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Showing papers on "Piecewise published in 1980"


Journal ArticleDOI
TL;DR: In this article, a monotone piecewise bicubic interpolation algorithm was proposed for data on a rectangular mesh, where the first partial derivatives and first mixed partial derivatives are determined by the mesh points.
Abstract: In a 1980 paper [SIAM J. Numer. Anal., 17 (1980), pp. 238–246] the authors developed a univariate piecewise cubic interpolation algorithm which produces a monotone interpolant to monotone data. This paper is an extension of those results to monotone $\mathcal{C}^1 $ piecewise bicubic interpolation to data on a rectangular mesh. Such an interpolant is determined by the first partial derivatives and first mixed partial (twist) at the mesh points. Necessary and sufficient conditions on these derivatives are derived such that the resulting bicubic polynomial is monotone on a single rectangular element. These conditions are then simplified to a set of sufficient conditions for monotonicity. The latter are translated to a system of linear inequalities, which form the basis for a monotone piecewise bicubic interpolation algorithm.

2,174 citations


Journal ArticleDOI
TL;DR: Two algorithms for parametric piecewise polynomial evaluation and generation are described and are shown to generalize to new algorithms for obtaining curve and surface intersections and for the computer display of parametric curves and surfaces.
Abstract: Two algorithms for parametric piecewise polynomial evaluation and generation are described. The mathematical development of these algorithms is shown to generalize to new algorithms for obtaining curve and surface intersections and for the computer display of parametric curves and surfaces.

538 citations


Journal ArticleDOI
TL;DR: A class of piecewise continuous mappings with positive slope, mapping the unit interval into itself is studied in this article, where the asymptotic behavior depends sensitively on initial data in that the rotation number is either a nowhere continuous function of initial data, or else it is a constant on all but a Cantor set of the unit intervals.
Abstract: A class of piecewise continuous mappings with positive slope, mapping the unit interval into itself is studied. Families of 1-1 mappings depending on some parameter have periodic orbits for most parameter values, but have an infinite invariant set which is a Cantor set for a Cantor set of parameter values. Mappings which are not 1-1 exhibit chaotic behavior in that the asymptotic behavior as measured by the rotation number covers an interval of values. The asymptotic behavior depends sensitively on initial data in that the rotation number is either a nowhere continuous function of initial data, or else it is a constant on all but a Cantor set of the unit interval.

180 citations


Journal ArticleDOI
TL;DR: In this article, a general numerical method to solve two-point seismic-ray tracing problems in a heterogeneous isotropic medium and how to solve them numerically is presented.
Abstract: A study of two-point seismic-ray tracing problems in a heterogeneous isotropic medium and how to solve them numerically will be presented in a series of papers. In this Part 1, it is shown how a variety of two-point seismic-ray tracing problems can be formulated mathematically as systems of first-order nonlinear ordinary differential equations subject to nonlinear boundary conditions. A general numerical method to solve such systems in general is presented and a computer program based upon it is described. High accuracy and efficiency are achieved by using variable order finite difference methods on nonuniform meshes which are selected automatically by the program as the computation proceeds. The variable mesh technique adapts itself to the particular problem at hand, producing more detailed computations where they are needed, as in tracing highly curved seismic rays. A complete package of programs has been produced which use this method to solve two- and three-dimensional ray-tracing problems for continuous or piecewise continuous media, with the velocity of propagation given either analytically or only at a finite number of points. These programs are all based on the same core program, PASVA3, and therefore provide a compact and flexible tool for attacking ray-tracing problems in seismology. In Part 2 of this work, the numerical method is applied to two- and three-dimensional velocity models, including models with jump discontinuities across interfaces.

128 citations


Journal ArticleDOI
TL;DR: It is shown that under reasonable assumptions, solving a multistage stochastic program with recourse is equivalent to solving a nested sequence of piecewise quadratic programs and the algorithm presented in an earlier report is extended to the multistages situation.
Abstract: We consider a multistage stochastic program with recourse, with discrete distribution, quadratic objective function and linear inequality constraints. We show that under reasonable assumptions, solving such a program is equivalent to solving a nested sequence of piecewise quadratic programs and we extend the algorithm presented in an earlier report to the multistage situation. Finally, we consider the application of the method to an energy investment problem and report on the results of numerical experiments.

