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Function Approximation
Published in John D. Ross, Kendall C. Richards, Introductory Analysis, 2020
John D. Ross, Kendall C. Richards
Function approximation is a branch of mathematics that seeks to find, among a family of functions, the function that best approximates a given function f. Function approximation is necessary in many applied settings, including numerical computation of functions, and is of broad interest to applied mathematicians and computer scientists. In this Extended Exploration, we will examine a number of ways we can carefully and systematically approximate functions, beginning with Taylor polynomials (which are well-known to calculus students) and Lagrange Interpolation polynomials (which are less so).
Computational fluid dynamics
Published in Amithirigala Widhanelage Jayawardena, Fluid Mechanics, Hydraulics, Hydrology and Water Resources for Civil Engineers, 2021
Amithirigala Widhanelage Jayawardena
The FEM has been developed on the basis that a continuum can be represented by an assemblage of subdivisions called finite elements. These elements are considered interconnected at joints called nodes. Piecewise approximations are made by assuming the unknown function to vary within each finite element in terms of their nodal values according to some interpolation functions. In some literature, the term shape function, displacement function and displacement model are used synonymously. Generally, the interpolation functions are assumed to be polynomials of a given degree.
Continuum problems
Published in Gianni Comini, Stefano Del Giudice, Carlo Nonino, Finite Element Analysis in Heat Transfer, 2018
Gianni Comini, Stefano Del Giudice, Carlo Nonino
It has already been remarked that the finite element method requires the preliminary replacement of the unknown solution ϕ, throughout the solution domain Ω, by an approximation ϕ^. The problem of approximation is not a new one in engineering and many techniques are available to find approximating functions that are smooth enough for easy evaluations, derivations and integrations within the domain.6,7
A simplified Bixon–Jortner–Plotnikov method for fast calculation of radiationless transfer rates in symmetric molecules
Published in Molecular Physics, 2023
A. I. Martynov, A. S. Belov, V. K. Nevolin
The rates of internal conversion, intersystem crossing, and radiative transition were evaluated using the formulas (21)–(23). Molecular parameters for the evaluation were obtained as described in the Computational details section. The result of such calculation strongly depends on the density of states. All three approximations for density of states were used for computations, the Gauss, Hybrid, and Pekarian functions. Each approximation simplifies the calculation in a different way and affects the accuracy differently which determines the choice of a function. The IC and ISC rates obtained by these methods are shown in Tables 3 and 4. The Hybrid function was used with the threshold of 0.01. The reason behind this choice is that this threshold is small enough for density to not become rarefied, but it's not too small for TSL group to become empty.
Traffic state prediction using conditionally Gaussian observed Markov fuzzy switching model
Published in Journal of Intelligent Transportation Systems, 2023
Zied Bouyahia, Hedi Haddad, Stéphane Derrode, Wojciech Pieczynski
However, while computing different quantities of interest, one arrives at integrals on which are impossible to compute exactly and, therefore, approximations are required. Given that parameter estimation is a hard problem in the general case, we simplify it by making the following approximation: we assume that is divided into F + 2 “discrete fuzzy levels”: Thus the distribution of is a distribution on