Showing papers on "Rate of convergence published in 1978"

The convergence of variable metric methods for nonlinearly 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).

R‐matrix solution of coupled equations for inelastic scattering

Abstract: The R‐matrix propagation method for solving the close coupled equations for inelastic scattering is presented. The method is shown to be fast, accurate, and stable without the need for stabilizing transformations. We demonstrate the use of this method for two model close coupling problems, and investigate the rate of convergence of the solutions.

Maximum norm estimates in the finite element method on plane polygonal domains. I

Abstract: The finite element method is considered when applied to a model Dirichlet problem on a plane polygonal domain. Rate of convergence estimates in the maximum norm, up to the boundary, are given locally. The rate of convergence may vary from point to point and is shown to depend on the local smoothness of the solution and on a possible pollution effect. In one of the applications given, a method is proposed for calculating the first few coefficients (stress intensity factors) in an expansion of the solution in singular functions at a corner from the finite element solution. In a second application the location of the maximum error is determined. A rather general class of non-quasi-uniform meshes is allowed in our present investigations. In a subsequent paper, Part 2 of this work, we shall consider meshes that are refined in a systematic fashion near a corner and derive sharper results for that case. 0. Introduction. Let Q2 be a bounded simply connected domain in the plane with boundary U2 consisting of a finite number of straight line segments meeting at vertices v;, j = 1, . . . , M, of interior angles 0 < oil S < aM < 27r (in a suitable ordering). We shall consider the Dirichlet problem -Au = f in Q2, (0.1) u= 0 on Q, where f is a given function, which for simplicity we assume to be smooth. To solve the problem (0.1) numerically, let Sh = Sh(2), 0 < h < 1, denote a one-parameter family of finite dimensional subspaces of H1 (2) n w2 (Q). We have in mind piecewise polynomials of a fixed degree on a sequence of partitions of Q2. In our considerations the partitions do not have to be quasi-uniform, not even locally (cf. examples in Section 9). Let uh E Sh be the approximate solution of (0.1) defined by the relation (0.2) A(uh, X) = (f X) for all X Esh. Here A(v, w) = fI Vv VW dx, and (v, w) = fvw dx. We wish to obtain local estimates up to the boundary in the maximum norm for the error u uh. Although our present assumptions allow meshes that are refined near a corner, in the subsequent paper, Part 2, we shall investigate the error in more detail in that case, and obtain sharper results. The general results derived in the present papei will be essential in those investigations. Received April 25, 1977. AMS (MOS) subject classifications (1970). Primary 65N30, 65N15. *This work was supported in part by the National Science Foundation. Copyright i 1978, American Mathematical Society

An adaptive lattice structure for noise-cancelling applications

Abstract: This paper describes an adaptive filter structure which may be used in multi-channel noise-cancelling applications. The proposed structure differs from those presented previously in that it incorporates a lattice filter framework, rather than tapped-delay-lines. The successive orthogonalization provided by the lattice offers advantages in adaptive convergence rate which cannot be achieved with tapped-delay-lines. In the sections below, we present an explicit description of the general noise-cancelling lattice structure, together with the appropriate adaptive algorithms.

Choosing the window width when estimating a density

Abstract: where Xl,...,Xn are independent identically distributed real observations, a is a kernel function and h(n) is a sequence of window widths, assumed to tend to zero as n tends to infinity. The kernel estimator has been widely discussed; for a survey see Rosenblatt (1971). When applying the method in practice it is of course necessary to choose a kernel and a window width. The choice of kernel was considered by Epachenikov (1969) who showed that there is in some sense an optimal kernel, which is part of a parabola, but that any reasonable kernel gives almost optimal results. Therefore the choice of kernel is not as important a problem in practice as might be supposed. It is quite satisfactory to choose a kernel for computational convenience, as below, or for any other attractive reason, such as, for example, the argument leading to the quadratic spline kernel used by Boneva, Kendall & Stefanov (1971) in their 'spline transform' technique. While the choice of kernel does not seem to lead to much difficulty, at least for reasonably large sample sizes, the choice of window width is quite a different matter. The results of Silverman (1978) show that the kernel estimate is uniformly consistent under quite mild conditions on the rate of convergence of the window width to zero, but that the rate of consistency can be very slow. The very interesting practical work of Boneva et al. (1971) shows that the estimates can change dramatically under quite small variations in window width. Thus there seems to be considerable need for objective methods of determining the wiindow width appropriate to a given sample. The method developed below, while being subjective to some extent, goes a long way towards resolving this difficulty in certain cases. First the method is described and some applications to sets of data are considered. The application of the method to multivariate data is then discussed. Finally, the theoretical justification of the method is obtained.

