Showing papers on "Piecewise published in 1976"
TL;DR: In this paper, it was shown that the minimum angle condition is not essential and that no angle is too close to $180 √ √ Ω(n) √ n.
Abstract: The finite element procedure consists in finding an approximate solution in the form of piecewise linear functions, piecewise quadratic, etc. For two-dimensional problems, one of the most frequently used approaches is to triangulate the domain and find the approximate solution which is linear, quadratic, etc., in every triangle. A condition which is considered essential is that the angle of every triangle, independent of its size, should not be small. In this paper it is shown that the minimum angle condition is not essential. What is essential is the fact that no angle is too close to $180^ \circ $.
691 citations
TL;DR: In this paper, a new method for the solution of nonlinear periodic networks has been developed, where the network is decomposed into a minimum number of linear and nonlinear subnetworks.
Abstract: A new method for the solution of nonlinear periodic networks has been developed. It avoids the time domain solution of the dynamic equations. In the proposed method, the network is decomposed into a minimum number of linear and nonlinear subnetworks. Only frequency domain solutions of the linear subnetworks are required. It is shown that considerable reduction in the size of the computational problem can be achieved by taking advantage of the linearities present in the network.
357 citations
163 citations
TL;DR: An L/sup 1/ estimate of the gradient of the error in the finite element approximation of the Green's function is proved that is optimal for all degrees.
Abstract: Uniform estimates for the error in the finite element method are derived for a model problem on a general triangular mesh in two dimensions. These are optimal if the degree of the piecewise polynomials is greater than one. Similar estimates of the error are also derived in L/sup p/. As an intermediate step, an L/sup 1/ estimate of the gradient of the error in the finite element approximation of the Green's function is proved that is optimal for all degrees.
162 citations
TL;DR: In this article, two methods of fitting piecewise multiple regression models are presented, one based on dynamic programming and the other based on hierarchical procedure, which is suitable for very long sequences of data.
Abstract: Two methods of fitting piecewise multiple regression models are presented. One, based on dynamic programming, yields maximum‐likelihood estimators and is suitable for sequences of moderate length. A second, hierarchical, procedure yields approximations to the maximum‐likelihood estimators and is suitable for very long sequences of data. Both methods have computational requirements that are linear in the number of segments.
101 citations
TL;DR: In this paper, it was shown that the nodal parameters for the cubic Clough-Tocher macroelement uniquely determined a piecewise polynomial of degree 3, and a quartic macroelement for degree 4.
Abstract: A short proof is given of Ciarlet’s result that the nodal parameters for the cubic Clough–Tocher macroelement uniquely determine a $C^1 $ piecewise polynomial of degree 3. Also, a quartic macroelement is introduced and it is shown that the nodal parameters for this element uniquely determine a $C^1 $ piecewise polynomial of degree 4.
64 citations
TL;DR: In this article, a finite algorithm using the piecewise strategy for large-scale bilinear programming problems was developed, which consists of systematically generating a sequence of expanding polytopes with the global optimum within each polytope being known.
Abstract: The paper deals with bilinear programming problems and develops a finite algorithm using the “piecewise strategy” for large-scale systems. It consists of systematically generating a sequence of expanding polytopes with the global optimum within each polytope being known. The procedure then stops when the final polytope contains the feasible region.
54 citations
TL;DR: For general 2 × 2 genuinely nonlinear conservation laws and isentropic gas dynamics equations, not necessarily convex, this paper proved uniqueness theorems of the Cauchy problem for piecewise continuous solutions with a finite number of centered rarefaction waves in each compact set.
41 citations
TL;DR: In this paper, the Ritz-Galerkin method was extended to the general setting of L-splines, and these methods were then contrasted with familiar tensor product techniques in application of the ritz-galerkin method for approximately solving elliptic boundary value problems.
Abstract: This paper considers the theoretical development of finite dimensional bivariate blending function spaces and the problem of implementing the Ritz-Galerkin method in these approximation spaces. More specifically, the approximation theoretic methods of polynomial blending function interpolation and approximation developed in [2, 11---13] are extended to the general setting of L-splines, and these methods are then contrasted with familiar tensor product techniques in application of the Ritz-Galerkin method for approximately solving elliptic boundary value problems. The key to the application of blending function spaces in the Ritz-Galerkin method is the development of criteria which enable one to judiciously select from a nondenumerably infinite dimensional linear space of functions, certain finite dimensional subspaces which do not degrade the asymptotically high order approximation precision of the entire space. With these criteria for the selection of subspaces, we are able to derive a virtually unlimited number of new Ritz spaces which offer viable alternatives to the conventional tensor product piecewise polynomial spaces often employed. In fact, we shall see that tensor product spaces themselves are subspaces of blending function spaces; but these subspaces do not preserve the high order precision of the infinite dimensional parent space.
