Showing papers on "Piecewise published in 1986"
TL;DR: In this paper, the optimal control of piecewise deterministic processes with state space constraint is studied under appropriate assumptions, and it is shown that the optimal value function is the only viscosity solution on the open domain which is also a supersolution on the closed domain.
Abstract: Optimal control of piecewise deterministic processes with state space constraint is studied. Under appropriate assumptions, it is shown that the optimal value function is the only viscosity solution on the open domain which is also a supersolution on the closed domain. Finally, the uniform continuity of the value function is obtained under a condition on the deterministic drift.
632 citations
TL;DR: In this article, the authors proved Lp stability and error estimates for the discontinuous Galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain.
Abstract: We prove Lp stability and error estimates for the discontinuous Galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain. Using finite element analysis techniques, we obtain L2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials. Lp estimates for p * 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree. The latter estimates are proved by combining finite difference and finite element analysis techniques.
606 citations
TL;DR: In this article, a new method for solving the close coupled equations of inelastic scattering is presented, based on Johnson's log derivative algorithm, and uses the same quadrature for the solution of the corresponding integral equations.
Abstract: A new method for solving the close coupled equations of inelastic scattering is presented. The method is based on Johnson’s log derivative algorithm, and uses the same quadrature for the solution of the corresponding integral equations. However it differs from the original method in the use of a piecewise constant diagonal reference potential. This results in a reduction in matrix operations at subsequent energies, and an improved convergence of the solution with respect to the number of grid points. These advantages are clearly demonstrated when our method is applied to an atom–diatom rotational excitation problem.
580 citations
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TL;DR: In this paper, a series of one-dimensional surfaces are fit to each window and the surface description is accepted, which is adequate in the least square sense and has the fewest parameters.
Abstract: An edge in an image corresponds to a discontinuity in the intensity surface of the underlying scene. It can be approximated by a piecewise straight curve composed of edgels, i.e., short, linear edge-elements, each characterized by a direction and a position. The approach to edgel-detection here, is to fit a series of one-dimensional surfaces to each window (kernel of the operator) and accept the surface-description which is adequate in the least squares sense and has the fewest parameters. (A one-dimensional surface is one which is constant along some direction.) The tanh is an adequate basis for the stepedge and its combinations are adequate for the roofedge and the line-edge. The proposed method of step-edgel detection is robust with respect to noise; for (step-size/?noise) ? 2.5, it has subpixel position localization (?position < ?) and an angular localization better than 10°; further, it is designed to be insensitive to smooth shading. These results are demonstrated by some simple analysis, statistical data, and edgelimages. Also included is a comparison of performance on a real image, with a typical operator (Difference-of-Gaussians). The results indicate that the proposed operator is superior with respect to detection, localization, and resolution.
495 citations
TL;DR: This paper constructs an hierarchy of uniformly high-order accurate approximations of any desired order of accuracy which are tailored to be essentially nonoscillatory.
420 citations
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01 Sep 1986
TL;DR: The basic facts about the B(arycentric, -ernstein, -ezier) form of a multivariate polynomial are recorded and proved and the description of this form in terms of difference operators is stressed.
Abstract: : The basic facts about the B(arycentric, -ernstein, -ezier) form of a multivariate polynomial are recorded and, in part, proved. These include: evaluation (de Casteliau's algorithm), differentiation and integration, product, degree raising, change of the underlying simplex, and the behavior on the boundary of the underlying simplex, with application to the construction of smooth pp functions on a given triangulation. Some effort has gone into making the notation fully reflect the symmetries and structure of this form. In particular, the description of this form in terms of difference operators is stressed. Keywords: Piecewise polynomials; Linear interpolation.
239 citations
31 Aug 1986
TL;DR: A method is developed for surface-fitting from sampled data based on an adaptive subdivision approach, a technique previously used for the design and display of free-form curved surface objects, which is simple in concept, yet realizes efficient data compression.
Abstract: A method is developed for surface-fitting from sampled data Surface-fitting is the process of constructing a compact representation to model the surface of an object based on a fairly large number of given data points In our case, the data is obtained from a real object using an automatic three-dimensional digitizing system The method is based on an adaptive subdivision approach, a technique previously used for the design and display of free-form curved surface objects Our approach begins with a rough approximating surface and progressively refines it in successive steps in regions where the data is poorly approximated The method has been implemented using a parametric piecewise bicubic Bernstein-Bezier surface possessing G1 geometric continuity An advantage of this approach is that the refinement is essentially local reducing the computational requirements which permits the processing of large databases Furthermore, the method is simple in concept, yet realizes efficient data compression Some experimental results are given which show that the representation constructed by this method is faithful to the original database
169 citations
TL;DR: A fully automatic tolerance-based surface offset capability may be developed by providing feedback between stages (3) and (2), successive degrees of parametric sub-division being determined by the errors from prior approximations until the desired accuracy is achieved.
