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Concept# Approximation

Summary

An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word approximation is derived from Latin approximatus, from proximus meaning very near and the prefix ad- (ad- before p becomes ap- by assimilation) meaning to. Words like approximate, approximately and approximation are used especially in technical or scientific contexts. In everyday English, words such as roughly or around are used with a similar meaning. It is often found abbreviated as approx.
The term can be applied to various properties (e.g., value, quantity, image, description) that are nearly, but not exactly correct; similar, but not exactly the same (e.g., the approximate time was 10 o'clock).
Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.
In science, approximation can refer to using a simpler process or model when the correct mod

Official source

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In this thesis we will deal with the creation of a Reduced Basis (RB) approximation of parametrized Partial Differential Equations (PDE) for three-dimensional problems. The the idea behind RB is to decouple the generation and projection stages (Ofﬂine/Online computational proce- dures) of the approximation process in order to solve parametrized (PDE) in a fast, cheap and reliable way. The RB method, especially applied to 3D problems, allows great computational savings with respect to the clas- sical Galerkin Finite Element (FE) Method. The standard FE method is typically ill suited to (i) iterative contexts like in optimization, sensitivity analysis and many queries in general and (ii) real time evaluation. We consider both coercive and noncoercive PDEs. For each class we discuss the steps to set up a RB approximation, either from an analytical and a numerical point of view. Then we present the applications of the RB method to three different problems of engineering interest and applica- bility: (i) a steady thermal conductivity problem in heat transfer; (ii) a linear elasticity problem; (iii) Stokes ﬂows with emphasis on geometrical and physical parameters.

2010In this thesis, I define and explain the notion of punctual analytical uniform development (DUAP) versus moments or cumulants approximation of punctual uniforms analyticals 's class statistics. Hence, I derive "truncated" DUAP's version for numerical computation and implementation which called finite DUAP (F-DUAP). Using F-DUAP approximation lead to an error which was estimated. Due to nature's one of axiom of UAP statistics, the concept extension of DUAP method, to an other class statistic is limited. So, a new local theoretical concept was defined named analytical uniform development (DUA). This generalization let all derived DUAP's theorems become more general. Automatic differentiation and F-DUAP allow the implementation of DUAP or DUA method's on computer: I write the CUMAD and CUMADG codes that make the methods of a practical use. By, the programme CUMAD, I valid the utility of DUAP method, when I applied it to the approximation to moments of "weighted sum of squares statistic".

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