The Fourier method for nonsmooth initial data

Abstract: Application of the Fourier method to very general linear hyperbolic Cauchy problems having nonsmooth initial data is considered, both theoretically and computationally. In the absence of smoothing, the Fourier method will, in general, be globally inaccurate, and perhaps unstable. Two main results are proven: the first shows that appropriate smoothing techniques applied to the equation gives stability; and the second states that this smoothing combined with a certain smoothing of the initial data leads to infinite order accuracy away from the set of discontinuities of the exact solution modulo a very small easily characterized exceptional set. A particular implementation of the smoothing method is discussed; and the results of its application to several test problems are presented, and compared with solutions obtained without smoothing. Introduction. In recent years the Fourier method for the numerical approximation of solutions to hyperbolic initial value problems has been used quite successfully. In fact, if the initial function is C°° and the coefficients of the equation are constant the method converges arbitrarily fast, i.e. is limited in practice only by the method of time discretization which is chosen. This is the reason that the Fourier method is caled "infinite order" accurate. However, the situation is drastically different when the initial function is not smooth. We take as a model the one space dimension scalar problem ut = ux to be solved for 2ir periodic u on the interval n < x < n with initial values

Linearly constrained minimax optimization

Abstract: We present an algorithm for nonlinear minimax optimization subject to linear equality and inequality constraints which requires first order partial derivatives. The algorithm is based on successive linear approximations to the functions defining the problem. The resulting linear subproblems are solved in the minimax sense subject to the linear constraints. This ensures a feasible-point algorithm. Further, we introduce local bounds on the solutions of the linear subproblems, the bounds being adjusted automatically, depending on the quality of the linear approximations. It is proved that the algorithm will always converge to the set of stationary points of the problem, a stationary point being defined in terms of the generalized gradients of the minimax objective function. It is further proved that, under mild regularity conditions, the algorithm is identical to a quadratically convergent Newton iteration in its final stages. We demonstrate the performance of the algorithm by solving a number of numerical examples with up to 50 variables, 163 functions, and 25 constraints. We have also implemented a version of the algorithm which is particularly suited for the solution of restricted approximation problems.

Convergent nonnegative matrices and iterative methods for consistent linear systems

Abstract: In this paper we study linear stationary iterative methods with nonnegative iteration matrices for solving singular and consistent systems of linear equationsAx=b. The iteration matrices for the schemes are obtained via regular and weak regular splittings of the coefficients matrixA. In certain cases when only some necessary, but not sufficient, conditions for the convergence of the iterations schemes exist, we consider a transformation on the iteration matrices and obtain new iterative schemes which ensure convergence to a solution toAx=b. This transformation is parameter-dependent, and in the case where all the eigenvalues of the iteration matrix are real, we show how to choose this parameter so that the asymptotic convergence rate of the new schemes is optimal. Finally, some applications to the problem of computing the stationary distribution vector for a finite homogeneous ergodic Markov chain are discussed.

Asymptotic $L^\infty $-Error Estimates for Linear Finite Element Approximations of Quasilinear Boundary Value Problems

Abstract: The authors consider linear finite element approximations of quasilinear boundary value problems and obtain the almost optimal rate of convergence $O(h^2 |\ln h|^q ),q = q(n)$, with respect to the $L^\infty $-norm.

Approximation of the Biharmonic Equation by a Mixed Finite Element Method

Abstract: “Optimal” order of convergence estimates are derived for a new mixed element approximation for the biharmonic problem.

One-step methods of any order for ordinary differential equations with discontinuous right-hand sides

Abstract: The right-hand sides of a system of ordinary differential equations may be discontinuous on a certain surface. If a trajectory crossing this surface shall be computed by a one-step method, then a particular numerical analysis is necessary in a neighbourhood of the point of intersection. Such an analysis is presented in this paper. It shows that one can obtain any desired order of convergence if the method has an adequate order of consistency. Moreover, an asymptotic error theory is developed to justify Richardson extrapolation. A general one-step method is constructed satisfying the conditions of the preceding theory. Finally, a simplified Newton iteration scheme is used to implement this method.