165 citations
TL;DR: In this paper, a set of basic properties postulated for the underlying production possibility sets are derived from linear programming formulations for piecewise log-linear estimation of efficient production surfaces, which permits the identification of increasing marginal products, and estimation of the classical S-shaped production functions.
Abstract: Linear programming formulations for piecewise loglinear estimation of efficient production surfaces are derived from a set of basic properties postulated for the underlying production possibility sets Unlike the piecewise linear model of Banker, Charnes, and Cooper Banker R D, A Charnes, W W Cooper 1984 Models for the estimation of technical and scale inefficiencies in data envelopment analysis Management Sci30 September 1078-1092, this approach permits the identification of increasing marginal products, and estimation of the classical S-shaped production functions Methods are also provided for estimating technical inefficiencies and other production characteristics, such as rates of substitution and transformation, marginal products, returns to scale and most productive scale sizes on the basis of observed production data The results of a simulation study are reported to illustrate the usefulness of this estimation method in empirical applications when there are a priori reasons to expect increasing marginal products in some regions of the production function A modified model is provided to extend this analysis to the situation of non-competing outputs addressed by the bi-extremal model of Banker, Charnes, Cooper, and Schinnar Banker R D, A Charnes, W W Cooper, A Schinnar 1981 A bi-extremal principle for frontier estimation and efficiency evaluation Management Sci27 December 1370-1382
135 citations
TL;DR: It is shown that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in L1(R) to within O(N-2) by a piecewise linear function with O(fN) nodes; the nodes are moved according to the method of characteristics.
Abstract: We show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in L1(R) to within O(N-2) by a piecewise linear function with O(fN) nodes; the nodes are moved according to the method of characteristics. We also show that a previous method of Dafermos, which uses piecewise constant approxima- tions, is accurate to O(N-1). These numerical methods for conservation laws are the first to have proven convergence rates of greater than O(fN-1/2). 1. Introduction. It is well-known that the solution of the hyperbolic conservation law,
128 citations
31 Aug 1986
TL;DR: It is found that convolution of a signal with any piecewise polynomial kernel of degree n--1 can be computed by integrating the signal n times and point sampling it several times for each output sample.
Abstract: Many applications of digital filtering require a space variant filter - one whose shape or size varies with position. The usual algorithm for such filters, direct convolution, is very costly for wide kernels. Image prefiltering provides an efficient alternative. We explore one prefiltering technique, repeated integration, which is a generalization of Crow's summed area table.We find that convolution of a signal with any piecewise polynomial kernel of degree n--1 can be computed by integrating the signal n times and point sampling it several times for each output sample. The use of second or higher order integration permits relatively high quality filtering. The advantage over direct convolution is that the cost of repeated integration filtering does not increase with filter width. Generalization to two-dimensional image filtering is straightforward. Implementations of the simple technique are presented in both preprocessing and stream processing styles.
TL;DR: In this paper, a piecewise continuous Ljapunov function is used for stability in systems of ordinary differential equations with impulse effect. But the approach presented is based on the specially introduced piecewise continuously LjAPunov functions.
TL;DR: This article describes and presents examples of some techniques for the representation and interactive design of surfaces based on a parametric surface representation that user v-spline curves and extends their application to a rectangular grid of control points.
Abstract: This article describes and presents examples of some techniques for the representation and interactive design of surfaces based on a parametric surface representation that user v-spline curves. These v-spline curves, similar in mathematical structure to v-splines, were developed as a more computationally efficient alternative to splines in tension. Although splines in tension can be modified to allow tension to be applied at each control point, the procedure is computationally expensive. The v-spline curve, however, uses more computationally tractable piecewise cubic curves segments, resulting in curves that are just as smoothly joined as those of a standard cubic spline. After presenting a review of v-splines and some new properties, this article extends their application to a rectangular grid of control points. Three techniques and some application examples are presented.
TL;DR: The empirical orthogonal functions (EOF) as mentioned in this paper can be considered as a mean square estimation technique of unknown values within a random process or field, and the optimization of error variance leads to a Fredholm integral equation.
