Showing papers on "Piecewise published in 1978"
Book•
1 Jan 1978
TL;DR: This book presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines as well as specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting.
Abstract: This book is based on the author's experience with calculations involving polynomial splines. It presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines. After two chapters summarizing polynomial approximation, a rigorous discussion of elementary spline theory is given involving linear, cubic and parabolic splines. The computational handling of piecewise polynomial functions (of one variable) of arbitrary order is the subject of chapters VII and VIII, while chapters IX, X, and XI are devoted to B-splines. The distances from splines with fixed and with variable knots is discussed in chapter XII. The remaining five chapters concern specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting. The present text version differs from the original in several respects. The book is now typeset (in plain TeX), the Fortran programs now make use of Fortran 77 features. The figures have been redrawn with the aid of Matlab, various errors have been corrected, and many more formal statements have been provided with proofs. Further, all formal statements and equations have been numbered by the same numbering system, to make it easier to find any particular item. A major change has occured in Chapters IX-XI where the B-spline theory is now developed directly from the recurrence relations without recourse to divided differences. This has brought in knot insertion as a powerful tool for providing simple proofs concerning the shape-preserving properties of the B-spline series.
12,342 citations
TL;DR: In this paper, a class of piecewise continuous, piecewise C1 transformations on the interval J c R with finitely many discontinuities n are shown to have at most n invariant measures.
Abstract: A class of piecewise continuous, piecewise C1 transformations on the interval J c R with finitely many discontinuities n are shown to have at most n invariant measures. 1. The way phenomena or processes evolve or change in time is often described by differential equations or difference equations. One of the sim- plest mathematical situations occurs when the phenomenon can be described by a single number and when this number can be estimated purely as a function of the previous number. That is, when the number xn+x can be written as xn+x = t(x") where t maps an interval J c R into itself. For x E J, let t°(x) denote x and t"+x(x) denote t(t"(x)) for n = 0,1.We will sayp E J is a periodic point with period n if p = t"(p) andp ¥= rk(p) for 1 1. In this paper we assume t is piecewise continuous and piecewise twice continuous differentiable. We also assume that a t ^ > 1 where Jx — j x E /, — t(x) exists \.
282 citations
TL;DR: In this article, an extension of Walsh functions to the analysis of time-varying linear systems is made by the introduction of the product matrix of Walsh vector and its transpose, and the operational property of product matrix.
Abstract: Extension of Walsh functions to the analysis of time-varying linear systems is made by the introduction of the product matrix of Walsh vector and its transpose, and the operational property of product matrix. The operational matrix for the backward integration of Walsh functions is first introduced. Therefore, the state transition matrix of optimal control of linear time-varying systems with quadratic performance index can be integrated approximately using Walsh functions. The solution of the state transition matrix leads to piecewise constant gains equally distributed.
77 citations
1 Oct 1978
TL;DR: In this article, the authors present a method of numerically integrating differential equations containing time delays via block-pulse functions, and the resulting solutions are piecewise constant with minimal mean square error.
Abstract: This paper presents a method of numerically integrating differential equations containing time delays via block-pulse functions. The resulting solutions are piecewise constant with minimal mean-square error. The method is based on the use of a newly developed `delay matrix? and enlarges the scope of useful applications of block-pulse functions to problems of dynamic systems. Examples of linear and nonlinear delay systems are included.
76 citations
TL;DR: In this paper, the authors developed a design technique for approximating nonseparable frequency characteristics by sums and products of separable transfer functions, called the piecewise separable decomposition of the characteristic.
Abstract: The present paper develops a design technique for approximating nonseparable frequency characteristics by sums and products of separable transfer functions. This approximation is called the "piecewise separable" decomposition of the characteristic. In the design technique, the desired filter with half-plane symmetry (radial symmetry) is obtained by shifting a low-pass characteristic in the frequency domain, and by combining these shifted characteristics. Also the paper includes design approaches for the four-quadrant symmetry filters. Two examples illustrate the technique of the paper.
47 citations
TL;DR: In this article, a collocation solution of creeping Newtonian flow through periodically constricted tubes is obtained, where the profile of the wall of the type of tube considered is piecewise continuous, composed of symmetric parabolic segments.
