Showing papers on "Piecewise published in 1978"

A practical guide to splines

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.

Ergodic transformations from an interval into itself

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 \.

Analysis and optimal control of time-varying linear systems via Walsh functions

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.

Analysis and synthesis of dynamic systems containing time delays via block-pulse functions

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.

Accuracy and Resolution in the Computation of Solutions of Linear and Nonlinear Equations

Abstract: Publisher Summary This chapter describes the accuracy and resolution in the computation of solutions of linear and nonlinear equations. In many problems, one is presented with piecewise initial data whose discontinuities occur along surfaces. According to the theory of hyperbolic equations, solutions with such initial data are themselves piecewise with their discontinuities occurring across characteristic surface. At points away from the discontinuities, the truncation error is small. In these regions, it is reasonable to use difference approximations of high order accuracy, except for the danger that the large truncation error at the discontinuities propagates into the smooth region. According to the theory of hyperbolic conservation laws, solutions of systems of the form are in general discontinuous. The discontinuities, called shocks, need not be present in the initial values but arise spontaneously and their speed of propagations is governed by the Rankine–Hugoniot jump relation. It is more difficult to construct accurate approximations of discontinuous solutions of nonlinear equations than of linear equations.

Some metric properties of piecewise monotonic mappings of the unit interval

Abstract: In this note, the result of Lasota and Yorke on the existence of invariant measures for piecewise C2 functions is extended to a larger class of piecewise continuous functions. Also the result of Li and Yorke on the existence of ergodic measures for piecewise C2 functions is extended for the above class of functions. Introduction. In a joint paper by Lasota and Yorke [7], they proved the existence of absolutely continuous invariant measures for t a piecewise C2 mapping of the unit interval with inixS[0l]\\r'(x)\\ > 1. Later Li and Yorke [8] proved the existence of ergodic measures for the same type of mapping. In this note, it will be shown that these results can be extended to a certain class of piecewise C1 mappings. Also at the end of this note, an attempt will be made to explain why there has been a sudden interest in studying piecewise monotonie mappings. Existence of invariant and of ergodic measures. Denote by (L„ || • ||,), the space of all functions / defined on [0, 1] for which |/| is integrable, and by m Lebesgue measure on [0, 1]. Let t: [0, 1]h»[0, 1] be a measurable nonsingular transformation, i.e., if A is measurable, m(A) = 0 implies m(r~x(A)) = 0. Given t, the Frobenius-Perron operator PT: Z.,1—»i, is given by the formula: The operator PT is linear, continuous, and satisfies the following conditions: (a) PT is positive: / > 0 => PJ > 0; (b) PT preserves integrals: Çpjdm=Çfdm, /EL,; •'o •'o (c) P(t„) = P\" where t\" denotes the nth iterate of t; (d) PJ = / iff the measure d¡i = f dm is invariant under t, i.e., ¡l(t~\\A)) = n(A) for each measurable A. Definition 1. A transformation t: [0, 1]h>P will be called piecewise Received by the editors October 17, 1977. AMS (MOS) subject classifications (1970). Primary 28A65; Secondary 58F15. © American Mathematical Society 1979

Design of two-dimensional recursive digital filters

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.

Considerations for efficient wire/surface modeling

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.

Error Analysis for Convex Separable Programs: The Piecewise Linear Approximation and The Bounds on The Optimal Objective Value

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 ...

Collocation solution of creeping newtonian flow through periodically constricted tubes with piecewise continuous wall profile

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.

Resolution Analysis for Discrete Systems

Abstract: summary. Treatments of geophysical inverse problems have tended to polarize into approaches intended to generate models either described by piecewise continuous functions or with some prior discretization. The two approaches are here developed in parallel, and the ideas of a trade-off between the anticipated error and the attainable level of detail in the model estimate are extended to the discrete case, either with even or uneven discretization. An alternative approach to specifying the potential resolution of a model is to establish upper and lower bounds on parameter values. Linear programming methods are extended to determine bounds which allow for subjective limits on parameter values. For a non-linear system the possible resolution may be investigated by estimation procedures based on the full set of successful solutions obtained by Monte-Carlo inversion.

Quantum rate theory for a symmetric double‐well potential

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.

Piecewise convex programs

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.

Superconvergence of piecewise polynomial Galerkin approximations, for Fredholm integral equations of the second kind

Abstract: Piecewise polynomial Galerkin approximations for Fredholm integral equations of the second kind are shown to posses superconvergence properties in some circumstances.

Studies on Piecewise-Linear Approximations of Piecewise-C1 Mappings in Fixed Points and Complementarity Theory

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...

Collocation Methods for Linear Elliptic Problems

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.

Piecewise polynomial basis functions for configuration interaction and many-body perturbation theory calculations. The radial limit of helium

Abstract: Highly accurate configuration interaction and many‐body perturbation theory calculations are shown to be possible with piecewise polynomial basis functions In a preliminary calculation on the helium atom ground state, a new upper bound of −28790286 au is found for the radial limit

Higher order methods for a singularly perturbed problem

Abstract: A finite element method using piecewise polynomials of degree ?k is used to approximate the problem ?u?+u?=f, ?>0 a small parameter. A very irregular mesh is used. On this mesh error estimates of order0(h k+1) are obtained uniformly in ?,h the maximum stepsize, fork?2. The condition number of the system of linear equations one has to solve in order to get the approximation is estimated. Extension of the results to more complicated problems is briefly indicated. Finally, a numerical example is given.

Triangulations by Reflections with Applications to Approximation

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.

Piecewise constant solutions of integral equations via Walsh functions

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.

Finite Element Approximation for First Order Systems

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.

Numerical inversion of the Radon transform

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.