scispace - formally typeset
Search or ask a question

Showing papers on "Rate of convergence published in 1980"


Journal ArticleDOI
TL;DR: In this article, a quadratically convergent MC-SCF procedure is described, which is based on the direct minimization of the energy, and the convergence radius is much improved by taking into account in the energy expansion those parts of third and higher order terms which account exactly for the orthonormality constraints imposed on the orbitals.
Abstract: A quadratically convergent MC–SCF procedure is described which is based on the direct minimization of the energy. In comparison to the Newton–Raphson technique, which has previously been applied by several authors for orbital optimization, the convergence radius is much improved by taking into account in the energy expansion those parts of third and higher order terms which account exactly for the orthonormality constraints imposed on the orbitals. The nonlinear equations which define the improved orbitals are solved iteratively by a simple adaption of the Gauss–Seidel method. The coefficients of the configuration expansion can be optimized simultaneously with the orbitals, a necessary requirement for over‐all quadratic convergence. The removal of redundant variables as well as useful approximations for the optimization of core orbitals are discussed. The convergence of the method is demonstrated to be much superior to classical Fock operator techniques and MC–SCF methods which are based on the generalized Brillouin theorem. The formalism is carried down to matrix operations and shows a simple structure.

336 citations


Journal ArticleDOI
TL;DR: In this paper, a unified analysis of three extrapolation methods: the scalar and vector \epsilon -algorithms and the minimum polynomial extrapolation algorithm is presented.
Abstract: The problem of computing the periodic steady-state response can be formulated as solving a nonlinear equation of the form z = F(z) where F(z) Is the solution vector for the nonlinear network after one period of integration from the initial vector z The convergence of the sequence y_0 , y_1 , \cdots generated by Y_{r+1} = F(y_r) can be accelerated by extrapolation methods This paper presents a unified analysis of three extrapolation methods: the scalar and vector \epsilon -algorithms and the minimum polynomial extrapolation algorithm The main result of the paper is the theorem giving conditions for quadratic convergence of the extrapolation methods To obtain this result the methods are studied for linear problems (where F is a linear function) and the error propagation properties are investigated For autonomous systems a function called G similar to F can be defined In order to obtain quadratic convergence from the extrapolation methods, the derivatives of F and G must be Lipschitz continuous The appendixes give sufficient conditions for the Lipschitz continuity A discussion of practical problems related to the implementation of the extrapolation methods is based on the convergence theorem and the error analysis The performance of the extrapolation methods is demonstrated and compared with other methods for steady-state analysis by four examples, two autonomous and two nonautonomous Extrapolation methods are very easy to implement, and they are efficient for the steady-state analysis of nonlinear circuits with few reactive elements giving rise to slowly decaying transients

185 citations



Journal ArticleDOI
TL;DR: In this paper, the integrability of last exit times and the number of boundary crossings of the partial sums is analyzed for a set of i.i.d. random variables indexed by the positive integer lattice points, and convergence rates for moderate deviations are derived.
Abstract: For a set of i.i.d. random variables indexed by $Z^d_+, d \geqslant 1$, the positive integer $d$-dimensional lattice points, convergence rates for moderate deviations are derived, i.e., the rate of convergence to zero of, for example, certain tail probabilities of the partial sums, are determined. As an application we obtain results on the integrability of last exit times (in a certain sense) and the number of boundary crossings of the partial sums.

84 citations


Journal ArticleDOI
TL;DR: In this paper, the authors derived the order of convergence for six numerical methods that have been proposed for the slab geometry, multigroup, discrete-ordinates neutron transport equations, and illustrated the results by means of a simple test problem.
Abstract: The order of convergence, as the spatial cell widths tend to zero, is derived for six numerical methods that have been proposed for the slab geometry, multigroup, discrete-ordinates neutron transport equations. Our results, which in two cases differ from earlier experimental results, are illustrated by means of a simple test problem.