Abstract: A collocation solution of creeping Newtonian flow through periodically constricted tubes is obtained. The profile of the wall of the type of tube considered is piecewise continuous, composed of symmetric parabolic segments. A transformation of the domain of interest into a rectangular one is obtained, which allows satisfaction of all boundary conditions. The collocation solution gives the stream function in terms of the new independent variables and can easily be converted to the original cylindrical coordinates. Axial and radial velocity components are obtained in analytical form, and the pressure drop is calculated from a volume integration of the viscous dissipation function as well as from line integration of the Navier-Stokes equation. The results are compared with the finite-difference solution by Payatakes et al. (1973b) and are found in good agreement. Differences between the two solutions are attributed mainly to discretization error in the finite-difference solution. The analytical expressions obtained from the collocation solution can be used together with porous media models of the constricted unit cell type for the modeling of processes taking place in granular porous media.
44 citations
TL;DR: In this paper, the authors established bounds on the possible deviation of the optimal objective value of a separable, convex program from its piecewise linear approximation based on a given subdivision interval.
Abstract: The bounds have been established on the possible deviation of the optimal objective value of a separable, convex program from the optimal objective value of a program which is its piecewise linear approximation based on a given subdivision interval. By a separable, convex program is meant a separable program with proper convexity-concavity properties which imply that any local optimum is also a global optimum. It is further shown that these bounds are actually attainable, and therefore, cannot be improved in general. Some examples are provided.Naturally, the inquiry requires some study of the piecewise linear approximation itself. The bounds on the function error are determined based on the assumptions—the boundedness of the first, or the second, or both the derivatives—about the original function. Some results are derived for a given piecewise linear function, to determine the nature of the original function with minimum error and satisfying certain conditions; these results would be applicable in those ...
43 citations
1 Dec 1978
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.
42 citations
TL;DR: In this article, a generalization of the Prager-Nagtegaal superposition principle for the optimal plastic design of structures subject to more than two alternate loads is presented.
36 citations
TL;DR: In this article, a quantum rate theory is presented for a symmetric double-well potential which is defined by a piecewise quadratic function, based on stationary states which are decomposed into left and right moving states.
Abstract: A quantum rate theory is presented for a symmetric double‐well potential which is defined by a piecewise quadratic function. The theory is based on stationary states which are decomposed into‐ and left‐moving states. The flux and transmission coefficients for the latter are found in terms of parabolic cylinder functions and are thermally averaged. Close analogies between the quantum and classical formulations are found when an appropriate phase space representation is used. The theory shows good agreement with experimental results for the diffusion of hydrogen and deuterium in niobium, but the agreement is poorer for the same process with the host metal vanadium and the theory does not predict the observed anomalous isotope effect for this process in palladium.
35 citations
TL;DR: In this article, the Barnhill-Gregory Boolean sum interpolant is used to construct a piecewise, interpolating C 1 surface through a set of arbitrarily spaced three-dimensional points {( x i, y i, z i ), i = 1,..., N }.
TL;DR: In this paper, the authors proposed a finite hybrid element (FHE) method to compute the ideal MHD spectrum of an axisymmetric plasma, which consists in extending the number of variables in the Lagrangian by considering derivatives of the displacements as additional variables and then introducing auxiliary constraints between the variables and their derivatives.
TL;DR: Under nonsingularity conditions on piecewise linearizations of PC1-mappings with the use of their partial derivatives and conditions on the diameter and thickness of simplices on which PL approximations are affine, one-to-one correspondence between solutions of a system of nonlinear equations and its PL approxIMations is investigated.
Abstract: A PC1-mapping is a continuous mapping from a subset P of Rn into Rn, where P is partitioned into n-dimensional compact convex polyhedra. such that the restriction to each polyhedron is continuously differentiable. Based on the classical results on PL (piecewise linear) approximations of PC1-mappings given by Whitehead and an extension of the inverse function theorem to PC1-mappings, the following are investigated under nonsingularity conditions on piecewise linearizations of PC1-mappings with the use of their partial derivatives and conditions on the diameter and thickness of simplices on which PL approximations are affine: (1) One-to-one correspondence between solutions of a system of nonlinear equations and its PL approximations. (2) Monotone convergence of the continuous deformation method. (3) A lower bound of the convergence speed of the continuous deformation method. (4) Conditions which characterize local uniqueness of solutions to the nonlinear complementarity problem. (5) One-to-one correspondenc...