Abstract: Some current uses of empirical orthogonal functions (EOF) are briefly summarized, together with some relations with spectral and principal component analyses. Considered as a mean square estimation technique of unknown values within a random process or field, the optimization of error variance leads to a Fredholm integral equation. Its kernel is the autocorrelation function, which in many practical cases is only known as discrete values of interstation correlation coefficients computed from a sample of independent realizations. The numerical solution in one or two dimensions of this integral equation is approximated in a new and more general framework that requires, in practice, the a priori choice of a set of generating functions. Developments are provided for piecewise constant, facetlike linear, and thin plate type spline functions. The first part of the paper ends with a review of the mapping, archiving and stochastic simulating possibilities of the EOF method. A second part includes a case s...
TL;DR: In this paper, the existence of weak solutions for the Cauchy problem for the Navier-Stokes equations for one-dimensional, isentropic flow when the initial velocity is in L 2 and the initial density is in BV is proved.
Abstract: We prove the global existence of weak solutions for the Cauchy problem for the Navier-Stokes equations for one-dimensional, isentropic flow when the initial velocity is in L2 and the initial density is in L2 ∩ BV. Solutions are obtained as limits of approximations obtained by building heuristic jump conditions into a semi-discrete difference scheme. This allows for a rather simple analysis in which pointwise control is achieved through piecewise H1 and total variation estimates.
01 Jan 1986
TL;DR: It is shown that an optimal stationary policy exists in relaxed controls under certain continuity assumptions and that a discounted infinite horizon control problem can be reformulated as a discrete-time Markov decision problem.
Abstract: Controlled piecewise-deterministic Markov processes have deterministic trajectories punctuated by random jumps, at which the sample path is right-continuous. By considering the sequence of states visited by the process at its jump times, it is shown that a discounted infinite horizon control problem can be reformulated as a discrete-time Markov decision problem (the ‘positive’ case). Under certain continuity assumptions it is shown that an optimal stationary policy exists in relaxed controls.
TL;DR: In this article, the authors give a method for constructing conservative Scheffe-type simultaneous confidence bands for curvilinear regression functions over finite intervals based on the use of a geometric inequality.
Abstract: This paper gives a method for constructing conservative Scheffe-type simultaneous confidence bands for curvilinear regression functions over finite intervals. The method is based on the use of a geometric inequality giving an upper bound for the uniform measure of the set of points within a given distance from y, an arbitrary piecewise differentiable path with finite length in $S^{k-1}$, the unit sphere in $R^k$. The upper bound is obtained by "straightening" the path so that it lies in a great circle in $S^{k-1}$.
TL;DR: In this article, the asymptotic rate of exits of Gaussian vector processes with continuously differentiable sample paths into intersections of failure domains with piecewise twice differentiable boundaries is derived.
TL;DR: It is suggested that a modified form of Bernstein-Bezier may be preferable for CAGD applications (as well as in applications of piecewise polynomials in data fitting and numerical solution of boundary-value problems).
23 Jun 1986
TL;DR: The Small-Signal Frequency Response Theory (SFRT) as discussed by the authors is a theory for calculating the output spectrum of ideal dc-to-dc converter systems with system coefficients piecewise constant in time, for a given spectrum of the signal injected into the control-input, in the small-signal limit.
Abstract: Small-Signal Frequency Response Theory is a theory for calculating the output spectrum of ideal dc-to-dc converter systems, i.e. systems with system coefficients piecewise constant in time, for a given spectrum of the signal injected into the control-input, in the small-signal limit. This theory, unlike other methods, can be applied to both resonant and PWM converters, and gives analytic results in closed form for ideal converters. This paper discusses the special case of ideal two-switched-network converter systems in PWM, programmed, and bang-bang operation. For the examples under study, theoretical prediction and experimental results are found to differ by at most 2dB in amplitude and 10 degrees in phase at most frequencies up to three times the switching frequency. Examples are given in this paper for which the theory gives the correct prediction, while other methods fail.
TL;DR: In this paper, the optimal order error estimates for continuous-time and discrete-time collocation methods were derived for a semilinear parabolic initial-boundary value problem in R \times ( {0,T} ], where R is the unit square.
