Showing papers on "Piecewise linear function published in 1986"
TL;DR: In this article, a nonlinear relationship between electricity sales and temperature is estimated using a semiparametric regression procedure that easily allows linear transformations of the data and accommodates introduction of covariates, timing adjustments due to the actual billing schedules, and serial correlation.
Abstract: A nonlinear relationship between electricity sales and temperature is estimated using a semiparametric regression procedure that easily allows linear transformations of the data. This accommodates introduction of covariates, timing adjustments due to the actual billing schedules, and serial correlation. The procedure is an extension of smoothing splines with the smoothness parameter estimated from minimization of the generalized cross-validation criterion introduced by Craven and Wahba (1979). Estimates are presented for residential sales for four electric utilities and are compared with models that represent the weather using only heating and cooling degree days or with piecewise linear splines.
954 citations
TL;DR: In this paper, a method for adaptive stabilization without a minimum-phase assumption and without knowledge of the sign of the high-frequency gain is developed, which leads to a guarantee of Lyapunov stability and an exponential rate of convergence for the state.
Abstract: In this paper, we develop a method for adaptive stabilization without a minimum-phase assumption and without knowledge of the sign of the high-frequency gain. In contrast to recent work by Martensson [8], we include a compactness requirement on the set of possible plants and assume that an upper bound on the order of the plant is known. Under these additional hypotheses, we generate a piecewise linear time-invariant switching control law which leads to a guarantee of Lyapunov stability and an exponential rate of convergence for the state. One of the main objectives in this paper is to eliminate the possibility of "large state deviations" associated with a search Over the space of gain matrices which is required in [8].
318 citations
TL;DR: A new approach to determination of mapping functions for registration of digital images is presented, where the images are divided into triangular regions by triangulating the control points and the overall mapping function is obtained by piecing together the linear mapping functions.
297 citations
TL;DR: This work presents an algorithm that finds a piecewise linear curve with the minimal number of segments required to approximate a curve within a uniform error with fixed initial and final points.
Abstract: Two-dimensional digital curves are often uniformly approximated by polygons or piecewise linear curves. Several algorithms have been proposed in the literature to find such curves. We present an algorithm that finds a piecewise linear curve with the minimal number of segments required to approximate a curve within a uniform error with fixed initial and final points. We compare our optimal algorithm to several suboptimal algorithms with respect to the number of linear segments required in the approximation and the execution time of the algorithm.
293 citations
01 Nov 1986-Graphical Models \/graphical Models and Image Processing \/computer Vision, Graphics, and Image Processing
TL;DR: This paper considers the problem of approximating a piecewise linear curve by another whose vertices are a subset of the vertices of the former, and shows that an optimum solution of this problem can be found in a polynomial time.
Abstract: In cartography, computer graphics, pattern recognition, etc., we often encounter the problem of approximating a given finer piecewise linear curve by another coarser piecewise linear curve consisting of fewer line segments. In connection with this problem, a number of papers have been published, but it seems that the problem itself has not been well modelled from the standpoint of specific applications, nor has a nice algorithm, nice from the computational-geometric viewpoint, been proposed. In the present paper, we first consider (i) the problem of approximating a piecewise linear curve by another whose vertices are a subset of the vertices of the former, and show that an optimum solution of this problem can be found in a polynomial time. We also mention recent results on related problems by several researchers including the authors themselves. We then pose (ii) a problem of covering a sequence of n points by a minimum number of rectangles with a given width, and present an O(n long n )-time algorithm by making use of some fundamental established techniques in computational geometry. Furthermore, an O(mn (log n ) 2 )-time algorithm is presented for finding the minimum width w such that a sequence of n points can be covered by at most m rectangles with width w . Finally, (iii) several related problems are discussed.
190 citations
TL;DR: This paper presents a unified parameter optimization algorithm for constructing prototype canonical piecewise-linear models of p-n junction diodes, bipolar transistors, MOSFET's, and GaAs FET's.