105 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered the problem of finding a finite element approximation to the solution of a linear elliptic boundary value problem on a square, and established an approximation in a function space consisting of tensor products of piecewise polynomials of degree not greater than r.
Abstract: Collocation at Gaussian quadrature points as a means of determining a $C^1 $ finite element approximation to the solution of a linear elliptic boundary value problem on a square is studied. Optimal order $L^2 $ and $H^1 $ error estimates are established for approximation in a function space consisting of tensorproducts of $C^1 $ piecewise polynomials of degree not greater that r, where $r \geqq 3$.

90 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the finite element method is bounded in Lo0, on polygonal domains, by a discrete weak maximum principle of the form 11uhIllL 42.
Abstract: Let Q2 be a polygonal domain in the plane and Shr(92) denote the finite element space of continuous piecewise polynomials of degree 2) defined on a quasi-uniform triangulation of Q2 (with triangles roughly of size h) It is shown that if uh E Sh(n) is a "discrete harmonic function" then an a priori estimate (a weak maximum principle) of the form 11uhIllL 42) This says that (modulo a logarithm for r = 2) the finite element method is bounded in Lo0, on plane polygonal domains 0 Introduction and Statement of Results The purpose of this paper is to discuss some estimates for the finite element method on polygonal domains In particular, we shall consider the validity of (for want of a better terminology) a "discrete weak maximum principle" for discrete harmonic functions and then use this result to discuss the boundedness in Loo of the finite element projection In this part we shall discuss the case of a quasi-uniform mesh In Part II we shall concern ourselves with meshes which are refined near points Let us first formulate the problems we wish to consider and state our results References to other work in the literature which are relevant to our considerations will be given as we go along For simplicity let Q2 be a simply connected (this is not essential) polygonal domain in R2 with boundary 3Q2 and maximal interior angle a, 0 < a < 2ir, where we emphasize that in general Q2 is not convex On Q2 we define a -family of finite element spaces For simplicity of presentation we shall restrict ourselves to a special but important class of piecewise polynomials For each 0 < h < 1, let Th denote a triangulation of Q2 with triangles having straight edges We shall assume that each triangle r is contained in a sphere of radius h and contains a sphere of radius yh for some positive constant y We shall also assume that the family {Tn } of triangulations Received September 19, 1978 AMS (MOS) subject classifications (1970) Primary 65N30, 65N15 *This work was supported in part by the National Science Foundation ? 1980 American Mathematical Society 0025-571 8/80/0000-0004/$0475 77 This content downloaded from 1575539253 on Wed, 08 Jun 2016 05:26:01 UTC All use subject to http://aboutjstororg/terms

84 citations


Journal ArticleDOI
TL;DR: This hierarchical modeling approach introduces criteria for segmentation, clustering, and identification of character substrings which reduce the need for ad hoc parameters or subjective decisions and is compatible with a generalized classification procedure.
Abstract: Describes a technique for quantitative analysis of EEG signals which is based on a hierarchy of models. These models include 1) recursively estimated autoregressive model, 2) piecewise stationary autoregressive model, 3) composite source model, and 4) character string and syntactic models. This hierarchical modeling approach introduces criteria for segmentation, clustering, and identification of character substrings which reduce the need for ad hoc parameters or subjective decisions. The hierarchical representation is therefore highly operator independent, provides significant data compression of complex signals, and is compatible with a generalized classification procedure. Examples of the application of this modeling approach to clinical patient EEG data illustrate the system capabilities.

73 citations


Journal ArticleDOI
TL;DR: An analytical solution to the kinematic wave approximation for unsteady flow routing is presented in this article, which allows time-dependent lateral inflow with piecewise spatial uniformity and can be applied to complex Kinematic cascades.
Abstract: An analytical solution to the kinematic wave approximation for unsteady flow routing is presented. The model allows time-dependent lateral inflow with piecewise spatial uniformity and can be applied to complex kinematic cascades. Kinematic shocks are considered as manifestations of higher-order effects such as rnonoclinal flood waves, bores, etc. Within the context of kinematic approximation therefore we retain their dynamic effects by routing the discontinuities as they appear. Certain simplifying assumptions are made which permit closed form solutions and an efficient numerical algorithm, based on the method of characteristics, is employed. The resulting model, called an approximate shock-fitting scheme, preserves the effect of the shocks without the usual computational complications and compares favorably with an implicit finite difference solution. The efficiency and accuracy of the new method are illustrated by computing a variety of unsteady flows, ranging from simple cascades to complex natural watersheds.

