Showing papers on "Piecewise linear function published in 1989"

The Nonparametric Approach

Abstract: The nonparametric approach to the measurement of productive efficiency can be specified either through a flexible form of the production function which satisfies the efficiency hypothesis, or a set-theoretic characterization of an efficient isoquant. In the first case the production frontier can be of any general shape satisfying some very weak conditions like quasi-concavity or monotonicity, although in most empirical and applied work piecewise linear or, log-linear functions have been frequently used. Thus both Farrell and Johansen applied linear programming models in the specification of the production frontier. Farrell’s efficiency measure is based on estimating by a sequence of linear programs (LPs) a convex hull of the observed input coefficients in the input space. Two features of Farrell efficiency make it very useful in applied research. One is that it is completely data-based i.e., it uses only the observed inputs and outputs of the sample units while assuming production functions to be homogeneous of degree one. Hence it has many potential applications for the public sector units, where for most of the inputs and outputs the price data are not available. For example consider educational production functions for public schools, where outputs such as test scores in achievement tests are only proxy variables for learning; inputs such as average class size, experience of teachers or ethnic background of students do not have observed market prices. Secondly, Farrell’s method uses a set of LP models to estimate the efficiency parameters, so that the production frontier appears as piecewise linear functions. Nonnegativity conditions on the parameter estimates can therefore be easily incorporated.

A comparison of adaptive refinement techniques for elliptic problems

Abstract: Adaptive refinement has proved to be a useful tool for reducing the size of the linear system of equations obtained by discretizing partial differential equations. We consider techniques for the adaptive refinement of triangulations used with the finite element method with piecewise linear functions. Several such techniques that differ mainly in the method for dividing triangles and the method for indicating which triangles have the largest error have been developed. We describe four methods for dividing triangles and eight methods for indicating errors. Angle bounds for the triangle division methods are compared. All combinations of triangle divisions and error indicators are compared in a numerical experiment using a population of eight test problems with a variety of difficulties (peaks, boundary layers, singularities, etc.). The comparison is based on the L-infinity norm of the error versus the number of vertices. It is found that all of the methods produce asymptotically optimal grids and that the number of vertices needed for a given error rarely differs by more than a factor of two.

Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach

Abstract: We propose a discrete mapping to study the qualitative properties of the dynamics of biological or other complex networks, involving switchlike interactions between the various elements. The stable steady states and the limit cycles of the piecewise-linear equations which model such networks are derived from the dynamical behaviour of the proposed discrete mapping. Furthermore, this mapping takes into account not only the logical structure of the network but also the parameters used in the description.

Fitting piecewise linear regression functions to biological responses

Abstract: An iterative approach was achieved for fitting piecewise linear functions to nonrectilinear responses of biological variables. This algorithm is used to estimate the parameters of the two (or more) regression functions and the separation point(s) (thresholds, sensitivities) by statistical approximation. Although it is often unknown whether the response of a biological variable is adequately described by one rectilinear regression function or by piecewise linear regression function(s) with separation point(s), an F test is proposed to determine whether one regression line is the optimal fitted function. A FORTRAN-77 program has been developed for estimating the optimal parameters and the coordinates of the separation point(s). A few sets of data illustrating this kind of problem in the analysis of thermoregulation, osmoregulation, and the neuronal responses are discussed.

The generalized linear complementarity problem applied to the complete analysis of resistive piecewise-linear circuits

Abstract: An important application of complementarity theory consists in solving sets of piecewise-linear equations and hence in the analysis of piecewise-linear resistive circuits. The authors show how a generalized version of the linear complementarity problem can be used to analyze a broad class of piecewise-linear circuits. Nonlinear resistors that are neither voltage nor current controlled can be allowed, and no restrictions on the linear part of the circuit have to be made. As a second contribution, the authors describe an algorithm for the solution of the generalized complementarity problem and show how it can be applied to yield a complete description of the DC solution set as well as of driving-point and transfer characteristics. >

On optimal interpolation triangle incidences

Abstract: The problem of determining optimal incidences for triangulating a given set of vertices for the model problem of interpolating a convex quadratic surface by piecewise linear functions is studied An exact expression for the maximum error is derived, and the optimality criterion is minimization of the maximum error The optimal incidences are shown to be derivable from an associated Delaunay triangulation and hence are computable in $O(N\log N)$ time for N vertices

Transfer Functions for Efficient Calculation of Multidimensional Transient Heat Transfer

Abstract: Finite difference or finite element methods reduce transient multidimensional heat transfer problems into a set of first-order differential equations when thermal physical properties are time invariant and the heat transfer processes are linear. This paper presents a method for determining the exact solution to a set of first-order differential equations when the inputs are modeled by a continuous, piecewise linear curve. For long-time solutions, the method presented is more efficient than Euler, Crank-Nicolson, or other classical techniques.