TL;DR: A cutting hyperplane method is proposed, which successively considers the various cells of the decomposition, checks whether the cell contains an optimal solution to the problem, and imposes a convexity cut which rejects the whole cell from the feasibility region, which is shown to be finitely convergent.
Abstract: A piecewise convex program is a convex program such that the constraint set can be decomposed in a finite number of closed convex sets, called the cells of the decomposition, and such that on each of these cells the objective function can be described by a continuously differentiable convex function.
In a first part, a cutting hyperplane method is proposed, which successively considers the various cells of the decomposition, checks whether the cell contains an optimal solution to the problem, and, if not, imposes a convexity cut which rejects the whole cell from the feasibility region. This elimination, which is basically a dual decomposition method but with an efficient use of the specific structure of the problem is shown to be finitely convergent.
The second part of this paper is devoted to the study of some special cases of piecewise convex program and in particular the piecewise quadratic program having a polyhedral constraint set. Such a program arises naturally in stochastic quadratic programming with recourse, which is the subject of the last section.
TL;DR: Piecewise polynomial Galerkin approximations for Fredholm integral equations of the second kind are shown to posses superconvergence properties in some circumstances in this paper.
Abstract: Piecewise polynomial Galerkin approximations for Fredholm integral equations of the second kind are shown to posses superconvergence properties in some circumstances.
TL;DR: In this paper, four gridding methods are analyzed and their performance compared by application to the same data set: (1) weighted average of closest points, (2) weighted averaging of three closest points with some azimuth control, (3) local polynomial surface fitting, and (4) piecewise cubic interpolation along profiles.
TL;DR: In this paper, piecewise Hermite cubic polynomials are applied to linear elliptic problems subject to Dirichlet and Neumann boundary conditions on rectangular domains and a priori estimates are obtained for the error of approximation.
Abstract: Collocation methods based on piecewise Hermite cubic polynomials are applied to linear elliptic problems subject to Dirichlet and Neumann boundary conditions on rectangular domains. A priori estimates are obtained for the error of approximation.
1 Jan 1978
TL;DR: This paper develops algorithms for sequential triangulations which may be used in such numerical approximation tasks as numerical integration, piecewise approximation of functions, and finite element methods.
Abstract: Triangulations of subsets of ℝn, or simplicial coverings of subsets of ℝn may be used as a helpful device in such numerical approximation tasks as numerical integration [14], [15], piecewise approximation of functions, and finite element methods [17], [18]. Triangulations also play a crucial role in combinatorial algorithms for the approximation of fixed points of mappings [3], [4], [5], [9], [13]. Recently the authors have shown [1] that triangulations of ℝn can be generated by performing reflections of vertices of simplices across edges in a certain prescribed way. Pivoting between simplices by reflections permits the generation of sequences of simplices with elementary programming, minimal storage, and versatile starting. Our objective here is to develop algorithms for sequential triangulations which may be used in the above mentioned applications.
TL;DR: In this paper, a finite element procedure for the numerical approximation of certain classes of boundary value problems for first order partial differential equations is examined, and it is shown that convergence is not optimal under weak regularity assumptions.
Abstract: In this paper we examine a finite element procedure for the numerical approximation of the solution of certain classes of boundary value problems for first order partial differential equations. Our first result shows, under.a weak regularity assumption, that the error in the $L_2 $ norm is $O(h^{K - 1} )$ for piecewise polynomials of degree $K - 1$; thus, convergence is not optimal in $L_2 $. We show by an example that there is, in fact, a loss in $L_2 $ norm. However, under stronger regularity assumptions, which apply primarily to elliptic systems, we show that convergence in $L_2 $ is optimal, e.g., with piecewise linear elements the error is of $O(h^2 )$. Numerical examples indicating the effectiveness of the method are given.