72 citations


Journal ArticleDOI
TL;DR: In this paper, a method of extending computation beyond the limit of the initial normal interval in Walsh series analysis of dynamical systems is presented, where the solution may be extended over contiguous normalized intervals of time to any length.
Abstract: This paper presents a method of extending computation beyond the limit of the initial normal interval in Walsh series analysis of dynamical systems. The solution may be extended over contiguous normalized intervals of time to any length. The procedure avoids operational matrices of prohibitively large size and reduces the computational effort to a minimum while retaining accuracy. As a by-product of the method, the solution at the segment junctions is also available in addition to the one in piecewise constant form. The formula also provides information regarding the stability of the resulting progression.

67 citations



Journal ArticleDOI
TL;DR: In this article, it has been shown that the Pontrjagin maximum principle can be used as an assumption on a closed-loop control to generate open-loop optimal controls.

Journal ArticleDOI
TL;DR: In this article, the most significant aspects of a moment method surface patch/wire formulation are speed, accuracy, convergence, and versatility, and techniques for improving these parameters are discussed and applied to a solution based on the piecewise sinusoidal reaction formulation.
Abstract: The most significant aspects of a moment method surface patch/wire formulation are speed, accuracy, convergence, and versatility. Techniques for improving these parameters are discussed and applied to a solution based on the piecewise sinusoidal reaction formulation.

Journal ArticleDOI
TL;DR: In this article, the LP formulation for finite zero-sum games with incomplete information using Bayesian mixed strategies is given, and general properties for the value of such games, such as concave-convex properties and piecewise bilinearity are derived.
Abstract: This paper gives the LP formulation for finite zero sum games with incomplete information using Bayesian mixed strategies. This formulation is then used to derive general properties for the value of such games, the well known concave-convex property but also the “piecewise bilinearity”. These properties may offer considerable help for computational purposes but also provide structural guidelines for the analysis of special classes of games with incomplete information.

Journal ArticleDOI
TL;DR: In this article, it was shown that the transformation x↦gbx+α (mod 1) (β>1, 0≦α<1) on [0, 1] has unique maximal measure.
Abstract: It is shown that the transformation x↦gbx+α (mod 1) (β>1, 0≦α<1) on [0, 1] has unique maximal measure.

Journal ArticleDOI
TL;DR: Directed canonical analysis as discussed by the authors is an extension of the general form of canonical analysis, which is a method for reducing the dimensionality of multivariate data sets with minimum loss of discriminatory variance.
Abstract: Directed canonical analysis is presented as an extension of the general form of canonical analysis, which is a method for reducing the dimensionality of multivariate data sets with minimum loss of discriminatory variance. The reduction takes the form of a linear transformation, y = Cx, that condenses the discriminatory variance onto a relatively few, high-variance orthogonal discriminant axes. Canonical analysis is developed as an analog to the one-way MANOVA. The directed extension allows user-specified contrasts to define linear relationships that are known or suspected to exist within the data. The linear transformation C is defined by means of the symmetric canonical form of the matrix eigenproblem. Canonical and principal components transformations and various distance classifiers were applied to 3 representative remotely sensed MSS data sets. Results indicate that use of a piecewise maximum likelihood classifier with the directed canonical discriminant axes will give the best overall combination of classification accuracy and computational efficiency if adequate sample sizes are available to estimate category statistics. For small sample sizes, piecewise Euclidean distance with the general canonical axes is recommended. In canonically transformed space, Euclidean distance is equivalent to the Mahalanobis classifier.

Journal ArticleDOI
01 Sep 1980
TL;DR: In this paper, the development of theory and algorithms relating to interpolation to data by functions which preserve the monotonicity and/or convexity of the data is presented.
Abstract: The development of theory and algorithms relating to interpolation to data by functions which preserve the monotonicity and/or convexity of the data is presented. The functions used for interpolation are polynomials, piecewise polynomials, polynomial splines and exponential splines. The techniques emphasized are the shape-preserving splines of the author, although some discussion of alternate techniques is given.