Considerable attention is devoted to the analysis of several specific finite dimensional blending function spaces, solution of the discretized problems, choice of bases, ordering of unknowns, and concrete numerical examples. In addition, we extend these notations to boundary value problems defined on planar regions with curved boundaries.
40 citations
TL;DR: Cubic spline smoothing as mentioned in this paper is a popular method for agricultural data smoothing, where spline functions are defined piecewise and can represent any variable arbitrarily well over wide ranges of the other.
Abstract: Agronomic data frequently requires smoothing in order to obtain a reliable functional relationship for interpolating, predicting, or determining the rate of change of one variable with respect to another. To test whether cubic spline functions could provide satisfactory smoothing, the necessary equations were derived, computer programs written, and several sets of soil temperature and water content data were smoothed. Cubic spline smoothing displayed the following, advantages: 1) Because spline functions are defined piecewise, they can represent any variable arbitrarily well over wide ranges of the other. 2) The data can be obtained at unequal intervals, so high sampling rates can be used where changes are rapid and low rates where they are slow. 3) Additionally, the gradients derived from cubic spline functions are smoothly joined parabolas, not the abruptly joined straightline segments characteristic of parabolic spline smoothing.
33 citations
TL;DR: In this paper, the solution of multidimensional bound state problems in quantum mechanics by the piecewise analytic method introduced by Gordon is discussed in detail and numerical techniques necessary to calculate the wavefunctions and matrix elements are presented.
Abstract: The solution of multidimensional bound state problems in quantum mechanics by the piecewise analytic method introduced by Gordon is discussed in detail. The numerical techniques necessary to calculate the wavefunctions and matrix elements are presented. An efficient least squares method for fitting the potential energy function to spectroscopic frequencies is described. Tests of the accuracy of the piecewise analytic method have been performed and the results are given. The techniques presented here are especially advantageous for systems in which the motion along the several coordinates is strongly coupled (beyond the perturbation theory limit).
TL;DR: Piecewise analytic scattering wavefunctions are constructed for colinear models of inelastic and reactive scattering systems, using piecewise analytic vibrational basis sets and reaction coordinates defined by conformal transformations as discussed by the authors.
Abstract: Piecewise analytic scattering wavefunctions are constructed for colinear models of inelastic and reactive scattering systems, using piecewise analytic vibrational basis sets and reaction coordinates defined by conformal transformations. Various approximate and exact methods for computing basis transformation matrices are discussed. Also presented are a compact formalism for extracting transmission and reflection coefficients (S‐matrix elements) from the wavefunctions, and a method for transforming the wavefunctions from one coordinate system to another. Model potential surfaces used include two models of inelastic scattering (harmonic oscillator–exponential repulsion and harmonic oscillator–Lennard‐Jones) and a colinear Porter–Karplus H3 surface. Transmission and reflection probabilities from converged close‐coupled calculations on the latter surface from threshold to −3.1 eV are presented. The number of closed channels used and the calculated probabilities are consistent with previously reported calculat...
TL;DR: In this article, an analog of the well-known Jackson-Bernstein-Zygmund theory on best approximation by trigonometric polynomials is developed for approximation methods which use piecewise polynomial functions.
Abstract: An analog of the well-known Jackson-Bernstein-Zygmund theory on best approximation by trigonometric polynomials is developed for approximation methods which use piecewise polynomial functions. Interpolation and best approximation by polynomial splines, Hermite and finite element functions are examples of such methods. A direct theorem is proven for methods which are stable, quasi-linear and optimally accurate for sufficiently smooth functions. These assumptions are known to be satisfied in many cases of practical interest. Under a certain additional assumption, on the family of meshes, an inverse theorem is proven which shows that the direct theorem is sharp.
TL;DR: In this paper, the control of a continuous linear plant disturbed by white plant noise is considered, where the control is constrained to be a piecewise constant function of time, and the cost function is the integral of quadratic error terms in the state and control, thus penalizing errors at every instant of time.