73 citations


Journal ArticleDOI
TL;DR: In this paper, the inverse Stefan problem is solved numerically by a generalized Gauss-Newton method introduced by Osborne and Watson [19] under some assumptions on the parameter space.
Abstract: The inverse Stefan problem can be understood as a problem of nonlinear approximation theory which we solved numerically by a generalized Gauss-Newton method introduced by Osborne and Watson [19]. Under some assumptions on the parameter space we prove its quadratic convergence and demonstrate its high efficiency by three numerical examples.

72 citations


Journal ArticleDOI
TL;DR: A numerical algorithm which is capable of finding local optima systematically at the quadratic rate of convergence is developed from a detailed analysis of the nature of trajectories and critical points.
Abstract: A new method is presented for finding a local optimum of the equality constrained nonlinear programming problem. A nonlinear autonomous system is introduced as the base of the theory instead of usual approaches. The relation between critical points and local optima of the original optimization problem is proved. Asymptotic stability of the critical points is also proved. A numerical algorithm which is capable of finding local optima systematically at the quadratic rate of convergence is developed from a detailed analysis of the nature of trajectories and critical points. Some numerical results are given to show the efficiency of the method.

71 citations


Journal ArticleDOI
TL;DR: A particular member of this algorithm class is shown to have a Q-superlinear rate of convergence under standard assumptions on the objective function.
Abstract: A new class of algorithms for unconstrained optimization has recently been proposed by Davidon [Conic Approximations and Collinear Scalings for Optimers, SIAM J. Num. Anal., to appear.]. This new method called “optimization by collinear scaling” is derived here as a natural extension of existing quasi-Newton methods. The derivation is based upon constructing a collinear scaling of the variables so that a local quadratic model can interpolate both function and gradient values of the transformed objective function at the latest two iterates. Deviation of the function values from quadratic behavior as well as gradient information influences the updating process. A particular member of this algorithm class is shown to have a Q-superlinear rate of convergence under standard assumptions on the objective function. The amount of computation required per update is essentially the same as for existing quasi-Newton methods.

67 citations


Journal ArticleDOI
TL;DR: In this article, a high-accuracy approximation to the mth-order linear ordinary differential equation Mu = f is presented, where the coefficients of Mn, In are obtained "locally" by solving a small linear system for each group of stencil points in order to make the approximation exact on a linear space S of dimension L + 1.
Abstract: This paper analyzes a high-accuracy approximation to the mth-order linear ordinary differential equation Mu = f. At mesh points, U is the estimate of u; and U satisfies MnU = Inf, where MnU is a linear combination of values of U at m + 1 stencil points (adjacent mesh points) and Inf is a linear combination of values of f at J auxiliary points, which are between the first and last stencil points. The coefficients of Mn, In are obtained "locally" by solving a small linear system for each group of stencil points in order to make the approximation exact on a linear space S of dimension L + 1. For separated two-point boundary value problems, U is the solution of an nby-n linear system with full bandwidth m + 1. For S a space of polynomials, existence and uniqueness are established, and the discretization error is O(h L+l ); the first m 1 divided differences of U tend to those of u at this rate. For a general set of auxiliary points one has L = J + m; but special auxiliary points, which depend upon M and the stencil points, allow larger L, up to L = 2J + m. Comparison of operation counts for this method and five other common schemes shows that this method is among the most efficient for given convergence rate. A brief selection from extensive experiments is presented which supports the theoretical results and the practicality of the method.

64 citations


Journal ArticleDOI
TL;DR: The probability of correct detection is shown to be a monotonely increasing function of the underlying fundamental signal-to-noise ratio response of the Kalman filter estimate of the failure mode state to a particular magnitude of failure.
Abstract: Real-time failure detection for systems having linear stochastic dynamical truth models has been posed in terms of two confidence region sheaths in [1]-[3]. One confidence region sheath is about the expected nominal no-failure trajectory; the other is about the Kalman estimate of the state(s) being monitored for failures. The implementation of a necessary and sufficient test of whether these two confidence regions of elliptical cross section are disjoint at any time instant is shown to result in a scalar test statistic that is compared to a prespecified decision threshold at each check-time in making failure/no-failure decisions. The motivating theoretical basis of the test statistic is briefly discussed, the implementation equations and theoretical milestones previously encountered in guaranteeing algorithm convergence and establishing convergence rate are Summarized, then the details are presented for: 1) the derivation and analytic evaluation of the expressions for the probabilities of false alarm and correct detection that serve as a basis for subsequent tradeoffs in setting the threshold level; and 2) the derivation of an expression for the decision threshold and a technique for its calculation from the covariance of the Kalman filter. The probability of correct detection is shown to be a monotonely increasing function of the underlying fundamental signal-to-noise ratio response of the Kalman filter estimate of the failure mode state to a particular magnitude of failure. Real data results are provided to illustrate application of this technique for the two-dimensional case to detect failures in an inertial navigation system having two-degree-of-freedom gyros. This is the application for which the technique was developed.

