Showing papers on "Piecewise published in 1984"
TL;DR: In this paper, the Lipschitz-Killing curvatures of smooth Riemannian manifolds for piecewise flat spaces have been studied in the special case of scalar curvature.
Abstract: We consider analogs of the Lipschitz-Killing curvatures of smooth Riemannian manifolds for piecewise flat spaces. In the special case of scalar curvature, the definition is due to T. Regge; considerations in this spirit date back to J. Steiner. We show that if a piecewise flat space approximates a smooth space in a suitable sense, then the corresponding curvatures are close in the sense of measures.
361 citations
TL;DR: In this article, the authors present a method to solve the economic power dispatch problem with piecewise quadratic cost functions, which allows for decentralizing the computations to multiple generators.
Abstract: This paper presents a method to solve the economic power dispatch problem with piecewise quadratic cost functions. The solution approach is hierarchical, which allows for decentral i zed computations. An advantage of this approach is the capability to optimize over a greater variety of operating conditions. Traditionally, one cost function for each generator is assumed. In this formulation multiple intersecting cost functions are assumed. This method has appl ication to fossil generation units capable of burning gas and oil , as well as other problems which result in multiple intersecting cost curves for a particular unit. The results show that the solution method is practical and valid for real-time application. The motivation for this research stems from the actual operational and planning problems of a large Southwestern Utility.
352 citations
TL;DR: In this article, the authors studied functional and functional differential equations and showed that functional equations are directly connected with difference equations of a discrete (for example, integer-valued) argument, the theory of which has been very intensively developed in the book and in numerous subsequent papers.
315 citations
TL;DR: In this paper, a method for producing monotone piecewise cubic interpolants to monotonous data is described, which is completely local and which is extremely simple to implement.
Abstract: A method is described for producing monotone piecewise cubic interpolants to monotone data which is completely local and which is extremely simple to implement.
294 citations
01 Apr 1984-Graphical Models \/graphical Models and Image Processing \/computer Vision, Graphics, and Image Processing
TL;DR: Results encouraged investigations into modeling the picture as a mosaic of patches where the gray-value function within each patch is described as a second-order bivariate polynomial of the pixel coordinates, facilitating the determination of threshold values related to a priori confidence limits.
Abstract: Modeling the image as a piecewise linear gray-value function of the pixel coordinates considerably improved a change detection test based previously on a piecewise constant gray-value function. These results encouraged investigations into modeling the picture as a mosaic of patches where the gray-value function within each patch is described as a second-order bivariate polynomial of the pixel coordinates. Such a more appropriate model allowed the assumption to be made that the remaining gray-value variation within each patch can be attributed to noise related to the sensing and digitizing devices, independent of the individual image frames in a sequence. This assumption made it possible to relate the likelihood test for change detection to well-known statistical tests ( t test, F test), facilitating the determination of threshold values related to a priori confidence limits.
213 citations
TL;DR: An interpolation scheme is described for values of position, gradient and Hessian at scattered points in three variables that reproduces polynomials of degree up to three exactly.
194 citations
TL;DR: In this paper, the authors generalize the Takagi function to a nowhere differentiable continuous function and clarify the close relation between Lebesgue's singular function with de Rham's functional equation.
Abstract: In this paper, we generalize the Takagi function which is known as a nowhere differentiable continuous function, and we clarify the close relation between the Takagi function and Lebesgue’s singular function with de Rham’s functional equation.
116 citations
TL;DR: In this paper, the spectral radius of the Perron-Frobenius operator for piecewise expanding transformations is derived and the speed of convergence to equilibrium in such one-dimensional systems is analyzed.
Abstract: We determine the essential spectral radius of the Perron-Frobenius-operator for piecewise expanding transformations considered as an operator on the space of functions of bounded variation and relate the speed of convergence to equilibrium in such one-dimensional systems to the greatest eigenvalues of generalized Perron-Frobenius-operators of the transformations (operators which yield singular invariant measures).
114 citations
113 citations
TL;DR: In this article, the authors studied the superconvergence phenomenon when solving a 2nd order elliptic problem by the usual linear elements and showed that the convergence rate of the averaged gradient to an exact gradient in the L 2-norm can locally be higher even by one than that of the original piecewise constant discrete gradient.
Abstract: We study a superconvergence phenomenon which can be obtained when solving a 2nd order elliptic problem by the usual linear elements. The averaged gradient is a piecewise linear continuous vector field, the value of which at any nodal point is an average of gradients of linear elements on triangles incident with this nodal point. The convergence rate of the averaged gradient to an exact gradient in theL 2-norm can locally be higher even by one than that of the original piecewise constant discrete gradient.
