Showing papers on "Piecewise published in 1974"

Segmentation of Plane Curves

Abstract: Piecewise approximation is described as a way of feature extraction, data compaction, and noise filtering of boundaries of regions of pictures and waveforms. A new fast algorithm is proposed which allows for a variable number of segments. After an arbitrary initial choice, segments are split or merged in order to drive the error norm under a prespecified bound. Results of computer experiments with cell outlines and electrocardiograms are reported.

Interior estimates for Ritz-Galerkin methods

Abstract: Interior a priori error estimates in Sobolev norms are derived from interior RitzGalerkin equations which are common to a class of methods used in approximating solutions of second order elliptic boundary value problems. The estimates are valid for a large class of piecewise polynomial subspaces used in practice, which are defined on both uniform and nonuniform meshes. It is shown that the error in an interior domain 2 can be estimated with the best order of accuracy that is possible locally for the subspaces used plus the error in a weaker norm over a slightly larger domain which measures the effects from outside of the domain Q. Additional results are given in the case when the subspaces are defined on a uniform mesh. Applications to specific boundary value problems are given. 0. Introduction. There are presently many methods which are available for computing approximate solutions of elliptic boundary value problems which may be classified as Ritz-Galerkin type methods. Many of these methods differ from each other in some respects (for example, in how they treat the boundary conditions) but have much in common in that they have what may be called "interior Ritz-Galerkin equations" which are the same. Here we shall be concerned with finding interior estimates for the rate of convergence for such a class of methods which are consequences of these interior equations. Let us briefly describe, in a special case, the type of question we wish to consider. Let &2 be a bounded domain in RN with boundary M2 and consider, for simplicity, the problem of finding an approximate solution of a boundary value problem (0.1) \u =f in Q2, (0.2) Au= g on U2, where A is some boundary operator. Suppose now that we are given a one-parameter family of finite-dimensional subspaces Sh (0 < h < 1) of an appropriate Hilbert space in which u lies and that, for each h, we have computed an approximate solution Uh c Sh to u using some Ritz-Galerkin type method. Here we have in mind, for example, methods such as the "engineer's" finite element method [8], [22], the Aubin-Babuska penalty method [2], [4], the methods of Nitsche [12], [13] or the Received October 15, 1973. AMS (MOS) subject classifications (1970). Primary 65N30, 65N15. Copyright i 1974, American Mathematical Society

Mesh selection for discrete solution of boundary problems in ordinary differential equations

Abstract: In order to use finite difference approximations with non-uniform meshes in boundary value problems, it is necessary to develop procedures for mesh selection. In this paper we extend techniques that have been used in piecewise polynomial approximation which permit the construction of equidistributing meshes. By this term we mean meshes on which the local truncation error of the method is approximately constant in some norm. Improved error estimates for methods which use equidistributing meshes are obtained.

A computer simulation study of Hertzian cone crack growth

Abstract: A general method for simulating on a computer the growth of the cone-shaped fracture that forms under Hertzian contact loading is outlined The program involves an incrementing procedure in which both contact circle and cone crack are grown in piecewise manner, according to suitable rate equations The contact circle expands at a rate determined by the mode of indenter loading, and thereby sets up a time-varying stress field Appropriate fracture-mechanics criteria are then invoked to calculate the response of the growing crack to the contact stresses Effects of loading mode, specimen environment and temperature, size and location of the initial flaw from which the cone crack nucleates, are investigated systematically The computer predictions compare favourably with available experimental data The results are discussed in the light of previous theoretical treatments of the Hertzian fracture problem, and some new features in the crack-growth characteristics are pointed out Calculations are made specifically for normal contact loading on glass, but ready extension of the program to other loading situations and materials in envisaged

System for processing data signals for insertion in television signals

Abstract: In any television transmission system there are periods during which the wide-band picture signals are absent. The invention provides a signal generator system for the transmission of digital information representing a series of pages of alphanumeric or graphical symbols piecewise during these periods. At least one peripheral generating device is provided for supplying digitized data to a main storage device (conveniently of the magnetic disc type) and transfer means is operable to read out portions of the data from the main storage device and store them temporarily in a second storage device. Discharge means is operable to discharge portions of the data from the second storage device at intervals to provide output signals for insertion in a television signal during periods as aforesaid.

