Models and Algorithms for Optimal Piecewise-Linear Function Approximation

Abstract

Piecewise-linear functions can approximate nonlinear and unknown functions for which only sample points are available. This paper presents a range of piecewise-linear models and algorithms to aid engineers to find an approximation that fits best their applications. The models include piecewise-linear functions with a fixed and maximum number of linear segments, lower and upper envelopes, strategies to ensure continuity, and a generalization of these models for stochastic functions whose data points are random variables. Derived from recursive formulations, the algorithms are applied to the approximation of the production function of gas-lifted oil wells.

The improvement of crack propagation modelling in triangular 2D structures using the extended finite element method

“…The crack is a straight line or a curve in the real application. Therefore, the original crack can always be approximated by the piecewise linear function . Then, it is convenient to acquire the crack‐tip elements and crack surface elements by using the novel geometric method.…”

“…Therefore, the original crack can always be approximated by the piecewise linear function. 27 Then, it is convenient to acquire the crack-tip elements and crack surface elements by using the novel geometric method. To clearly illustrate the novel method, it is expressed in the following mathematic forms.…”

A Seemingly Polynomial-Time Algorithm for Optimal Curve Fitting by Segmented Straight Lines

“…We will present an algorithm that relies on dynamic programming to find optimal solutions to (3). To this end, introduce V (i, m, y) as the minimal cost over the subinterval [t i , t N ], using m segments, as a function of the value y at t i , i.e.…”

Proposal of Simple and Accurate Two-Parametric Approximation for the Q-Function

“…In particular, with the goal to derive closed form formulas enabling a straightforward performance analysis of the quantizers designed for the Gaussian source, some complex forms of the -function approximation have recently been applied in [21,22]. Also, in a number of papers, it has been pointed out that certain difficulties appear due to the nonexisting closedform solution for the inverse -function and other related 2 Mathematical Problems in Engineering special functions, for instance, in [29][30][31][32][33]. This has motivated the research presented in [29,30,33], where piecewise linear solutions have been proposed to overcome somewhat this problem, where, as a consequence, the performance degradation has been noticed.…”

“…Also, in a number of papers, it has been pointed out that certain difficulties appear due to the nonexisting closedform solution for the inverse -function and other related 2 Mathematical Problems in Engineering special functions, for instance, in [29][30][31][32][33]. This has motivated the research presented in [29,30,33], where piecewise linear solutions have been proposed to overcome somewhat this problem, where, as a consequence, the performance degradation has been noticed. As indicated in [19,28], with the suitable approximation of the -function this problem can be better solved, so that this has directed the research we presented here.…”