99 citations
TL;DR: In this paper, the p-version of the finite element method has been studied in the context of polynomial approximation in weighted Sobolev spaces, which are identified as the domains of positive real powers of the Legendre operator.
Abstract: The purpose of this article is to present the approximation theory underlying the p-version of the finite element method. By exploiting the relationship between polynomial approximation and certain weighted Sobolev spaces, which are identified as the domains of positive real powers of the Legendre operator, this one-dimensional result is generalized via a tensor product construction to yield a nonconforming piecewise polynomial approximation result in the usual unweighted Sobolev spaces on triangulated domains of $R^n $. It is then shown that essentially the same result holds for approximation by conforming piecewise polynomials provided that the function being approximated possesses the same degree of conformality across the common boundaries of adjacent simplices and the same homogeneous boundary conditions. Inverse results are given for the special case of approximation in $L_2 $.
TL;DR: The Aumann-Shapley A-S prices are used here to allocate costs among destinations in a way that each destination will pay its “real part” in the total transportation costs.
Abstract: The Aumann-Shapley A-S prices are axiomatically determined on certain classes of piecewise continuously differentiable cost functions. One of these classes consists of all cost functions derived from the transportation problems and some of their generalizations. These prices are used here to allocate costs among destinations in a way that each destination will pay its “real part” in the total transportation costs. An economic transportation model is presented in which the A-S prices are compatible with consumer demands. Finally an algorithm is provided to calculate both the optimal solution and the associated A-S prices for transportation problems.
TL;DR: In this article, the numerical solution of the initial value problem for the wave equation is considered for the case when the equation coefficients are piecewise smooth, and the numerical phase velocity has the same accuracy as the difference approximation would if the coefficients in the differential equation were smoooth.
Abstract: The numerical solution of the initial value problem for the wave equation is considered for the case when the equation coefficients are piecewise smooth. This problem models linear wave propagation in a medium in which the properties of the medium change discontinuously at interfaces. Convergent difference approximations can be found that do not require the explicit specification of the boundary conditions at interfaces in the medium and hence are simple to program. Although such difference approximations typically can only be expected to be first-order accurate, the numerical phase velocity has the same accuracy as the difference approximation would if the coefficients in the differential equation were smoooth. This is proved for the one-dimensional case and demonstrated numerically for an example in two space dimensions in which the interface is not aligned with the computational mesh.
TL;DR: In this paper, Galerkin's method using piecewise polynomials as basis functions is used to solve the problem of singularities in the distribution and the kernel of the integral equation.
Abstract: A harmonic function in the interior of a polygon is the double layer potential of a distribution satisfying a second kind integral equation. This may be solved numerically by Galerkin's method using piecewise polynomials as basis functions. But the corners produce singularities in the distribution and the kernel of the integral equation; and these reduce the order of convergence. This is offset by grading the mesh, and the orders of convergence and superconvergence are restored to those for a smooth boundary.
TL;DR: In this article, the authors considered feedback, two-person, zero-sum differential games, and obtained two inequalities for the directional derivatives of the nonsmooth value function, and showed that these inequalities, together with the boundary conditions, constitute necessary and sufficient conditions which the value function must satisfy.
Abstract: We consider feedback, two-person, zero-sum differential games. We obtain two inequalities for the directional derivatives of the nonsmooth value function. We show that these inequalities, together with the boundary conditions, constitute necessary and sufficient conditions which the value function must satisfy. In the region where the value function is differentiable, the inequalities become the well-known main equation of differential game theory (Isaacs-Bellman equation). The results obtained here may be useful in the approximation of the value function by piecewise smooth splines and also in the classification of singular surfaces.
TL;DR: In this article, the use of a non-conventional linear programming technique, involving a piecewise differentiable penalty function minimization, is presented in connection with power system Constrained Economic Dispatch (CED) problems.
Abstract: The use of a non-conventional linear programming technique, involving a piecewise differentiable penalty function minimization, is presented in this paper in connection with power system Constrained Economic Dispatch (CED) problems.
TL;DR: Two Galerkin type piecewise polynomial approximation procedures based on bilinear forms with different weight functions are analyzed and compared in this article, where the optimal order error estimates are proved and numerical results are presented.