Abstract: This paper considers the control of a continuous linear plant disturbed by white plant noise when the control is constrained to be a piecewise constant function of time; ie a stochastic sampled-data system The cost function is the integral of quadratic error terms in the state and control, thus penalizing errors at every instant of time while the plant noise disturbs the system continuously The problem is solved by reducing the constrained continuous problem to an unconstrained discrete one It is shown that the separation principle for estimation and control still holds for this problem when the plant disturbance and measurement noise are Gaussian
TL;DR: In this paper, a piecewise maximization of the Hamiltonian and a limiting process utilizing a penalty function of the control variables are proposed to obtain suboptimal singular and/or bang-bang solutions of typical lumped and distributed parameter control problems.
Abstract: A new and simple computational algorithm for obtaining suboptimal singular and/or bang-bang solutions of typical lumped and distributed parameter control problems is proposed. The algorithm is based on the piecewise maximization of the Hamiltonian and a limiting process utilizing a penalty function of the control variables. Theoretical developments and computational applications of the algorithm to several linear lumped parameter control problems are presented. Extensions and applications of the algorithm to nonlinear and distributed parameter systems are given in Part II.
TL;DR: A fast scheme using a sphtand-merge procedure is described for functions of both one and two variables for piecewise constant and piecewise linear approximations without continuity constraints.
Abstract: Algorithms for piecewise approximations with variable breakpomts can be used for picture segmentation. The high computational requirements of many of the algorithms whmh search for optimal solutions make them unsuitable for such apphcations A fast scheme using a sphtand-merge procedure is described for functions of both one and two variables. Examples of its application on pictures are given. The present implementation is only for piecewise constant and piecewise linear approximations without continuity constraints. I t can be readily extended to higher order approximations without constraints. Theoretically the spht-and-merge procedure can also be used with continuity constraints.
TL;DR: The optimal control problem is thereby reduced to a linearly constrained parameter optimization problem which can be solved efficiently using the quadratically convergent Gold-farb-Lapidus algorithm.
Abstract: The optimization of nonlinear systems subject to linear terminal state variable constraints is considered A technique for solving this class of problems is proposed that involves a piecewise polynomial parameterization of the system variables The optimal control problem is thereby reduced to a linearly constrained parameter optimization problem which can be solved efficiently using the quadratically convergent Gold-farb-Lapidus algorithm Illustrative numerical examples are presented
TL;DR: This paper deals with curve fitting by a piecewise cubic polynomial which is continuous with its first derivative and a suboptimal algorithm is applied to minimize the sum of squares of residuals.
Abstract: This paper deals with curve fitting by a piecewise cubic polynomial which is continuous with its first derivative. A knot is inserted successively until a certain criterion is satisfied. Then a suboptimal algorithm is applied to minimize the sum of squares of residuals.
TL;DR: In this article, the original formalism is rederived using Green's function techniques, and the first order perturbation integrals are rewritten in a more compact and numerically stable form.
Abstract: Methods presented previously [J. Chem. Phys. 51, 14 (1969); Methods Comput. Phys. 10, 81 (1971)] for constructing piecewise analytic (Airy function) solutions to the close‐coupled equations of molecular scattering and bound states are reviewed and developed further. The original formalism is rederived using Green’s function techniques. The first order perturbation integrals are rewritten in a more compact and numerically stable form. A quadrature is presented for integrating products of wavefunctions of unequal diagonal slopes. Numerically efficient techniques are presented for constructing single channel bound state functions and using them in matrix element and overlap integrals.
TL;DR: In this paper, the use of Walsh functions in the design of piecewise constant gains for the linear optimal control problem with quadratic performance criteria was considered in the context of solving a vector differential equation.
Abstract: The above paper considers the use of Walsh functions in the design of piecewise constant gains for the linear optimal control problem with quadratic performance criteria. The use of Walsh functions is illustrated in the context of solving a vector differential equation. A Walsh series is assumed for each rate variable and the unknown Walsh coefficients are determined by solving the resulting set of simultaneous equations. In this correspondence it is shown that the same solution can be easily obtained by proceeding directly with the piecewise constant approximation for the rate variables.
TL;DR: In this article, the orthogonality of piecewise continous eigenfunctions is used to solve boundary value problems in composite media by the method of separation of variables, which can be applied to solve the problem of transient temperature of a reactor pressure vessel during a postulated loss-of-coolant accident.
01 Jan 1976
TL;DR: Based on the Diakoptic Equation, an algorithm is developed for block diagonalizing and solving any given set of linear simultaneous equations and is then applied to block diagonalize and solve the Jacobian matrix and hence the load flow problem.
