arXiv 2102.022692021-02-03math.NAcs.NAmath.OC

局部Chebyshev逼近的数值微分

Numerical Differentiation using local Chebyshev-Approximation

Stefan H. Reiterer

备注：: This work was privately done during my study time on university. No other funding was involved and it should no associated with my current affiliation

摘要：

在应用数学中，特别是在最优化中，函数通常仅被提供为由软件包提供的所谓的“黑盒”，或者非常复杂的算法，这使得自动微分非常复杂甚至不可能。因此人们寻求导数的数值逼近。不幸的是，数值微分本身就是一项困难的任务，并且众所周知，它是数值不稳定的。有许多关于这个主题的作品，包括（全局）切比雪夫近似的使用。切比雪夫近似有一个很大的特性，即如果函数是光滑的，则它们收敛得非常快。然而，这些方法有几个缺点，因为在实践中的函数是不光滑的，和一个整体的近似需要许多功能评估。尽管如此，还是有希望的。由于在现实世界中的应用程序的功能是大多数时间光滑的有限点，角落或边缘。这促使使用局部切比雪夫方法，其中函数仅局部近似，因此切比雪夫近似仍然产生期望函数的快速近似。我们将在这项工作中研究这种方法，并将提供一个数值例子

英文摘要：

In applied mathematics, especially in optimization, functions are often only provided as so called "Black-Boxes" provided by software packages, or very complex algorithms, which make automatic differentation very complicated or even impossible. Hence one seeks the numerical approximation of the derivative. Unfortunately numerical differentation is a difficult task in itself, and it is well known that it is numerical instable. There are many works on this topic, including the usage of (global) Chebyshev approximations. Chebyshev approximations have the great property that they converge very fast, if the function is smooth. Nevertheless those approches have several drawbacks, since in practice functions are not smooth, and a global approximation needs many function evalutions. Nevertheless there is hope. Since functions in real world applications are most times smooth except for finite points, corners or edges. This motivates to use a local Chebyshev approach, where the function is only approximated locally, and hence the Chebyshev approximations still yields a fast approximation of the desired function. We will study such an approch in this work, and will provide a numerical example

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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