62 citations


Journal ArticleDOI
TL;DR: It is shown that a very general variant of this method converges linearly thus generalizing a result of I. N. Katz's method, which is widely used for solving problems of optimal location.
Abstract: Weiszfeld's method is widely used for solving problems of optimal location. It is shown that a very general variant of this method converges linearly thus generalizing a result of I. N. Katz.

Journal ArticleDOI
TL;DR: In this article, the so-called Babuska method of finite elements with Lagrange multipliers for numerically solving the problem Au = f in Q2, u = g on a92, Q2 C Rn, n > 2.
Abstract: We consider the so-called Babuska method of finite elements with Lagrange multipliers for numerically solving the problem Au = f in Q2, u = g on a92, Q2 C Rn, n > 2. We state a number of local conditions from which we prove the uniform stability of the Lagrange multiplier method in terms of a weighted, mesh-dependent norm. The stability conditions given weaken the conditions known so far and allow mesh refinements on the boundary. As an application, we introduce a class of finite element schemes, for which the stability conditions are satisfied, and we show that the convergence rate of these schemes is of optimal order.

Journal ArticleDOI
TL;DR: It is shown that a not necessarily bandlimited function f can be approximately represented by generalized sampling sums which originate from discretized convolution integrals known, e.g., in approximation theory.
Abstract: The classical Shannon sampling theorem is concerned with the representation of bandlimited signal functions by a sum built up from a countable number of samples. It is shown that a not necessarily bandlimited function f can be approximately represented by generalized sampling sums which originate from discretized convolution integrals known, e.g., in approximation theory. The rate of convergence of the new sums to f is precisely as good as that of the associated convolution integrals. This gives sufficient as well as matching necessary conditions for a certain rate of convergence.

Journal ArticleDOI
TL;DR: In this paper, the theory of block-by-block method for solving Volterra integral equations is extended to nonsingular VOLTERRA integro-differential equations and convergence is proved and a rate of convergence is found.
Abstract: The theory of a block-by-block method for solving Volterra integral equations is extended to nonsingular Volterra integro-differential equations. Convergence is proved and a rate of convergence is found. The convergence results obtained are analogous to those obtained by Weiss [121 for Volterra integral equations. Several numerical examples are included.

Journal ArticleDOI
TL;DR: Methods are developed to stabilize the performance of the reconstruction algorithms in the presence of noise and an analysis is given for the necessary condition for complete reconstruction in imaging situations involving a number of discrete inputs confined to limited angular range.
Abstract: The propagation of errors incurred in 3-D reconstructions with limited angular input performed by deconvolution and matrix inversion algorithms is analyzed. The convergence rate and noise properties of an iterative scheme that utilizes the finite extent of the object to recover the missing Fourier components in deconvolution are studied. Methods are developed to stabilize the performance of the reconstruction algorithms in the presence of noise. An analysis is given for the necessary condition for complete reconstruction in imaging situations involving a number of discrete inputs confined to limited angular range.