Abstract: Based on the Diakoptic Equation, an algorithm is developed for block diagonalizing and solving any given set of linear simultaneous equations. This algorithm is then applied to block diagonalize and solve the Jacobian matrix and hence the load flow problem. The algorithm is perfectly general and can be appliedto any set of linear simultaneous equations, symmetrical or asymmetrical and hence can be applied to other fields as well. Applied to the load flow problem, it requires to consider a few more additional right hand side vectors to the Jacobian, and further an inter-subdivision matrix is to be formed and solved. No matrix inversion is required in this method and hence sparsity can be exploited to the maximum extent. Practical steps are given, so that the algorithm can be implimented by others.
TL;DR: In this article, local support bases for a class of splines consisting piecewise of elements in the null space of a linear differential operator L with the component pieces tied smoothly together at the knots by requiring the continuity of certain ExtendedHermite-Birkhoff linear functionals are obtained.
TL;DR: In this article, necessary and sufficient conditions are obtained for bisingular integral operators on piecewise smooth Ljapunov curves with discontinuous coefficients in the -spaces with a weight to be Noetherian.
Abstract: Necessary and sufficient conditions are obtained for bisingular integral operators on piecewise smooth Ljapunov curves with discontinuous coefficients in the -spaces with a weight to be Noetherian. The Banach algebra generated by these operators is studied; a regularizer is constructed in the case of continuous coefficients.Bibliography: 40 titles.
TL;DR: System theory for holographic representation of linear space-variant systems is derived and the resulting piecewise isoplanatic approximation (PIA) is illustrated by example application to the invariant system, ideal magnifier, and Fourier transformer.
Abstract: System theory for holographic representation of linear space-variant systems is derived. The utility of the resulting piecewise isoplanatic approximation (PIA) is illustrated by example application to the invariant system, ideal magnifier, and Fourier transformer. A method previously employed to holographically represent a space-variant system, the discrete approximation, is shown to be a special case of the PIA.
01 Jan 1976
TL;DR: In this article, the authors define a rather general class of piecewise functions, and discuss some of its basic algebraic and structural properties, and specialize to a (still quite general) class of Tchebycheffian splines and discuss their zero properties and approximation powers.
Abstract: This paper is divided into two parts. In the first part we define a rather general class of piecewise functions, and discuss some of its basic algebraic and structural properties. In the second part, we specialize to a (still quite general) class of Tchebycheffian splines and discuss their zero properties and approximation powers.
TL;DR: In this article, a constructive method for finding a solution to the piecewise linear complementarily problem is given, which is essentially based on Lemke's algorithm for solving the linear complementary problem.
Abstract: This paper gives a constructive method for finding a solution to the piecewise linear complementarily problem. The construction is essentially based on Lemke’s algorithm for solving the linear complementarily problem, and the extended algorithm does indeed generate a piecewise linear almost complementary path. A necessary and sufficient condition for the algorithm to compute a solution is a slight modification of the condition assumed by Eaves [4] for the existence of solutions to general nonlinear cases. The results presented in this paper do not depend on the particular subdivision scheme used to define the piecewise linear functions.
TL;DR: In this paper, the controllability conditions for R n and R n null controllabilities were derived for two special cases of the problem of controllable admissible control inputs.
Abstract: An important problem in system theory is whether a dynamic system is controllable with inputs from the class of admissible controls. This paper treats the case whore the dynamic system is described by differential-difference equations and the class of admissible control inputs consists of piecewise constant functions. This situation arises in practice in the on-line control of some industrial processes. Euclidean space and function space controllability are studied, and controllability conditions which are both necessary and sufficient are derived for R n and R n null controllability. Simple algebraic conditions are also given for two special cases. In the study of function space null controllability the concept of j-controllability is introduced and a sufficiency theorem is stated in terms of this new concept.
TL;DR: In this paper, a piecewise maximization of the Hamiltonian and a limiting process utilizing a penalty function of the control variables for the computation of suboptimal singular and/or bang-bang control was proposed.
Abstract: A new and simple algorithm based on the piecewise maximization of the Hamiltonian and a limiting process utilizing a penalty function of the control variables for the computation of suboptimal singular and/or bang-bang control, previously developed and applied to linear lumped systems in Part I, is tested on four typical nonlinear lumped and distributed systems of particular chemical engineering interest. The computational results have shown that the proposed algorithm is a very effective method for solving singular and/or bang-bang control problems with high state dimensionality, extreme nonlinearity, and multiple controls.