Journal ArticleDOI
TL;DR: In this paper, an analytical technique has been suggested for predetermination of true saturation characteristic of transformers and reactors from the manufacturer data giving the conventional rms saturation characteristic, which has some advantages over that suggested by Talukdar et al.
Abstract: An analytical technique has been suggested for predetermination of true saturation characteristic of transformers and reactors from the manufacturer data giving the conventional rms saturation characteristic. The method has some advantages over that suggested by Talukdar et al.

Journal ArticleDOI
Karl Scherer1
TL;DR: In this paper, a general method of obtaining optimal global error bounds by scaling local error estimates is developed, which is reduced to the solution of a fixpoint problem, and the exact asymptotic behaviour of the approximation with n parameters is determined.
Abstract: In the second section a general method of obtaining optimal global error bounds by scaling local error estimates is developed. It is reduced to the solution of a fixpoint problem. In Sect. 3 we show more concrete error estimates reflecting a singularity of order ?. It is shown that under general circumstances an optimal global error bound is achieved by an (asymptotically) geometric mesh for the local error estimates. In the fourth section we specialize this to the best approximation ofg(x)x ? by piecewise polynomials with variable knots and degrees having a total numberN of parameters. This generalizes the result of R. DeVore and the author forg(x)=1. In the last section this problem is studied for the functione ?x on (0, ?). The exact asymptotic behaviour of the approximation withN parameters is determined toe qoN , whereq o=0.895486 ....

Journal ArticleDOI
TL;DR: The symbol map of Gohberg and Krupnik as mentioned in this paper for the closed algebra generated by singular integral operators with piecewise continuous coefficients is extended to the case of curves with corners.
Abstract: The symbol map of Gohberg and Krupnik [6] for the closed algebra generated by singular integral operators with piecewise continuous coefficients is extended to the case of curves with corners. This algebra includes the operator of the double layer potential on such curves.

Journal ArticleDOI
TL;DR: In this article, the principle of constant proportionality is introduced to provide an interesting interpretation for scaling multiple-choice data a la Guttman, and a small example is presented for discussion of the technique.
Abstract: The proposed method handles the classical method of reciprocal averages (MRA) in a piecewise (item-by-item) mode, whereby one can deal with smaller matrices and attain faster convergence to a solution than the MRA. A new concept “the principle of constant proportionality” is introduced to provide an interesting interpretation for scaling multiple-choice data a la Guttman. A small example is presented for discussion of the technique.

Book ChapterDOI
01 Jan 1980
TL;DR: In this paper, the authors presented the problem of approximating the function xβ, β > 0 by piecewise polynomials with variable knots and degree, and the asymptotic rate of approximation was determined by splines with free knots and variable degree in terms of the total number of free parameters.
Abstract: Publisher Summary This chapter discusses variableknot and variable degree spline approximation to xβ. The asymptotic rate of approximation of the function xβ is determined by splines with free knots and variable degree in terms of the total number of free parameters. There are several examples of functions that can be approximated more efficiently by splines with variable knots than by splines with fixed knots. These functions generally have singularities and the knots are bunched near the singularities. Another possible way to improve the approximation of a function is by varying the degree of the polynomials that make up the approximating spline. The chapter presents the problem of approximating the function xβ, β > 0 by piecewise polynomials with variable knots and degree.

Journal ArticleDOI
TL;DR: In this paper, a large, digitized data base is employed in a detailed examination of the mathematical form of the point rainfall-rate distribution function, and the optimum form is found to depend on both the data sampling rate and the rain rate limits considered.
Abstract: A large, digitized data base is employed in a detailed examination of the mathematical form of the point rainfall-rate distribution function. The optimum form is found to depend on both the data sampling rate and the rain rate limits considered. In general, the lognormal function appears to provide a very good approximation to the distribution. It is found, however, that a better fit is provided by piecewise power-law approximations to different portions of the distribution. As the sampling interval is reduced to the ultimate limit imposed by the tipping bucket itself, a single power relationship is found to provide the best fit over the range of rainfall rates from several mm/h to the observed upper limit.

