A modified Monte-Carlo quadrature. II.
A modified Monte-Carlo quadrature. II.
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修正的蒙特卡罗求积。
DOI：
10.1090/s0025-5718-1966-0210285-0
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发表时间：
1966
影响因子：
2
通讯作者：
S. Haber
中科院分区：
文献类型：
--
作者：
S. Haber
涉及计算过程适应正在集成的特定功能；因此需要对被积函数进行初步分析并编写特殊的积分程序。 （蒙特卡罗计算通常在自动计算机上完成。）在本文中，我们提出了一种改进的蒙特卡罗求积方法，其应用是完全自动的，并且产生 I 的估计，其方差稍微但通常显着地低于 J 的方差。 2. 过程。我们的方法是分层抽样的一种形式[2]；然而，区域 A（“群体”）被分成简单地根据其几何形状定义的子区域，而不是根据我们期望在其中找到的值来定义。我们定义 A 是 k 维区间的情况下的过程，即一维区间 (a', bi) 的笛卡尔积，i = 1, * - *, k。 （通过使用非直角坐标系，该方法可以应用于一些其他形状的区域。）每个区间（a'，b'）被细分点a' = ai，0 < aij < ... < ai，i = b'划分为子区间。这样做的方式是，对于每个 i，差异 ai+、- ai、j 是可通约的；最简单的是，它们都可能是
involves adaptation of the computation procedure to the particular function being integrated; thus it necessitates preliminary analysis of the integrand and the writing of a special integration program. (Monte-Carlo calculations are generally done on automatic computers.) In this paper we present a modified Monte-Carlo quadrature method whose application is completely automatic and which produces an estimate of I whose variance is slightly, but often significantly, lower than that of J. 2. The Procedure. Our method is a form of stratified sampling [2]; however the region A (the "population") is broken into subregions defined simply in terms of its geometry, rather than in terms of the values off we expect to find in them. We define the procedure in the case that A is a k-dimensional interval, i.e. the cartesian product of one-dimensional intervals (a', bi), i = 1, * - *, k. (By using other than cartesian coordinates, the method may be applied to regions of some other shapes.) Each interval (a', b') is divided into subintervals by the subdivision points a' = ai,0 < aij < ... < ai,i = b'. This is done in such a way that for each i the differences ai+, - ai,j are commensurable; most simply, all of them may be multiples of the