Approximation Methods
Approximation Methods
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DOI：
10.1017/9781108499996.009
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发表时间：
2020-09
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通讯作者：
Kenneth L. Judd
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作者：
Kenneth L. Judd
在本章中，我们将讨论一个在各种经济问题中都会遇到的非常重要的问题：函数的逼近。当计算真实函数的时间或复杂性成本太高时，或者当该函数未知并且我们只需要对其主要属性有一个粗略的了解时，通常会发生这样的问题。通常，唯一需要的是能够在一个或几个点上计算这个函数，并对所有其他值进行猜测。这就给我们留下了一些关于近似的局部或全局特性以及我们想要达到的精度水平的选择。正如我们将在不同的应用中看到的那样，选择方法通常是效率和计算简便的问题。在Judd [1998]之后，我们将考虑3种近似方法1。局部近似，它基本上利用了函数在一个点的值及其在同一点的导数的信息。然后，我们的想法是（希望）在基准点的邻域中获得函数的良好近似。2. L p近似，它实际上找到了一个很好的函数，它接近于我们想要在L p范数意义下计算的函数。理想情况下，我们需要整个函数的信息来找到一个好的近似，这通常是不可行的-或者这将使问题1
In this chapter, we deal with a very important problem that we will encounter in a wide variety of economic problems: approximation of functions. Such a problem commonly occurs when it is too costly either in terms of time or complexity to compute the true function or when this function is unknown and we just need to have a rough idea of its main properties. Usually the only thing that is required then is to be able to compute this function at one or a few points and formulate a guess for all other values. This leaves us with some choice concerning either the local or global character of the approximation and the level of accuracy we want to achieve. As we will see in different applications, choosing the method is often a matter of efficiency and ease of computing. Following Judd [1998], we will consider 3 types of approximation methods 1. Local approximation, which essentially exploits information on the value of the function in one point and its derivatives at the same point. The idea is then to obtain a (hopefully) good approximation of the function in a neighborhood of the benchmark point. 2. L p approximations, which actually find a nice function that is close to the function we want to evaluate in the sense of a L p norm. Ideally, we would need information on the whole function to find a good approximation , which is usually infeasible – or which would make the problem 1