjugate directions to minimize a nonlinear function subject to linear inequality constraints is presented, which converges to a stationary point assuming only first-order differentiability, and has ann-q step superlinear or quadratic rate of convergence with stronger assumptions.
Abstract: An iterative procedure is presented which uses conjugate directions to minimize a nonlinear function subject to linear inequality constraints. The method (i) converges to a stationary point assuming only first-order differentiability, (ii) has ann-q step superlinear or quadratic rate of convergence with stronger assumptions (n is the number of variables,q is the number of constraints which are binding at the optimum), (iii) requires the computation of only the objective function and its first derivatives, and (iv) is experimentally competitive with well-known methods.
TL;DR: In this article, a partial approximation of the physical finite element representation of circular arches as an assemblage of straight beam elements is compared with the mathematical model based on the partial approximation.
TL;DR: The results discussed highlight the operational aspects of multiplier methods and demonstrate their significant advantages over ordinary penalty methods.
TL;DR: In this article, the authors apply the method of non-discrete mathematical induction to Newton's process and prove that the rate of convergence is a positive number depending on the initial data.
Abstract: The author applies the method of nondiscrete mathematical induction (see [2---5]) which involves considering the rate of convergence as a function, not as a number, to Newton's process and proves that the rate of convergence is $$\omega (r) = \frac{{r^2 }}{{2(r^2 + d)^{1/2} }}$$ whered is a positive number depending on the initial data (see Theorem 2.3).
Abstract: Many data analysis problems in psychology may be posed conveniently in terms which place the parameters to be estimated on one side of an equation and an expression in these parameters on the other side. A rule for improving the rate of convergence of the iterative solution of such equations is developed and applied to four problems: the principal axis communality problem, individual differences multidimensional scaling,L
P norm multiple regression, andL
P norm factor analysis of a data matrix. The rule results in substantially faster solutions or in solutions where none would be possible without the rule.
TL;DR: A linear programming approach known as the differential correction algorithm, which has been shown by several authors to always converge in theory, is used, and it is obtained convergence in nearly every case, and quadratic convergence in most cases.
Abstract: We present a program which has given excellent results for uniform approximation of functions by polynomials, rational functions, generalized polynomials, and generalized rational functions. The algorithm is described in detail and several examples are discussed. The approximation is done over a finite point set, which is commonly a set of real numbers or points in the plane (in the latter case we are doing what is often known as surface fitting). Input to and output from the program is in tabular form. The method used is a linear programming approach known as the differential correction algorithm, which has been shown by several authors to always converge in theory (quadratically in some situations). In practice, we have obtained convergence in nearly every case, and quadratic convergence in most cases. The program can also be used for simultaneous approximation of several functions.
TL;DR: In this paper, three iterative methods for numerically solving the steady-state Navier-Stokes equations are presented, i.e., the Laplacian Driver (LAD), the Numerical Oseen (NOS), and the Split NOS method.
TL;DR: In this article, it was shown that for most purely competitive sequences of economies on a compact set of agents' characteristics, the core converges to the set of equilibria at least as fast as the inverse of the number of agents.
01 Jan 1975
TL;DR: In this paper, an upper bound for the rate of convergence of consistent estimators based on sample quantities is derived for the case when the underlying distribution is double-exponential and shown to coincide with the classical Pitman asymptotic relative efficiency.
Abstract: The rate at which the probability $P_\theta\{|t_n - \theta| \geqq \varepsilon\}$ of consistent estimator $t_n$ tends to zero is of great importance in large sample theory of point estimation. The main tools available at present for finding the rate are Bernstein-Chernoff-Bahadur's theorem and Sanov's theorem. In this paper, we give two new techniques for finding the rate of convergence of certain consistent estimators. By using these techniques, we have obtained an upper bound for the rate of convergence of consistent estimators based on sample quantities and proved that the sample median is an asymptotically efficient estimator in Bahadur's sense if and only if the underlying distribution is double-exponential. Furthermore, we have proved that the Bahadur asymptotic relative efficiency of sample mean and sample median coincides with the classical Pitman asymptotic relative efficiency.
01 Apr 1975
TL;DR: In this article, the authors considered the problem of atom-diatom elastic and rotationally and vibrationally inelastic collisions in the recently developed coupled-states approach in which the body-tixed coupled equations are approximated by the neglect of intermultiplet coupling and the eigenvalue of the orbital angular momentum operator.
Abstract: The study of atom-diatom elastic and rotationally and vibrationally inelastic collisions is considered in the recently developed coupled-states approach in which the body-tixed coupled equations are approximated by the neglect of intermultiplet coupling and the eigenvalue of the orbital angular momentum operator ??? is approximated by n e (e + 1) The rate of convergence with respect to molecular basis expansion for the coupled-states cross sections is presented for the He-H 2 collisions at 065 eV along with a comparison to the corresponding close-coupling cross section computed in the standard fixed-space coordinate system For elastic and rotationally inelastic scattering, the coupled-states and close-coupling convergence properties are similar while for vibrational transition cross sections, the convergence is irregular It is found in general that the coupled-states method is a more reliable approximation when fully converged cross section are computed
TL;DR: In this article, convergence theorems for a class of degenerate-kernel methods for numerical solution of Fredholm integral equations of the second kind were proved. And it was shown that the simplest of these methods has a faster rate of convergence than the simple method of moments, or Galerkin method, even though its computational requirements are almost identical.