Abstract: To take advantage of the remarkable computational efficiency of the canonical piecewise-linear approach for dc nonlinear electronic circuit analysis, the devices must be modeled by a canonical piecewise-linear model. This paper presents a unified parameter optimization algorithm for constructing such models. This algorithm is then applied to derive prototype canonical piecewise-linear models of p-n junction diodes, bipolar transistors, MOSFET's, and GaAs FET's. The canonical piecewise-linear model can be regarded as a universal model since its form remains unchanged for all devices. Only the coefficients differ from one device to another. For large-scale circuits, the canonical piecewise-linear representation has a decisive advantage over other representations in regard to the number of memory locations needed to specify the equations.
159 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
TL;DR: In the space of binary sequences, minimal sets, that is: sets invariant under the shift operation, that have no invariant proper subsets, are investigated as discussed by the authors, and the optimal minimal set is the optimal set for which the convergence of the running average to the rotation number is faster than for any other minimal set.
Abstract: In the space of binary sequences, minimal sets, that is: sets invariant under the shift operation, that have no invariant proper subsets, are investigated. In applications, such as a piecewise linear circle map and the Smale horseshoe in a mapping of the annulus, each of these sets is invariant under the mapping. These sets can be assigned a unique rotation number equal to the average of the number of ones in the sequences. One special minimal set is the optimal set for which the convergence of the running average to the rotation number is faster than for any other minimal set. These sequences are instrumental in analytically solving for the width of the parameter intervals for which members of a one parameter family of piecewise linear critical circle maps have rational rotation number.
46 citations
TL;DR: In this article, the authors dealt with a discontinuous control system that consists of two finite-dimensional linear systems; the first describes the motion in a given half-space, the second in the complementary half-Space Necessary and sufficient conditions for local controllability are obtained.
44 citations
TL;DR: In this paper, the authors investigate invariant circles for a one-parameter family of piecewise linear twist homeomorphisms of the annulus and classify them into families, showing that invariant circle of all types and rotation numbers occur.
Abstract: We investigate invariant circles for a one-parameter family of piecewise linear twist homeomorphisms of the annulus. We show that invariant circles of all types and rotation numbers occur and we classify them into families. We compute parameter ranges in which there are no invariant circles.
01 Jan 1986
TL;DR: In this paper, the differential calculus for convex, compact-valued multifunctions developed by Tyurin, Banks and Jacobs is used to give an equivalent description in terms of multifunctions of the class of functions which can be represented as the difference of two globally Lipschitzian convex functions.
Abstract: The differential calculus for convex, compact-valued multifunctions developed by Tyurin, Banks and Jacobs is used to give an equivalent description in terms of multifunctions of the class of functions which can be represented as the difference of two globally Lipschitzian convex functions. This approach is also used to develop a means of representing piecewise-linear continuous functions as the difference of two piecewise-linear convex functions in finite dimensions. This leads directly to a Minkowski duality theorem for piecewise-linear positively homogeneous continuous functions and equivalence classes of convex compact sets produced by convex compact polyhedrons: every piecewise-linear positively homogeneous continuous function may be uniquely characterized by its quasidifferential (as defined by Demyanov and Rubinov) at zero.
TL;DR: A BMDP program for obtaining near optimum piecewise linear regression equations and an idea intrinsic to the method is that restricting parameter space to a discrete set makes the difficult problems become standard problems.
Journal Article•
TL;DR: Two types of LCP solution algorithms are briefly described: pivoting algorithms and the modulus algorithm; it is shown that these algorithms have certain disadvantages if applied to the problem but can be overcome by the new methods to be presented.
Abstract: In the simulation of electronic circuits piecewise linear modelling yields a global circuit description which in principle can be used to solve for a circuit response in a finite number of steps. During the solution process a sequence of linear complementarily problems (LCP) has to be solved within the piecewise linear system description. The purpose of this paper is to present and discuss some new methods to solve this LCP for certain matrix classes.To start with, two types of LCP solution algorithms are briefly described: pivoting algorithms and the modulus algorithm. It is shown that these algorithms have certain disadvantages if applied to the problem as stated above. Those problems can be overcome by the new methods to be presented. The first one is a modified version of an iterative algorithm of 0. L. Mangasarian. The second one is a so-called simplicial method, based on a new integer labelling and an efficient labelling algorithm. Convergence conditions are given, as is a bound for the error in the...
TL;DR: In this article, two fully discrete finite element solutions of the nonstationary semiconductor equations were constructed, one was nonlinear, the other was partly linear, and both were shown to converge in a strong norm to the weak solution.