Journal ArticleDOI
TL;DR: In this paper, the use of a penalty method to enforce the constraint of incompressibility in nonlinear elasticity is described and a theoretical analysis of the associated mixed method and a new equivalence theorem are seen to lead to a way to retain positive definiteness.
Abstract: This paper describes the use of a penalty method to enforce the constraint of incompressibility in nonlinear elasticity. As an example, a problem involving the use of the Newton–Raphson method in conjunction with incremental loading and a successive mesh refinement scheme is presented. It is shown that during the incremental loading phase and the Newton–Raphson refinement on a fixed mesh, all tangent stiffness matrices are positive definite for the chosen energy density and load increment. But when the mesh is refined and the solution is interpolated as a starting value on the new mesh, the tangent stiffness matrix is indefinite. A theoretical analysis of the associated mixed method and a new equivalence theorem are seen to lead to a way to retain positive definiteness. The key is the use of an equivalent tangent stiffness matrix which is the reduced Hessian matrix. The numerical example shows that both positive definiteness and the quadratic convergence rate of the Newton–Raphson method are obtained.

Journal ArticleDOI
TL;DR: In this paper, a class of implementable penalty function and multiplier methods for nonconvex nonlinear programming problems is presented, which, under suitable assumptions, produce a sequence of points converging to a strong local minimum, regardless of the location of the initial guess.
Abstract: This paper deals with penalty function and multiplier methods for the solution of constrained nonconvex nonlinear programming problems. Starting from an idea introduced several years ago by Polak, we develop a class of implementable methods which, under suitable assumptions, produce a sequence of points converging to a strong local minimum for the problem, regardless of the location of the initial guess. In addition, for sequential minimization type multiplier methods, we make use of a rate of convergence result due to Bertsekas and Polyak, to develop a test for limiting the growth of the penalty parameter and thereby prevent ill-conditioning in the resulting sequence of unconstrained optimization problems.

Journal ArticleDOI
TL;DR: The Preconditioned Simultaneous Displacement iterative method (PSD method) is introduced in a new “computable” form for the numerical solution of linear systems of the form Au=b, where the matrix A is large and sparse.

Journal ArticleDOI
TL;DR: A triangulation is introduced for homotopy methods to compute fixed points on the unit simplex or inRn, which allows for factors of incrementation of more than two and can be accelerated without using restart methods.
Abstract: In this paper a triangulation is introduced for homotopy methods to compute fixed points on the unit simplex or inR n . This triangulation allows for factors of incrementation of more than two. The factor may be of any size and even different at each level. Also the starting point on a new level may be any gridpoint of the last found completely labelled subsimplex on the last level. So, the decision which new factor of incrementation and which starting point is used, can be made on the ground of previous approximations. Doing so, the convergence rate can be accelerated without using restart methods.

Journal ArticleDOI
TL;DR: Asymptotic upper and lower bounds for the uniform measure of the rate of convergence in the central limit theorem using a variety of norming constants were obtained in this article, and extensive generalizations of the classical characterizations of convergence were deduced in terms of series and order of magnitude conditions.
Abstract: Asymptotic upper and lower bounds are obtained for the uniform measure of the rate of convergence in the central limit theorem using a variety of norming constants. For many distributions the upper and lower bounds are of the same order of magnitude. As easy corollaries we deduce extensive generalizations of the classical characterizations of the rate of convergence in terms of series and order of magnitude conditions.

C. Jekeli1
01 May 1980
TL;DR: In this paper, the truncation theory as it pertains to the calculation of geoid undulations based on Stokes' integral, but from limited gravity data, is reexamined.
Abstract: The truncation theory as it pertains to the calculation of geoid undulations based on Stokes' integral, but from limited gravity data, is reexamined. Specifically, the improved procedures of Molodenskii et al. are shown through numerical investigations to yield substantially smaller errors than the conventional method that is often applied in practice. In this improved method, as well as in a simpler alternative to the conventional approach, the Stokes' kernel is suitably modified in order to accelerate the rate of convergence of the error series. These modified methods, however, effect a reduction in the error only if a set of low-degree potential harmonic coefficients is utilized in the computation. Consider, for example, the situation in which gravity anomalies are given in a cap of radius 10 deg and the GEM 9 (20,20) potential field is used. Then, typically, the error in the computed undulation (aside from the spherical approximation and errors in the gravity anomaly data) according to the conventional truncation theory is 1.09 m; with Meissl's modification it reduces to 0.41m, while Molodenskii's improved method gives 0.45 m. A further alteration of Molodenskii's method is developed and yields an RMS error of 0.33 m. These values reflect the effect of the truncation, as well as the errors in the GEM 9 harmonic coefficients. The considerable improvement, suggested by these results, of the modified methods over the conventional procedure is verified with actual gravity anomaly data in two oceanic regions, where the GEOS-3 altimeter geoid serves as the basis for comparison. The optimal method of truncation, investigated by Colombo, is extremely ill-conditioned. It is shown that with no corresponding regularization, this procedure is inapplicable.

