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.

Error Analysis of a Finite Element Method for the Space-Fractional Parabolic Equation

Abstract: We consider an initial boundary value problem for a one-dimensional fractional-order parabolic equation with a space fractional derivative of Riemann--Liouville type and order $\alpha\in (1,2)$. We study a spatial semidiscrete scheme using the standard Galerkin finite element method with piecewise linear finite elements, as well as fully discrete schemes based on the backward Euler method and the Crank--Nicolson method. Error estimates in the $L^2(D)$- and $H^{\alpha/2}(D)$-norm are derived for the semidiscrete scheme and in the $L^2(D)$-norm for the fully discrete schemes. These estimates cover both smooth and nonsmooth initial data and are expressed directly in terms of the smoothness of the initial data. Extensive numerical results are presented to illustrate the theoretical results.

Approximation Classes for Adaptive Methods

Abstract: Adaptive Finite Element Methods (AFEM) are numerical proce- dures that approximate the solution to a partial differential equation (PDE) by piecewise polynomials on adaptively generated triangulations. Only re- cently has any analysis of the convergence of these methods (10, 13) or their rates of convergence (2) become available. In the latter paper it is shown that a certain AFEM for solving Laplace's equation on a polygonal domain ⊂ R 2 based on newest vertex bisection has an optimal rate of convergence in the following sense. If, for some s > 0 and for each n = 1,2, . . ., the solu- tion u can be approximated in the energy norm to order O(n s ) by piecewise linear functions on a partition P obtained from n newest vertex bisections, then the adaptively generated solution will also use O(n) subdivisions (and floating point computations) and have the same rate of convergence. The question arises whether the class of functions A s with this approximation rate can be described by classical measures of smoothness. The purpose of the present paper is to describe such approximation classes A s by Besov smoothness.

Multistability analysis for recurrent neural networks with unsaturating piecewise linear transfer functions

Abstract: Multistability is a property necessary in neural networks in order to enable certain applications (e.g., decision making), where monostable networks can be computationally restrictive. This article focuses on the analysis of multistability for a class of recurrent neural networks with unsaturating piecewise linear transfer functions. It deals fully with the three basic properties of a multistable network: boundedness, global attractivity, and complete convergence. This article makes the following contributions: conditions based on local inhibition are derived that guarantee boundedness of some multistable networks, conditions are established for global attractivity, bounds on global attractive sets are obtained, complete convergence conditions for the network are developed using novel energy-like functions, and simulation examples are employed to illustrate the theory thus developed.