Abstract: In part I of the paper (see Zlamal [13]) finite element solutions of the nonstationary semiconductor equations were constructed. Two fully discrete schemes were proposed. One was nonlinear, the other partly linear. In this part of the paper we justify the nonlinear scheme. We consider the case of basic boundary conditions and of constant mobilities and prove that the scheme is unconditionally stable. Further, we show that the approximate solution, extended to the whole time interval as a piecewise linear function, converges in a strong norm to the weak solution of the semiconductor equations. These results represent an extended and corrected version of results announced without proof in Zlamal [14].
TL;DR: An optimal crashing policy for the project is evolved, and an extensive sensitivity analysis is reported to examine the effect of changing the priority structure of goals and unit crashing costs on the optimality of the solution.
TL;DR: Adaptive Zoom Tracking-Experimental Code (AZTEC) as discussed by the authors is a code to solve the one-dimensional gas dynamical equations in a variable area duct with specific implementation for plane, cylindrical, and spherical geometries.
TL;DR: In this article, Cook and Kress present a model for representing ordinal preference rankings, where the voter can express intensity or degree of preference, and the consensus of a set of m rankings is that ranking whose distance from this set is minimal.
Abstract: Cook and Kress Cook, Wade D., Moshe Kress. 1985. Ordinal ranking with intensity of preference. Management Sci.31 1 26-32. present a model for representing ordinal preference rankings, where the voter can express intensity or degree of preference. The consensus of a set of m rankings is that ranking whose distance from this set is minimal. The consensus problem is then shown to be an integer programming problem with a piecewise linear convex objective function. In the present note we prove that the constraint matrix for this integer problem is totally unimodular. In addition, it is shown that the problem can be expressed as an equivalent integer linear programming problem. These two facts allow us to represent the consensus problem as a linear programming model. To further facilitate an efficient solution procedure to the consensus problem, it is shown that the number of columns in the L.P. model can generally be reduced significantly. Computational results on a wide range of problems is presented.
01 Dec 1986
TL;DR: In this article, the authors developed a method for adaptive stabilization without a minimum phase assumption and without knowledge of the sign of the high frequency gain, which leads to a guaranteed Lyapunov stability and an exponential rate of convergence for the state.
Abstract: In this paper, we develop a method for adaptive stabilization without a minimum phase assumption and without knowledge of the sign of the high frequency gain. In contrast to recent work by Martensson [8], we include a compactness requirement on the set of possible plants and assume that an upper bound on the order of the plant is known. Under these additional hyphotheses, we generate a piecewise linear time-invariant switching control law which leads to a guarantee of Lyapunov stability and an exponential rate of convergence for the state. One of the main objectives in this paper is to eliminate the possibility of "large state deviations" associated with a search over the space of gain matrices which is required in [8].
TL;DR: In this article, it was shown that a chain of piecewise linear maps displays spatial disorder if the individual map shows temporal chaos, and that the spatial correlation length of the chain scales as the inverse of the square root of the temporal Lyapunov exponent.
TL;DR: In this article, error estimates for a fully discrete scheme for the numerical solution of the neutron transport equation in two-dimensional Cartesian geometry obtained by using a special quadrature rule for the angular variable and the discontinuous Galerkin finite element method with piecewise linear trial function for the space variable were derived.
Abstract: We derive error estimates for a fully discrete scheme for the numerical solution of the neutron transport equation in two-dimensional Cartesian geometry obtained by using a special quadrature rule for the angular variable and the discontinuous Galerkin finite element method with piecewise linear trial function for the space variable
TL;DR: In this paper, the generalized gradient of F is characterized, and necessary and sufficient conditions for a (strict) local minimization of F are given for regression models where the range of the dependent variable is restricted.
Abstract: The censored linear $l_1 $ approximation problem is to minimize the nonconvex piecewise linear function $F(x) = \sum _{i = 1}^m |y_i - \max (z_i ,x^T a_i )|$. The problem arises in regression models where the range of the dependent variable is restricted. Unlike the maximum likelihood and least squares estimators the censored $l_1 $ estimator provides a consistent estimator without an assumption that the errors are normally distributed.This paper presents a compact characterization of the generalized gradient of F, and necessary and sufficient conditions for a (strict) local minimizes of F. A reduced gradient algorithm for linear programming and $l_1 $ approximation is extended to provide a stable finite direct descent method, for calculating a local minimizes of F. This provides an efficient method of calculating the censored $l_1 $ estimator.