The dimension of piecewise polynomial spaces, and one-sided approximation

Abstract: Two separate problems are discussed. One is a question implicit in the whole theory of piecewise polynomials: suppose we consider the space S of all piecewise polynomials of degree p and of continuity class Cq, say on a given triangulation in the plane. Then what is the dimension of S, and what is a convenient basis for this space? The answer is known in a dozen special cases, but not in general. The second question has arisen in the approximation of variational inequalities, but is of independent interest. We are given a nonnegative function u on a domain Ω, and want to approximate it from below by a nonnegative spline or finite element uh: o ≤ uh ≤ u. We sketch a proof that under this constraint the usual order of approximation is still possible.

Computation of constrained optimal controls using parameterization techniques

Abstract: Recently, Sirisena [1] presented an algorithm for computing optimal controls by approximating the control profiles by piecewise polynomials. In this note, the algorithm is extended to treat optimal control problems with saturation-type control constraints and boundary constraints on the state variables.

Collocation for systems of boundary value problems

Abstract: Collocation with piecewise polynomial functions is used to solve a general nonlinear system of first order differential equations subject to multi-point linear constraints. Using (vector) piecewise polynomial functions of degreed, convergence of orderO(h d+1) is achieved. Moreover, Richardson's extrapolation can be applied to the solution values at mesh points for equal spacing. A numerical example which demonstrates these features is given.

Bicubic fundamental splines in plate bending

Abstract: This paper deals with solving the two-dimensional variational problem for plate bending by the Ritz method using bicubic fundamental splines. It is a piecewise polynomial method, very adaptable to practical numerical computation, and can be an alternative for the well-known Finite Element Method1.

Conjugate Distributions for Incomplete Observations

Abstract: Reliability data often include information that the failure event has not yet occurred for some items, while observations of complete lifetimes are available for other items. Bayesian analysis requires a conjugate posterior density function after any combination of complete and incomplete observations. This paper considers density functions having failure rate function consisting of a known function multiplied by an unknown scale factor. It is shown that a gamma family of priors is conjugate for both complete and incomplete experiments. A very flexible and convenient model results from the assumption of a piecewise constant failure rate function.

Sensitive Discount Optimality in Controlled One-Dimensional Diffusions

Abstract: In this paper we consider the problem of optimally controlling a diffusion process on a compact interval in one-dimensional Euclidean Space. Under the assumptions that the action space is finite and the cost rate, drift and diffusion coefficients are piecewise analytic, we present a constructive proof that there exist piecewise constant n-discount optimal controls for all finite n _ 1 and measurable oo-discount optimal controls. In addition we present a sequence of second order differential equations that characterize the coefficients of the Laurent series of the expected discounted cost of an n-discount optimal control. 1. Introduction. In this paper we consider the problem of optimally controlling a one-dimensional diffusion process on a compact interval, where the coefficients of the infinitesimal operator and the costs are piecewise analytic functions and the set of possible actions is finite. We extend Pliska's results [12] concerning the existence of piecewise constant optimal controls for the expected discounted cost criterion to the case where the discount rate is allowed to decrease to zero. The optimality criterion used for studying this problem is n-discount optimality. Our results and those of Pliska depend heavily on Mandl's work [10]. The criterion of n-discount optimality was introduced in the finite state and action Markovian decision problem in the case n = 0 and n = + oo by Blackwell [1] and extended to include all n ? - 1 by Veinott [16]. In these papers it is proved that there exist stationary n-discount optimal policies for all n. In the paper by Veinott [16] it is shown that these results are also valid for the continuous time model. In [2] Denardo applies this criterion to Markov renewal programming. Jaquette [9] introduces a new optimality criterion, moment optimality and using expansions analogous to ours proves that there exists moment optimal policies for all small interest rates. In order to prove the existence of n-discount optimal controls, we first prove the validity of the Laurent expansion of the expected discounted cost in powers of the discount rate. This approach was first used by Miller and Veinott [11] in the finite state and action case. In Section 3 we give a development of this theory. In Theorem 4 we present a recursive scheme for computing the terms