Abstract: Optimal order error estimates are derived for the continuous-time collocation method and a discrete-time collocation method (the Crank Nicolson collocation method) for approximating the solution of a semilinear parabolic initial-boundary value problem in $R \times ( {0,T} ]$, where R is the unit square. In each case, at each time level, the approximation is a $C^1 $ piecewise polynomial of degree $r \geqq 3$ defined by collocation at Gauss points in R. Optimal order error estimates are also derived for a family of discrete-time collocation methods for a semilinear second order hyperbolic initial-boundary value problem in $R \times ( {0,T} ]$.
TL;DR: In this article, the boundary integral formulation due to Burton and Miller is used to overcome the non-existence and non-uniqueness problems associated with classical integral equation formulations of this problem.
Abstract: This paper is concerned with the efficient determination of acoustic fields around arbitrary-shaped finite structures in an infinite three-dimensional acoustic medium. The boundary integral formulation due to Burton and Miller is used to overcome the non-existence and non-uniqueness problems associated with classical integral equation formulations of this problem. A class of numerical approximation schemes is developed and the results of applying these schemes to a number of test problems are discussed and compared. The choice of the parameters of the method is critically considered. In particular, the gains to be made, in terms of solution accuracy, by using higher-order approximations to the unknown boundary function, rather than the commonly employed piecewise constant representation, are examined in view of their additional computational costs.
TL;DR: The mesh-dependent variational framework introduced here is of special interest in the approxi- mation of the solution of optimal control problems governed by differential equations.
Abstract: This paper studies the approximation of the solution of nonlinear ordinary differen- tial equations by (discontinuous) piecewise polynomials of degree K and traces at the nodes of discretization. A mesh-dependent variational framework underlying this discontinuous ap- proximation is derived. Several families of one-step, hybrid and multistep schemes are obtained. It is shown that the convergence rate in the L2-norm is K + 1. The nodal-conver- gence rate can go up to 2 K + 2, depending on the particular scheme under consideration. The mesh-dependent variational framework introduced here is of special interest in the approxi- mation of the solution of optimal control problems governed by differential equations. 1. Introduction. The object of this paper is the study of (discontinuous) piecewise polynomial approximations to the solution of systems of nonlinear ordinary dif-
TL;DR: Methods for interpolating and approximating three-dimensional scattered data are presented, which consist of several local least squares approximations, followed by a piecewise bicubic Hermite interpolant to gridded data, and then optionally follow by a modified Shepard's method.
TL;DR: In this article, piecewise-polynomial methods for the approximation of Cauchy principal value integrals were proposed and a simple, efficient and numerically stable algorithm for the evaluation of the weights of the resulting piecewise polynomial quadratures was developed.
Abstract: In this paper, we propose piecewise-polynomial methods for the approximation of Cauchy principal value integrals and develop a simple, efficient and numerically stable algorithm for the evaluation of the weights of the resulting piecewise-polynomial quadratures. We present two examples to illustrate the advantages of these quadratures versus the Gauss–Jacobi quadratures.
TL;DR: A C 2 piecewise nonic interpolant defined over triangles is derived using the Bernstein-Bezier method to require only vertex data and to have seventh degree polynomial precision.
TL;DR: In this paper, the authors considered the classical diffraction problem for the wave propagation in the case where the propagation speed is piecewise constant, and the surface separating two media is unbounded.
Abstract: We consider the classical diffraction problem for the wave propagation in the case where the propagation speed is piecewise constant, and the surface separating two media is unbounded. The validity of the limiting absorption and amplitude principles is proved.
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TL;DR: The results indicate that the proposed operator is superior with respect to detection, localization, and resolution.
Abstract: An edge in an image corresponds to a discontinuity in the intensity surface of the underlying scene. It can be approximated by a piecewise straight curve composed of edgels, i.e., short, linear edge-elements, each characterized by a direction and a position. The approach to edgel-detection here, is to fit a series of one-dimensional surfaces to each window (kernel of the operator) and accept the surface-description which is adequate in the least squares sense and has the fewest parameters. (A one-dimensional surface is one which is constant along some direction.) The tanh is an adequate basis for the stepedge and its combinations are adequate for the roofedge and the line-edge. The proposed method of step-edgel detection is robust with respect to noise; for (step-size/?noise) ? 2.5, it has subpixel position localization (?position < ?) and an angular localization better than 10°; further, it is designed to be insensitive to smooth shading. These results are demonstrated by some simple analysis, statistical data, and edgelimages. Also included is a comparison of performance on a real image, with a typical operator (Difference-of-Gaussians). The results indicate that the proposed operator is superior with respect to detection, localization, and resolution.