Book ChapterDOI
Liqun Qi1
01 Jan 1980
TL;DR: In this paper, the Krawczyk-Moore algorithm was generalized to the problem of computing a sequence of interval vectors from a given initial interval vector, and a computational test for existence, uniqueness, and convergence was proposed.
Abstract: Publisher Summary This chapter describes the generalization of the Krawczyk–Moore algorithm It presents the Krawczyk–Moore algorithm for producing a sequence of interval vectors from a given initial interval vector It also reviews a computational test for existence, uniqueness, and convergence The proof of the convergence of an interval algorithm is similar to the proof of a theorem It is found that instead of F in the algorithm, a narrower interval matrix function g has been given The quadratic convergence may be proved under reasonable conditions, and it is clear that the results obtained may be applied to the interval procedure It is found that as long as Krawczyk–Moore algorithm converges to a point z , the point z is the unique solution of the f ( x ) = 0 in X (0)

Journal ArticleDOI
TL;DR: In this paper, the effect of numerical integration in conforming finite element methods for the general shell problem is investigated within the linear model of Koiter and sufficient conditions are given to ensure that the order of convergence is independent of whether or not numerical integration is used.



Journal ArticleDOI
TL;DR: In this paper, a recursive estimate of the stochastic structure of a stationary time series is constructed based on the assumption that the true structure is ARMA, i.e., has a rational spectrum.
Abstract: A recursive estimate of the stochastic structure of a stationary time series is constructed based on the assumption that the true structure is ARMA, i.e., has a rational spectrum. The estimate is recursive in the sense that each successive estimate is obtained from the previous one by a relatively simple adjustment, that could be effected in a "real time" situation. The procedure is basically that of updating a regression when all variates involved are constructed from previous estimates of the parameter vector. The strong convergence of the estimate to the true value is established as well as a result relating to the rate of convergence.

Journal ArticleDOI
TL;DR: In this paper, the convergence of a new mixed finite element approximation of the Navier-Stokes equations was studied, which uses low order Lagrange elements and leads to an optimal order of convergence for the velocity and the pressure.
Abstract: We study in this paper the convergence of a new mixed finite element approximation of the Navier-Stokes equations. This approximation uses low order Lagrange elements, leads to optimal order of convergence for the velocity and the pressure, and induces an efficient numerical algorithm for the solution of this problem.

Journal ArticleDOI
TL;DR: A modified barrier function algorithm is suggested which turns out to have superior scaling properties which make it preferable to the classical algorithm, even in the nondegenerate case, if extrapolation is to be used to accelerate convergence.
Abstract: In a previous paper the authors have shown that the classical barrier function has an O(r) rate of convergence unless the problem is degenerate when it reduces O(r½). In this paper a modified barrier function algorithm is suggested which does not suffer from this problem. It turns out to have superior scaling properties which make it preferable to the classical algorithm, even in the nondegenerate case, if extrapolation is to be used to accelerate convergence.

Journal ArticleDOI
01 Jan 1980
TL;DR: In this article, a rete of convergence for the Wong and Zakai approximation of solutions of stochastic differential equations is obtained, based on an approximation of Browniam Motion by Transport processes.
Abstract: A rete of convergence for the Wong and Zakai approximation of solutions of stochastic differential equations is obtained, based on an approximation of Browniam Motion by Transport processes

Journal ArticleDOI
TL;DR: The Chebychev explicit method can be extended to nonsymmetric operators L whose complex eigenvalues lie within an ellipse in the complex plane and results in high execution efficiency on a “pipeline” computer.