Abstract: Two Galerkin type piecewise polynomial approximation procedures based on bilinear forms with different weight functions are analyzed and compared. Optimal order error estimates are proved and numerical results are presented. 1. Introduction. In this paper we shall discuss Galerkin piecewise polynomial approximation methods for the singular two-point boundary value problem b (1.1) Lu(x) -u"(x) --u'(x) + q(x)u(x) =f(x), x E I- (0,1), x
TL;DR: A Clough-Tocher like interpolation scheme is described for values of position, gradient and hessian at scattered points in two variables that reproduces polynomials of degree up to three exactly.
TL;DR: In this paper, the representation and approximation of 3D and 4D surfaces is accomplished by means of local, piecewise defined, smooth interpolation methods on geometric domains of triangles or tetrahedra.
Abstract: The representation and approximation of three- and four-dimensional surfaces is accomplished by means of local, piecewise defined, smooth interpolation methods. In order to interpolate to arbitrarily located data, the schemes are defined on geometric domains of triangles or tetrahedra, respectively.
TL;DR: The application of Ortiz' recursive formulation of the Tau method to the construction of piecewise polynomial approximations to the solution of linear and nonlinear boundary value problems for ordinary differential equations proves efficient in the numerical treatment of problems with rapid functional variations, stiff and singularly per- turbed problems.
Abstract: This paper concerns the application of Ortiz' recursive formulation of the Tau method to the construction of piecewise polynomial approximations to the solution of linear and nonlinear boundary value problems for ordinary differential equations. A practical error estimation technique, related to the concept of correction in Zadunaisky's sense, is considered and used in the design of an adaptive approach to the Tau method. It proves efficient in the numerical treatment of problems with rapid functional variations, stiff and singularly per- turbed problems. A technique of increased accuracy at matching points of segmented Tau approximants is also discussed and successfully applied to several problems. Numerical examples show that, for a given degree of approximation, our segmented Tau approximant gives an accuracy comparable to that of the best segmented approximation of the exact solution by means of algebraic polynomials. 1. Introduction. We discuss the use of Ortiz' recursive formulation of the Tau method (23)-(25) in the numerical solution of boundary value problems for linear and nonlinear differential equations defined over an interval a < x < b. We con- sider global approximations over (a, b), with a single polynomial expression, and segmented forms based on a step-by-step formulation of the Tau method considered by Ortiz in (26). The Tau approximate solution of a differential problem defined by a differential operator D is represented in terms of the elements of a sequence Q of canonical polynomials. Such a sequence is uniquely determined by D, it is independent of the specific boundary conditions of the problem, and of the particular interval (a, b) in which the solution is required. These properties make possible the use of segmenta- tion within the framework and with the software (32) designed for the recursive formulation of the Tau method. The concept of correction, in Zadunaisky's sense (37) (see also Stetter (36)), is discussed in the context of the Tau method and related to a practical error estimation technique. This technique, based on Tau estimators introduced here, is systematically applied to all examples, linear or nonlinear. It is
TL;DR: In a plane polygonal domain with a reentrant corner, this article considered a homogeneous Dirichlet problem for Poisson's equation and the corresponding Galerkin finite element solutions in a family of piecewise polynomial spaces based on quasi-uniform triangulations with the diameter of each element.
Abstract: In a plane polygonal domain with a reentrant corner, consider a homogeneous Dirichlet problem for Poisson's equation -Au = f with f smooth and the corresponding Galerkin finite element solutions in a family of piecewise polynomial spaces based on quasi-uniform (uniformly regular) triangulations with the diameter of each element compara-
TL;DR: In this article, the existence of an invariant asymptotic measure for certain locally expanding maps of the unit cube in Ω(k) was proved. But the results of Lasota and Yorke on piecewise expanding maps to higher dimensions were not considered.
Abstract: By using a well known technique from classical statistical mechanics of one-dimensional lattice spin systems we prove existence of an absolutely continuous invariant asymptotic measure for certain locally expanding mapsT of the unit cube in ℝ
k
. We generalize herewith in a certain sense the results of Lasota and Yorke on piecewise expanding maps of the unit interval to higher dimensions. We show a Kuzmin-type theorem for these systems from which exponential approach to equilibrium and strong mixing properties follow.
TL;DR: In this paper, the collocation method with piecewise linear trial functions for systems of singular integral equations with Cauchy kernel and piecewise continuous coefficients is considered and necessary and sufficient conditions for the stability in L2 are given.