Improved theoretical and experimental models for the coaxial colinear antenna

Abstract: An improved theoretical model of the coaxial colinear (COCO) antenna is presented that takes into account different element lengths and power transfers between the antenna and the transmission lines. The antenna equation now contains an exact kernel instead of an approximate kernel and a piecewise constant basis function is used instead of a piecewise linear function, yielding faster results. More points are used for better accuracy and yet faster computations. The linear systems of equations of the theoretical model are solved using a preconditioned conjugate gradient method. Uniform and tapered current distributions are obtained experimentally and theoretically on end-fed coaxial colinear antennas. There is reasonable agreement between theory and measurements. The gains of a few COCO antennas relative to equivalent lengths of half-wave dipoles are given. >

Efficient implementation of piecewise linear activation function for digital VLSI neural networks

Abstract: A piecewise linear approximation to the sigmoidal neuron activation function is proposed, which maps compactly into a digital integrated circuit realisation.

Optimum Design of Large Sewer Networks

Abstract: A linear programming, diameter discretization, heuristic approach is presented for the optimum design of large gravity sewer networks. The mathematical model contains a nonlinear convex function relating pipeline diameter and slope, which is approximated by piecewise linear segments. This approach uses a modified Hazen‐Williams hydraulic model at part‐full flow conditions, along with a newly developed universal expression to determine the coefficient of roughness. Moreover, the hydraulic formulation contains a regression equation to determine Darcy's friction factor based on the depth of flow in the pipe. The developed model has been extensively and successfully used to design several large sewer networks.

Shape matching using curvature processes

Abstract: Shape matching is a fundamental problem of vision in general and interpretation of deforming shapes in particular. The objective of matching in this instance is to recover the deformation and therefore generalizes the notion of correlation, which aims to only produce a numerical measure of the similarity between two shapes. To address shape matching, we introduce a new representation of a closed 2D shape a cyclic sequence of the extended circular images of the convex and concave segments of its contour. This representation is then used to establish correspondences between segments of the two contours using dynamic programming. Finally, we compute a recovery of the differences between two similar contours in terms of the action of curvature process. Computation of convex and concave segments of the contours, given in piecewise linear form, is accomplished using the analytic representation of a local B-spline fit. We show the result of our deformation recovery scheme appliedto dynamic cloud silhouette analysis using hand-traced input from real satellite images.

Superconvergence of the gradient in piecewise linear finite-element approximation to a parabolic problem

Abstract: Some recent results concerning maximum-norm superconvergence of the gradient in piecewise linear finite-element approximations of an elliptic problem are carried over to a parabolic problem. Both the standard semidiscrete in space Galerkin method and the lumped mass modification are analyzed for both smooth and nonsmooth data situations, and the Crank–Nicolson discretizations in time of these procedures are considered as examples of completely discrete schemes.

Smooth piecewise quadric surfaces

Abstract: This paper is concerned with the construction of tangent plane continuous piecewise quadric surfaces that interpolate finite sets of essentially arbitrary points in IR 3 according to a given ‘topology’ which is described in terms of a piecewise linear interpolant. Moreover, within certain ranges depending on the topology and the location of data points, given normal directions at the points are also matched by the interpolating piecewise quadric surface.

Automated design of multiple-class piecewise linear classifiers

Abstract: A new method and a supporting theorem for designing multiple-class piecewise linear classifiers are described. The method involves the cutting of straight line segments joining pairs of opposed points (i.e., points from distinct classes) ind-dimensional space. We refer to such straight line segments aslinks. We show how nearly to minimize the number of hyperplanes required to cut all of these links, thereby yielding a near-Bayes-optimal decision surface regardless of the number of classes, and we describe the underlying theory. This method does not require parameters to be specified by users — an improvement over earlier methods. Experiments on multiple-class data obtained from ship images show that classifiers designed by this method yield approximately the same error rate as the bestk-nearest neighbor rule, while providing faster decisions.

An efficient method to find all solutions of piecewise-linear resistive circuits

Abstract: An efficient method is presented for solving a system of piecewise-linear equations, F(x)=y, each element of which is a piecewise-linear function of one variable. Let n denote the dimension of x and L the number of regions of x in which F is linear. Then it is shown that the multiplications required to solve L linear simultaneous equations obtained from F(x)=y are O(Ln) for large n. >

An exact formula for the measure dimensions associated with a class of piecewise linear maps

Abstract: An exact formula for the various measure dimensions of attractors associated with contracting similitudes is given. An example is constructed showing that for more general affine maps the various measure dimensions are not always equal.

A DSS oriented method for multiobjective linear programming problems

Abstract: Most of practical linear programming problems involve multiple and conflicting objectives. The paper presents an interactive method to approach this kind of problems. The main original aspect of this method lies in the fact that it combines the advantageous features of both the paradigms of Satisfactory Goals and Multiattribute Utility Assesment. It is a DSS oriented approach providing a ‘two level’ interaction: 1. (1) interactive assessment of the decision maker's utility function using the UTA ordinal regression model; 2. (2) interactive modification of the satisfaction levels. Piecewise linear optimazation techniques are used to determine, at each iteration, a new compromise solution over the set of efficient solutions.

