Piecewise linear function

About: Piecewise linear function is a research topic. Over the lifetime, 8133 publications have been published within this topic receiving 161444 citations.

Semiparametric estimates of the relation between weather and electricity sales

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.

On the Accuracy of the Finite Volume Element Method Based on Piecewise Linear Polynomials

Abstract: We present a general error estimation framework for a finite volume element (FVE) method based on linear polynomials for solving second-order elliptic boundary value problems. This framework treats the FVE method as a perturbation of the Galerkin finite element method and reveals that regularities in both the exact solution and the source term can affect the accuracy of FVE methods. In particular, the error estimates and counterexamples in this paper will confirm that the FVE method cannot have the standard O(h2) convergence rate in the L2 norm when the source term has the minimum regularity, only being in L2, even if the exact solution is in H2.

Section-wise piecewise-linear functions: Canonical representation, properties, and applications

Abstract: This paper presents a new closed form analytical formula for representing n-dimensional surfaces and scalar functions of n variables which are piecewise-linear over each cross section obtained by freezing any combination of n - 1 of the n coordinates. This new section-wise piecewise-linear representation can be easily programmed with efficient computer storage. It is a global representation in the sense that a single formula is used to compute for f(x 1 ,x 2 ,...,x n ) for all values of (x 1 , x 2 ,...., x n ). Since this representation is expressed in closed analytic form, it allows standard mathematical operations and manipulations to be carried out in theorectical studies, In particular, it led to the possibility of deriving explicit closed form expressions for system parameters and design formulas. Examples are given which illustrate the potential applications of this representation in the modeling and analysis of nonlinear devices, circuits and systems.

Mixed finite element methods for linear elasticity with weakly imposed symmetry

Abstract: In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approxima- tions to both stresses and displacements. The methods are based on a modified form of the Hellinger-Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.