Journal ArticleDOI
TL;DR: In this paper, the p-analytic mappings of simply or doubly connected domains on rectangles or circular rings are considered and error bounds are given for smooth and for piecewise constant functions.
Abstract: In this note we consider so called "p-analytic" mappings of simply or doubly connected domains on rectangles or circular rings. Real and imaginary parts of the mappings can be described by minimal-principles. By minimizing the corresponding functionals in a class of linear or bilinear finite elements we obtain an approximation of the mapping and also upper and lower bounds for the "p-module" of a domain with polygonal boundary. Error bounds are given for smooth and for piecewise constant functionsp. We present numerical experiments.


Journal ArticleDOI
TL;DR: The design, implementation, and performance of a real-time estimation algorithm, referred to in this paper as the sequential piecewise recursive (SPWR) algorithm, for the global-positioning system (GPS) low-dynamics navigation system is described.
Abstract: The design, implementation, and performance of a real-time estimation algorithm, referred to in this paper as the sequential piecewise recursive (SPWR) algorithm, for the global-positioning system (GPS) low-dynamics navigation system is described. The SPWR algorithm for this application was implemented in single precision arithmetic (32 bit, floating point). Numerical results are presented covariance and filter gains at a slower rate than the state measurement update, and it uses U-D factor formulation to perform covariance computations. The SPWR algorithm saves real-time processing requirements without appreciable degradation of filter performance. Another important feature of the SPWR algorithm is that it incorporates pseudorange and delta-range data from each GPS satellite sequentially for navigation solution. The SPWR algorithm, for this application, was implemented in single precision arithmetic (32 bit, floating point). Numerical results are presented.

Journal ArticleDOI
TL;DR: In this article, it was shown that the elements of the closed operator algebra generated by one-dimensional singular integral operators with piecewise continuous coefficients with a fixed finite set of points of discontinuity can be written as the sum of a singular integral operator, a compact operator, and generalized Mellin convolutions.
Abstract: It is shown that the elements of the closed operator algebra generated by one-dimensional singular integral operators with piecewise continuous coefficients with a fixed finite set of points of discontinuity can be written as the sum of a singular integral operator, a compact operator, and generalized Mellin convolutions. Their Gohberg-Krupnik symbol is given in terms of the Mellin transform. This gives an explicit construction of an operator with prescribed Gohberg—Krupnik symbol.

Journal ArticleDOI
TL;DR: In this paper, an exact solution method for the problem of optimally controlling a deterministic discrete linear system with piecewise quadratic cost functional containing a "dead-zone" is described.
Abstract: In this paper, we describe an exact solution method for the problem of optimally controlling a deterministic discrete linear system with piecewise quadratic cost functional containing a ‘dead-zone’. This model generalizes the well-known optimal control model for a linear system with (symmetric) quadratic objective functional. This more general problem is solved by using concepts and techniques from the theory of nonlinear programming, for example Kuhn-Tucker theory, duality, linear complementarity, and Lemke's algorithm. We apply the results to the solution of a 25-period deterministic pension funding problem which is modelled as a discrete-time optimal control problem. The controls for this problem are company contributions and investment amounts, and the state is the value of the pension fund.

01 Nov 1980
TL;DR: An idea due to J. Butland leads to an improved algorithm for piecewise monotonic piecewise cubic interpolation which is published and plans for software implementing the new algorithm are described.
Abstract: An idea due to J. Butland (Computer Graphics 80, Online Public., Ltd., 1980, 409-422) leads to an improved algorithm for piecewise monotonic piecewise cubic interpolation. The modifications to the algorithm published (Siam J. Numer. Anal. 17, 2 (April 1980), 238-246) and plans for software implementing the new algorithm are described. 4 figures.

Journal ArticleDOI
TL;DR: In this paper, the Galerkin method was used to solve a parabolic initial boundary value problem in one space variable, using piecewise polynomial functions and gave an alternative proof of superconvergence.
Abstract: We consider the Galerkin method to solve a parabolic initial boundary value problem in one space variable, using piecewise polynomial functions and give an alternative proof of superconvergence. Then by means of Lobatto quadrature, we obtain purely explicit vector initial value problems without loss in the order of accuracy, global or pointwise.