The Nonstationary Infinite Horizon Inventory Problem

Abstract: By allowing disposal in each period, a sequence of upper and lower bounds on the infinite horizon nonstationary periodic review inventory problem is obtained. The nth bounds depend only on knowledge of the demand distributions in the first n periods, giving planning horizon results. In general, calculation of any bound requires solution of an ordinary n period problem, although for the proportional cost case the solution for n = 1 and good approximations for n = 2 may easily be given. The upper and lower relative cost and policy bounds are monotonic in the horizon length and, under mild conditions, converge to a unique infinite horizon solution. An analytic expression is given for the asymptotic convergence rate, which is geometric even for the undiscounted case. The formulation given here is a concrete example of the general planning horizon formulation for infinite horizon dynamic programming models set forth in [Morton, T. 1975. Infinite horizon dynamic programming models—a planning horizon formulation...

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.

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.

Rates of Convergence

Abstract: In Section 7.1, rate of convergence is defined, and our approach to the rate problem discussed. The rates are developed (in Section 7.3) for three separate cases, two forms of the basic KW procedure and the basic RM procedure. These algorithms are discussed in Section 7.1 and are put into a form which will be useful in the subsequent development. The results of particular interest are the expressions for {Un} given by (7.1.3), (7.1.6) and (7.1.10), and the necessary (resp., best) values for 3 of (7.1.4) ((7.1.5), resp.), and similarly for the other algorithms Section 7.2 gives some notation and lists and discusses several sets of conditions. The rate of convergence theorems are stated and proved in Section 7.3. Section 7.4 contains a discussion of the value of averaging several observations per iteration and compares the basic KW procedure (Theorem 2.3. 5) with the KW procedure when the directions are chosen at random (Theorem 2.3.6).

Extended iterative methods for the solution of operator equations

Abstract: Given an iterative methodM 0, characterized byx (k+1=G 0(x( k )) (k?0) (x(0) prescribed) for the solution of the operator equationF(x)=0, whereF:X?X is a given operator andX is a Banach space, it is shown how to obtain a family of methodsM p characterized byx (k+1=G p (x( k )) (k?0) (x(0) prescribed) with order of convergence higher than that ofM o. The infinite dimensional multipoint methods of Bosarge and Falb [2] are a special case, in whichM 0 is Newton's method. Analogues of Theorems 2.3 and 2.36 of [2] are proved for the methodsM p, which are referred to as extensions ofM 0. A number of methods with order of convergence greater than two are discussed and existence-convergence theorems for some of them are proved. Finally some computational results are presented which illustrate the behaviour of the methods and their extensions when used to solve systems of nonlinear algebraic equations, and some applications currently being investigated are mentioned.

Improved difference schemes for the Dirichlet problem of Poisson's equation in the neighbourhood of corners

Abstract: The standard 5-point difference scheme for the model problem Δu=f on a special polygonal domain with given boundary values is modified at a few points in the neighbourhood of the corners in such a way that the order of convergence at interior points is the same as in the case of a smooth boundary. As a side result improved error bounds for the usual method in the neighbourhood of corners are given.

Discretization of Volterra integral equations of the first kind (II)

Abstract: In the present paper integral equations of the first kind associated with strictly monotone Volterra integral operators are solved by projecting the exact solution of such an equation into the spaceS m (?1) (Z N ) of piecewise polynomials of degreem?0, possessing jump discontinuities on the setZ N of knots. Since the majority of "direct" one-step methods (including the higher-order block methods) result from particular discretizations of the moment integrals occuring in the above projection method we obtain a unified convergence analysis for these methods; in addition, the above approach yields the tools to deal with the question of the connection between the location of the collocation points used to determine the projection inS m (?1) (Z N ) and the order of convergence of the method.

Convergence Rate in Adaptive Arrays.

Abstract: : A generalized expression has been derived for the weight jitter noise in the output of a Gram-Schmidt self orthogonalizing network of any dimensionality. In earlier quarterly reports on this contract, it was shown that these networks transform the inputs to a set of independent variables, the networks can be utilized readily in either pilot signal or power inversion arrays for communications, and that this transformation provides fast convergence in cases of disparate eigenvalues where the more conventional circuits converge slowly. An alternative method of implementing a self-orthogonalizing array, using full feedback along each column, is more tolerant to failures at nodes in the network, but generates more weight jitter noise in the output for a given convergence rate.