Abstract: Convergence theorems are proved for a recently proposed class of degenerate-kernel methods for the numerical solution of Fredholm integral equations of the second kind. In particular, it is shown that the simplest of these methods has a faster rate of convergence than the simple method of moments, or Galerkin method, even though its computational requirements are almost identical.
TL;DR: In this article, a sequence of empirical Bayes estimators for a density function and its derivatives, which are not assumed to be uniformly bounded, using classes of kernel functions is proposed.
Abstract: In this paper we construct sequences of estimators for a density function and its derivatives, which are not assumed to be uniformly bounded, using classes of kernel functions. Utilizing these estimators, a sequence of empirical Bayes estimators is proposed. It is found that this sequence is asymptotically optimal in the sense of Robbins (Ann. Math. Statist. 35 (1964) 1-20). The rates of convergence of the Bayes risks associated with the proposed empirical Bayes estimators are obtained. It is noted that the exact rate is $n^{-q}$ with $q \leqq \frac{1}{3}$. An example is given and an explicit kernel function is indicated.
01 Jan 1975
TL;DR: In this paper, the authors present an analysis and a comparison of penalty and multiplier methods for constrained minimization, and show that multiplier methods alleviate to a substantial extent the traditional disadvantages of penalty methods (ill-conditioning, slow convergence), while retaining all of their attractive features.
Abstract: The purpose of this paper is to present an analysis and a comparison of penalty and multiplier methods for constrained minimization. Global convergence and rate of convergence results are given which show that multiplier methods alleviate to a substantial extent the traditional disadvantages of penalty methods (ill-conditioning, slow convergence), while retaining all of their attractive features. At the same time a global duality framework is constructed in the absence of convexity. Within this framework multiplier methods may be viewed as gradient methods for maximizing a certain dual functional. This interpretation leads to sharper rate of convergence results and motivates efficient modifications of the multiplier iteration.
TL;DR: In this paper, the convergence of a general iterative technique for finding the zeros of G:D?H 2 subject to the general constraintP(x)=x, whereP:D?D is established.
Abstract: LetH 1 andH 2 denote Hilbert spaces and suppose thatD is a subset ofH 1. This paper establishes the local and linear convergence of a general iterative technique for finding the zeros ofG:D?H 2 subject to the general constraintP(x)=x, whereP:D?D. The results are then applied to several classes of problems, including those of least squares, generalized eigenvalues, and constrained optimization. Numerical results are obtained as the procedure is applied to finding the zeros of polynomials in several variables.
TL;DR: Application of direct iterations, based on convergent splittings, to the eigenvalue problem of large sparse symmetric matrices is discussed, and a general convergence proof is given.
TL;DR: In this article, the convergence of the series is studied for integrals of the type where is a wide-sense stationary process (or a quasi-stationary process) and sufficient conditions are obtained for convergence in the class of all sequences having a given rate of decrease of the correlation function.
Abstract: Convergence almost everywhere of series is studied, where is a wide-sense stationary sequence (or a quasi-stationary sequence). Sufficient conditions are obtained for convergence of the series, which are also necessary in the class of all sequences having a given rate of decrease of the correlation function.Analogous results are also valid for integrals of the type where is a wide-sense stationary process.Bibliography: 12 titles.
TL;DR: The present paper gives the upper and the lower bounds for beta(xi,n).
Abstract: The easily computable asymptotic power of the locally asymptotically optimal test of a composite hypothesis, known as the optimal C(α) test, is obtained through a “double” passage to the limit: the number n of observations is indefinitely increased while the conventional measure ξ of the error in the hypothesis tested tends to zero so that ξnn½ → τ ≠ 0. Contrary to this, practical problems require information on power, say β(ξ,n), for a fixed ξ and for a fixed n. The present paper gives the upper and the lower bounds for β(ξ,n). These bounds can be used to estimate the rate of convergence of β(ξ,n) to unity as n → ∞. The results obtained can be extended to test criteria other than those labeled C(α). The study revealed a difference between situations in which the C(α) test criterion is used to test a simple or a composite hypothesis. This difference affects the rate of convergence of the actual probability of type I error to the preassigned level α. In the case of a simple hypothesis, the rate is of the order of n-½. In the case of a composite hypothesis, the best that it was possible to show is that the rate of convergence cannot be slower than that of the order of n-½ ln n.
TL;DR: A numerical method is described by which a 1D MHD non-linear lagrangian system may be relaxed adiabatically to pressure equilibrium by an iterative process, appropriate for problems involving a slow passage through a sequence of quasi-equilibrium states.
TL;DR: It is shown that every cluster point of the sequence {x"j} constructed by the method is a stationary point and the rate of convergence is (n-q)-step superlinear.