TL;DR: As an extension to the breakpoint hopping algorithm developed in reference 1, the algorithm presented in this paper efficiently solves the d.c. operating point(s) problem and for tracing the driving-point and transfer characteristics of an extremely broad class of non-linear resistive circuits.
Abstract: As an extension to the breakpoint hopping algorithm developed in reference 1, the algorithm presented in this paper efficiently solves the d.c. problem for finding the d.c. operating point(s) and for tracing the driving-point and transfer characteristics of an extremely broad class of non-linear resistive circuits. In particular, bipolar and MOS transistor circuits are included. A user-friendly C program has been written to implement this algorithm where the input format for describing the circuit is compatible with the SPICE program.
TL;DR: In this paper, a nonconforming combined approach of the Ritz-Galerkin method and the finite element method is presented for solving Laplace's boundary value problems with singularities.
Abstract: For solving Laplace's boundary value problems with singularities, a nonconforming combined approach of the Ritz-Galerkin method and the finite element method is presented. In this approach, singular functions are chosen to be admissible functions in the part of a solution domain where there exist singularities; and piecewise linear functions are chosen to be admissible functions in the rest of the solution domain. In addition, the admissible functions used here are constrained to be continuous only at the element nodes on the common boundary of both methods. This method is nonconforming; however, the nonconforming effect does not result in larger errors of numerical solutions as long as a suitable coupling strategy is used.
TL;DR: This algorithm consists of a finite sequence of cycles, derived from the simplex method, characteritic of linear programming, and the line search, characteristic of nonlinear programming, which reduces the required storage and amount of calculation.
Abstract: A compact algorithm is presented for solving the convex piecewise-linear-programming problem, formulated by means of a separable convex piecewise-linear objective function (to be minimized) and a set of linear constraints. This algorithm consists of a finite sequence of cycles, derived from the simplex method, characteritic of linear programming, and the line search, characteristic of nonlinear programming. Both the required storage and amount of calculation are reduced with respect to the usual approach, based on a linear-programming formulation with an expanded tableau. The tableau dimensions arem×(n+1), wherem is the number of constraints andn the number of the (original) structural variables, and they do not increase with the number of breakpoints of the piecewise-linear terms constituting the objective function.
TL;DR: In this paper, a piecewise linear second order ordinary differential equation which is topologically equivalent to the sine-Gordon equation is considered and the behavior of the periodic solutions is examined, and it is shown that subharmonic motions with period n times that of the disturbance appear via saddle-node bifurcations for all n = 1, 2, 3, etc.
TL;DR: In this paper, a system of nonlinear equations governed by more than one parameter is discussed, with particular attention to bifurcation behaviour, and the procedure adopted is to add to the original system of n equations (m − 1) further equations, in the case of m parameters, and to seek solution curves in Rn+m to this augmented system.
Abstract: Systems of nonlinear equations governed by more than one parameter are discussed with particular attention to bifurcation behaviour. The procedure adopted is to add to the original system of n equations (m − 1) further equations, in the case of m parameters, and to seek solution curves in Rn+m to this augmented system. Two types of additional equations are considered: one describes a piecewise linear path in the space of parameters, and the second constrains the solution curve to be a locus of singular points. These ideas are all subsequently applied to the systems of equations arising from finite element approximations of boundary value problems in nonlinear elasticity. The behaviour of a nonlinear elastic thick-walled cylinder subjected to internal pressure and axial extension is discussed.
TL;DR: In this paper, the distance between a piecewise linear approximation and a general curve is given for the approximate computation of the metric on general curves, and an algorithm has been successfully implemented in Fortran.
Abstract: An algorithm is given for computing a metric on piecewise linear curves. A bound for the distance between a piecewise linear approximation and a general curve allows the approximate computation of the metric on general curves. The algorithm has been successfully implemented in Fortran.
TL;DR: In this paper, the authors studied two bifurcations which occur at the same parameter value and showed that the three orbits created in this compound bifurbation are the principal periodic orbits of a homoclinic bifurlcation seen in the system.