Two-phase algorithm for nonlinear constraint problems

Abstract: An algorithm is described which under appropriate hypotheses solves the general nonlinear programming problem with nonlinear constraints. The computational implementation is based on the iterative use of any available package which solves the linearly constrained problem with a nonlinear objective function. Large, sparse problems with nonlinear constraints can be solved efficiently by this algorithm, when a suitable linear constraint package is used. The algorithm consists of two phases. The first (Phase I) uses an external squared penalty function to find a point x 1 , close to a local minimum. Starting with x 1 , the algorithm then solves a sequence of linearly constrained problems (Phase II). Selected nonlinear constraints are linearized for each such Phase II iteration. With suitable assumptions, convergence from any initial point, with quadratic convergence in Phase II, is shown. The practical implementation of this algorithm is described, and its potential application to a model for the assessment of energy alternatives is discussed briefly.

Discrete least squares approximations for ordinary differential equations

Abstract: The application of the least squares method, using $C^q $ piecewise polynomials of order $k + m,k \geqq m,q \geqq m$, for obtaining approximations to an isolated solution of a nonlinear mth order ordinary differential equation, involves integrals which in practice need to be discretized. Using for this latter purpose the k-point Gaussian quadrature rule in each subinterval, the discrete least squares schemes obtained are close to collocation, on the same points, by piecewise polynomials from $C^{m - 1} $.We prove here that under smoothness assumptions similar to those made by de Boor and Swartz for the collocation procedure, i.e. that the solution be in $C^{m + 2k} $, an optimal global rate of convergence $O(|\Delta |^{k + m} )$ is obtained in the uniform norm for the discrete least squares schemes, provided that the partitions $\Delta $ are quasiuniform. In addition, a superconvergence rate of $O(|\Delta |^{2k} )$ is obtained at the knots for those derivatives l which satisfy $0 \leqq l \leqq 2(m - 1) - q$.

Moment Inequalities for Sn Under General Dependence Restrictions, with Applications.

Abstract: : Consider a sum composed of a sequence of random variables and a sequence of constants. This paper establishes moment inequalities with dependence restrictions imposed upon the random variables but not depending upon the constants. A further inequality of more complicated form is also established. The dependence restrictions considered are either of the weak multiplicative type or of related types, namely exchangeable sequences and strongly mixing sequences. Three applications are developed. One treats the almost sure convergence of series under mild dependence restrictions and finite limit conditions. Secondly, an improved technique is presented for the problem of establishing the rate of convergence in the central limit theorem for simple linear rank statistics. Finally, the central limit theorem for strongly mixing summands is treated.

The Order of Convergence of General Linear Methods for Ordinary Differential Equations

Abstract: General linear methods, used for the numerical solution of ordinary differential equations, may be characterized by a pair of matrices $(A,B)$. In this article, the order of convergence of such methods is obtained by defining a sequence of norms associated with an order vector. The order of consistency is defined in terms of these norms and it is shown that this definition is sufficient to establish the order of convergence of conventional stable methods. The general case requires additional properties which are also defined in terms of the norms. Some examples of unconventional methods are given.

The flow between rotating coaxial disks

Abstract: Numerical approximations of nonunique solutions of the Navier-Stokes equations are obtained for steady viscous incompressible axisymmetric flow between two infinite rotating coaxial disks. For example, nineteen solutions have been found for the case when the disks are rotating with the same speed but in opposite direction. Bifurcation and perturbed bifurcation phenomena are observed. An efficient method is used to compute solution branches. The stability of solutions is analyzed. The rate of convergence of Newton's method at singular points is discussed. In particular, recovery of quadratic convergence at "normal limit points" and bifurcation points is indicated. Analytical construction of some of the computed solutions using singular perturbation techniques is discussed.

The analytic theory of policy iteration

Abstract: Publisher Summary The policy iteration method of dynamic programming was studied in an abstract setting by Puterman and Brumelle. Motivated by continuous time examples, they viewed the dynamic programming problem as that of obtaining a zero for the optimality equation. They showed that the policy iteration procedure is equivalent to Newton–Kantorovich iteration. Under the assumptions that suprema are obtained and that certain transition-type operators possess nonpositive inverses, this equivalence was exploited to present a convergence theory for policy iteration, including rates of convergence and error bounds. This chapter reviews the work of Puterman and Brumelle. It explores some implications of relaxing their assumptions and presents further applications of this theory. Discrete time problems with general state spaces are also studied in the chapter. The chapter also discusses the results for a discounted, finite state problem including rate of convergence calculations.