Abstract: We consider the collocation method with piecewise linear trial functions for systems of singular integral equations with Cauchy kernel and piecewise continuous coefficients Necessary and sufficient conditions for the stability in L2 are given The results are obtained in the case of a closed Ljapunov curve as well as in the case of an interval The proof of the main theorem is based on a modification of the Banach algebra technique established in the local principle by Gohberg and Krupnik [2] Our results extend those obtained by Prosdorf and Schmidt [9, 10] from the case of continuous coefficients and unit circle to the case of piecewise continuous coefficients
TL;DR: It is proved that algorithm presented generates for curves of some class, called spirals, the minimal number of circular arcs in Tchebycheff norm, which evaluates the published methods of piecewise circular approximation.
Abstract: The approximation of plane curves by smooth piecewise circular arc curves in Tchebycheff norm is analysed. The algorithm of approximation is proposed. It is proved that algorithm presented generates for curves of some class, called spirals, the minimal number of circular arcs. Results are used to evaluate the published methods of piecewise circular approximation.
TL;DR: A detailed analysis is performed for a finite element method applied to the general one-dimensional convection diffusion problem, shown to be quasi-optimal, provided that the input data is piecewise smooth—a reasonable assumption in practice.
Abstract: A detailed analysis is performed for a finite element method applied to the general one-dimensional convection diffusion problem. Piecewise polynomials are used for the trial space. The test space is formed by locally projecting L-spline basis functions onto “upwinded” polynomials. The error is measured in the $L_p$ mesh dependent norm. The method is shown to be quasi-optimal, provided that the input data is piecewise smooth—a reasonable assumption in practice. A posteriors error estimates are derived having the property that the effectivity index $\theta = $ (error estimate/true error) converges to one as the maximum mesh size goes to zero. These error estimates are composed of locally computable error indicators, providing for an adaptive mesh refinement strategy. Numerical results show that $\theta $ is nearly one even on coarse meshes, and optimal rates of convergence are attained by the adaptive procedure. The robustness of the algorithm is tested on a nonlinear turning point problem modeling flow th...
TL;DR: A new type of parametrically defined space curve is described, which allows for the convenient control over local shape attributes while maintaining global second order geometric continuity.
TL;DR: In this paper, it was shown that these impulse response expressions are quite good approximations to many practical filters, such as low-pass, high-pass and band-stop filters, Hilbert transformers, and differentiators.
Abstract: Recursive filter structures have been found for FIR filters with piecewisepolynomial or piecewise-(polynomial · sinusoid) impulse responses. The amount of arithmetic required for these filters is proportional to the number of piecewise sections in their impulse responses rather than the actual filter lengths. In this paper, it is shown that these impulse response expressions are quite good approximations to many practical filters. Low-pass filters, high-pass filters, narrowband, band-pass, and band-stop filters, Hilbert transformers, and differentiators all have impulse responses which can be approximated by these forms, and a long filter impulse response consists of only a few piecewise sections with greatly reduced arithmetic requirements. Though this technique is based on time-domain approximation, a frequency-domain optimization to select filter parameters is presented with excellent results.
TL;DR: Use of the M‐B chararteristics is shown to guarantee iteration stability because the slope approaches unity as μ tends to infinity, and isoparametric elements of second order are used for both geometry and sources to ensure high solution fidelity.
Abstract: An interface formulation, successfully used for the solution of magnetostatic fields in piecewise homogeneous media, is extended to nonlinear media problems. The advantage of boundary integral techniques, in the reduction of problem dimensionality, is retained. This nonlinear boundary element method (BEM) algorithm alternates solution of the augmented interface equation with satisfaction of the constitutive media relations. Use of the M‐B chararteristics is shown to guarantee iteration stability because the slope approaches unity as μ tends to infinity. Isoparametric elements of second order are used for both geometry and sources to ensure high solution fidelity. Galerkin’s method is employed for discretization to matrix form. Each iteration requires only the product of the inverted system matrix with the augmented RHS. Examples include saturation of square and circular cylinders of high permeability in a uniform field. It is shown that with an optimal relaxation factor of 0.65, only 3–6 iterations are ...
TL;DR: A comparison study of several recently published segmentation procedures, regarding mainly their performance as measured by the ability to detect and localize segment boundaries correctly.
TL;DR: A collocation procedure with polynomials and piecewise polynomial approximation is considered for second order functional differential equations with two side-conditions to optimize the rate of convergence.
Abstract: A collocation procedure with polynomial and piecewise polynomial approximation is considered for second order functional differential equations with two side-conditions. The piecewise polynomials are taken in the classC
1 and reduce to polynomials of increasing degree on each interval of a suitable assigned partition. Appropriate choices of the partition are made, according to the jump discontinuities in the derivatives caused by the functional argument, in order to optimize the rate of convergence.