A Dual Method for Certain Positive Semidefinite Quadratic Programming Problems

Abstract: This paper presents a dual active set method for minimizing a sum of piecewise linear functions and a strictly convex quadratic function, subject to linear constraints. It may be used for direction finding in nondifferentiable optimization algorithms and for solving exact penalty formulations of (possibly inconsistent) strictly convex quadratic programming problems. An efficient implementation is described extending the Goldfarb and Idnani algorithm, which includes Powell's refinements. Numerical results indicate excellent accuracy of the implementation.

A simple algorithm for finding all solutions of piecewise-linear networks

Abstract: An efficient algorithm is presented for the searching of all solutions of a piecewise-linear (PWL) resistive network. Let P/sub /n be the number of largest segments of PWL resistors, and P/sub /n/sub -1/ the second largest. The algorithm presented is better than the brute-force method by a factor of P/sub /n*P/sub /n/sub -1/ and the Chua and Ying method, the most efficient method at the present time, by a factor of at least P/sub /n/sub -1/. The algorithm is based on a PWL network theorem, which is also introduced. >

Structure and control of piecewise-linear systems

Abstract: Elementary algebraic topology is used in the study of the structure and optimal control of piecewise-linear and non-linear systems.

SharpL ∞ -error estimates for semilinear elliptic problems with free boundaries

Abstract: A numerical scheme to approximate a semilinear PDE involving a (singular) maximal monotone graph is analyzed inL ?. A preliminary regularization is combined with piecewise linear finite elements defined on a triangulation which is not assumed to be acute; the discrete maximum principle is thus avoided. Sharp pointwise error estimates are derived for both the smoothing and the discretization procedures. An optimal choice of the regularization parameter as a function of the mesh size leads to a sharp global rate of convergence. These error estimates for solutions, in conjunction with nondegeneracy properties of continuous problems, provide sharp interface error estimates. Two model examples are discussed: the obstacle problem and a combustion equation.

A unified treatment of superconvergent recovered gradient functions for piecewise linear finite element approximations

Abstract: Piecewise linear finite element approximations to two-dimensional Poisson problems are treated. For simplicity, consideration is restricted to problems having Dirichlet boundary conditions and defined on rectangular domains Ω which are partitioned by a uniform triangular mesh. It is also required that the solutions u ∈ H3 (Ω). A method is proposed for recovering the gradients of the finite element approximations to a root mean square accuracy of O(h2), both at element edge mid-points and element vertices, using simple averaging schemes over adjacent elements. Piecewise linear interpolants (respectively discontinuous and continuous) are then fitted to these recovered gradients, and are shown to be O(h2) estimates for ∇u in the L2-norm, and thus superconvergent. A discussion is given of the extension of the results to problems with more general region and mesh geometries, boundary conditions and with solutions of lower regularity, and also to other second-order elliptic boundary value problems, e.g. the problem of planar linear elasticity.

A Simplicial Algorithm for Stationary Point Problems on Polytopes

Abstract: A simplicial variable dimension restart algorithm for the stationary point problem or variational inequality problem on a polytope is proposed. Given a polytope C in Rn and a continuous function f: C → Rn, find a point x in C such that fx · x ≥ fx · x for any point x in C. Starting from an arbitrary point v in C, the algorithm generates a piecewise linear path of points in C. This path is followed by alternating linear programming pivot steps to follow a linear piece of the path and replacement steps in a simplicial subdivision of C. Within a finite number of function evaluations and linear programming pivot steps the algorithm finds an approximate stationary point. The algorithm leaves the starting point v along a ray pointing to one of the vertices w of C. The vertex w is obtained from the optimum solution of the linear programming problem maximize fv · x subject to x ∈ C.

On the number of solutions of piecewise-linear resistive circuits

Abstract: Topological conditions are proved which allow an upper bound to be found for the number of DC operating points of a piecewise-linear circuit. The circuits may be nonreciprocal. The conditions amount to checking whether certain voltage and current orientations exist on the graph of the circuit. This can be implemented on a computer and it can be used for designing piecewise-linear circuits with prescribed DC operating points. >

Singular perturbation in piecewise-linear systems

Abstract: A singular perturbation technique is developed that allows for a decoupling of a continuous piecewise-linear system into slow and fast subsystems. Under the assumption of asymptotic stability, the fast variable is found to decay in the boundary layer to its quasi-steady-state solution, which is given by a continuous implicit function of the slow variable. The solution is found using a finite step algorithm. Sufficient conditions for the approximation to be accurate to an order of O( mu ), where mu is a parameter of the system, are given. The technique is illustrated by a numerical example. >