Higher order compatible triangular finite elements

Abstract: For a triangulated polygon Ω, classes of piecewise rational functions are defined which are inC p?1 (Ω). A basis of these functions which contains all polynomials of degree 2p?1 or less is considerably smaller than a similar basis of piecewise polynomial functions.

Capacity-achieving input distributions for some amplitude-limited channels with additive noise (Corresp.)

Abstract: An additive noise channel wherein the noise is described by a piecewise constant probability density is shown to reduce to a discrete channel by means of an explicit construction. In addition, conditions are found which describe a class of continuous amplitude-limited channels for which the capacity-achieving input distribution is binary.

Global solution of the generalized Abel integral equation by implicit interpolation

Abstract: The construction of a (global) approximate solution for a given generalized Abel integral equation may be viewed as a problem of (implicit) interpolation in a prescribed linear space. In this paper, piecewise polynomials (extended spline functions) of a given degree and of class C are used to generate such an approximating function. Results on convergence and error bounds are given, and the practical application of this method is illustrated by a numerical example.

Inelastic Stability of Unbraced Building Frames

Abstract: Strain reversal is taken into account in terms of generalized stress and generalized strain. The stiffness matrix is so derived that it is applicable to any piecewise elastic regime in the elastic-plastic deformation of a member, the effect of the axial force being taken into account by means of approximate stability functions. Numerical computation for the smallest critical load can be executed by following the standard matrix procedure for calculating eigenvalues. The proposed analysis facilitates the study of not only the stability limit, but also those states beyond and remote from this point. The proposed analysis can be applied to the practical design of such structures and is demonstrated in detail by means of simple numerical examples. A comparison of the result with that obtained by means of the exact stability functions is made and found to be in good agreement. The influence of unloaded plastic hinges on the ultimate load is illustrated.

Digital hardware for approximating to the amplitude of quadrature pairs

Abstract: A method for calculating the square root of the sum of two squared variables by a piecewise linear approximation is presented. Simplicity is obtained by restricting the number of 1s in the hard-wired coefficients and by using well suited intervals for the linear approximation. To obtain a small minimax error, a constant gain is introduced and optimised along with the coefficients.

Zur approximation des transfiniten durchmessers bei bis auf ecken analytischen geschlossenen jordankurven

Abstract: LetC be a closed Jordan curve in the complex plane and letf(z)=dz+a0+a1z−1+… be the analytic function mapping |z|>1 schlicht onto the exterior ofC (d>0 is the transfinite diameter ofC). Similar to the Fekete points a point system will be defined calledextremal points. One can use the Fekete points or the extremal points to approximated. The author has proved [3] that in the case of an analytic closed Jordan curve the approximation ofd by means of extremal points is much better than the approximation ofd by the use of Fekete points. Here we show how to approximated by means of extremal points in the case of a piecewise analytic, closed Jordan curve possessing corners of openingαπ (0<α<2).

$L^\infty $-Error Bounds for Multivariate Lagrange Approximation

Abstract: This paper contains $L^\infty $-error bounds for piecewise Lagrange polynomial approximation over (i) a Cartesian product of polyhedral domains, (ii) a rectangular parallelepiped, and (iii) the Cartesian product of a domain of the type (i) and a domain of the type (ii). Bounds are given for the error as well as all its possible partial derivatives, in terms of characteristic mesh lengths of the partitions, for functions with specified smoothness properties.