TL;DR: In this paper, the double Walsh series is used to obtain the piecewiseconstant solutions of integral equations as well as convolution integral, and some special properties of the Walsh functions are derived.
Abstract: Double Walsh series is used to obtain the piecewise‐constant solutions of integral equations as well as convolution integral. Some special properties of the Walsh functions are derived. The Volterra and Fredholm integral equations are transformed, respectively, into simultaneous linear algebraic equations, which are then solved to obtain the piecewise constant solutions.
TL;DR: In this article, a stationary process y t, t ∈ R 1 is considered which is Markov between points of changeover from a stationary point process, and at these points it changes over according to a distribution dependent only on the value of y t just before the change is analysed.
Abstract: A stationary process y t , t ∈ R 1 is considered which is Markov between points of changeover from a stationary point process, and at these points it changes over according to a distribution dependent only on the value of y t just before the change is analysed. An explicit form of a rate-conservative principle is stated, and its relationship with formulae relating the distribution of the process at an instant t and the distribution at a point of changeover is shown. The theory is applied to discrete state processes and to processes which are generalizations of the Takacs processes, and is also applied in the theory of G / M / l and G / G / k queues to obtain relations between distributions of a process and some process imbedded in it.
TL;DR: In this article, the least square method with piecewise constant trial functions is investigated and an error estimate is derived, and an implementation using the fast Fourier transform is described and numerical results are reported.
Abstract: The problem of inverting the Radon transform, i.e. the reconstruction of a function inR 2 from its line integrals arises e.g. in computerized tomography and in nondestructive testing. In the present paper the least squares method with piecewise constant trial functions is investigated. An error estimate is derived. An implementation using the fast Fourier transform is described and numerical results are reported.
TL;DR: In this article, the authors study several ways of representing non-linear functions of a single argument in mixed integer, separable, and related programming terms, and show the problem size and optimization results of using the following techniques: separable programming, mixed integer programming with special ordered sets of type 1, linear programming with Special Ordered Sets of type 2 and mixed-integer programming using strategies based on the quasi-integrality of the binary variables.
TL;DR: In this paper, the authors consider a set of utility functions defined on P, the closure of the positive orthant of Rl that satisfy the differentiable monotonicity and differentiable convexity conditions and show that the demand function derived from such a utility function is piecewise smooth.
TL;DR: In this paper, the canonical perturbation method is applied to optimal control problems, and it is shown that the state and adjoint equations present interesting symmetries if the state equations are themselves of the Hamiltonian type, which is frequently the case if a mechanical system is to be controlled.
Abstract: In this paper the canonical perturbation method, which is widely used in analytical mechanics, is applied to optimal control problems. It is shown that the state and adjoint equations present additional interesting symmetries if the state equations are themselves of the Hamiltonian type, which is frequently the case if a mechanical system is to be controlled. The application of the canonical perturbation method to optimal control problems turns out to be particularly simple, if the optimal control is piecewise constant. Several examples are considered.
13 Apr 1978
TL;DR: A (smooth) increasing quadratic spline is constructed which interpolates the data points and preserves the convexity of the data.
Abstract: Let F be an arbitrary continuous cumulative distribution function of a single variable specified by a finite set of points. A (smooth) increasing quadratic spline is constructed which interpolates the data points and preserves the convexity of the data [2]. The spline is compared with Akima's piecewise cubic approximation [1] for several common distributions F.
TL;DR: A wide class of implicit one-step methods for the construction of global approximations to solutions of initial value problems is studied in this article, where error bounds are given for the general class of methods but emphasis is placed on methods based on Hermite interpolation, for which higher rates of convergence are obtained for special choices of interpolation points.
Abstract: A wide class of implicit one-step methods for the construction of global approximations to solutions of initial value problems is studied. Approximations more general than piecewise polynomials can be constructed to exploit certain characteristics of the differential equation. Error bounds are given for the general class of methods but emphasis is placed on methods based on Hermite interpolation, for which higher rates of convergence are obtained for special choices of interpolation points. Computational examples are presented.