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Piecewise

About: Piecewise is a research topic. Over the lifetime, 21064 publications have been published within this topic receiving 432096 citations. The topic is also known as: piecewise-defined function & hybrid function.


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01 Feb 1993
TL;DR: In this article, the authors present jump formulas for generalized functions in non-conservative form for systems in nonconservative form with piecewise C? characteristics, and the abstract theory of generalized functions.
Abstract: to generalized functions and distributions.- Multiplications of distributions in classical physics.- Elementary introduction.- Jump formulas for systems in nonconservative form. New numerical methods.- The case of several constitutive equations.- Linear wave propagation in a medium with piecewise C? characteristics.- The canonical Hamiltonian formalism of interacting quantum fields.- The abstract theory of generalized functions.

112 citations

Posted Content
TL;DR: An optimal combination for the polynomial spaces that minimize the number of unknowns in the numerical scheme without compromising the accuracy of the numerical approximation is presented.
Abstract: The novel idea of weak Galerkin (WG) finite element methods is on the use of weak functions and their weak derivatives defined as distributions. Weak functions and weak derivatives can be approximated by polynomials with various degrees. Different combination of polynomial spaces leads to different weak Galerkin finite element methods, which makes WG methods highly flexible and efficient in practical computation. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the numerical scheme, yet without compromising the accuracy of the numerical approximation. For illustrative purpose, the authors use second order elliptic problems to demonstrate the basic idea of polynomial reduction. A new weak Galerkin finite element method is proposed and analyzed. This new finite element scheme features piecewise polynomials of degree $k\ge 1$ on each element plus piecewise polynomials of degree $k-1\ge 0$ on the edge or face of each element. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete $H^1$ norm and the standard $L^2$ norm. In addition, the paper presents a great deal of numerical experiments to demonstrate the power of the WG method in dealing with finite element partitions consisting of arbitrary polygons in two dimensional spaces or polyhedra in three dimensional spaces. The numerical examples include various finite element partitions such as triangular mesh, quadrilateral mesh, honey comb mesh in 2d and mesh with deformed cubes in 3d. The numerical results show a great promise of the robustness, reliability, flexibility and accuracy of the WG method.

111 citations

Journal ArticleDOI
TL;DR: The optimization of both the interpolation points and the piecewise interpolating polynomials for the formation of the upper and lower envelopes of the signal reveal important characteristics of the method which where previously hidden, leading to significant performance improvements.
Abstract: Empirical mode decomposition (EMD) is a relatively new, data-driven adaptive technique for analyzing multicomponent signals. Although it has many interesting features and often exhibits an ability to decompose nonlinear and nonstationary signals, it lacks a strong theoretical basis which would allow a performance analysis and hence the enhancement and optimization of the method in a systematic way. In this paper, the optimization of EMD is attempted in an alternative manner. Using specially defined multicomponent signals, the optimum outputs can be known in advance and used in the optimization of the EMD-free parameters within a genetic algorithm framework. The contributions of this paper are two-fold. First, the optimization of both the interpolation points and the piecewise interpolating polynomials for the formation of the upper and lower envelopes of the signal reveal important characteristics of the method which where previously hidden. Second, basic directions for the estimates of the optimized parameters are developed, leading to significant performance improvements.

111 citations

Journal ArticleDOI
TL;DR: This novel a posteriori limiter, which has been recently proposed for the simple Cartesian grid case in 62, is able to resolve discontinuities at a sub-grid scale and is substantially extended here to general unstructured simplex meshes in 2D and 3D.

111 citations

Journal ArticleDOI
TL;DR: In this article, the evolution of wave functions that consist of polynomial segments, usually joined smoothly together, is considered for a non-relativistic particle and its evolution can be expressed in terms of antiderivatives of the propagator.
Abstract: For a non-relativistic particle, we consider the evolution of wave functions that consist of polynomial segments, usually joined smoothly together. These spline wave functions are compact (that is, they are initially zero outside a finite region), but they immediately extend over all available space as they evolve. The simplest splines are the square and triangular wave functions in one dimension, but very complicated splines have been used in physics. In general the evolution of such spline wave functions can be expressed in terms of antiderivatives of the propagator; in the case of a free particle or an oscillator, all the evolutions are expressed exactly in terms of Fresnel integrals. Some extensions of these methods to two and three dimensions are discussed.

111 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20251
2023917
20222,014
20211,089
20201,147
20191,106