A Modified Monte-Carlo Quadrature
نویسنده
چکیده
where "| A |" denotes the (fc-dimensional) volume of A. If the x¡ are regarded as independent (or at least pairwise independent) random variables, then the estimator / is a random variable whose mean is I and whose standard deviation is dN~112, where d2 = | A \ jA f2 — (JA f)2. (7 is the mean, and d is the standard deviation of the random variable | A ¡fix), where x is a random variable uniformly distributed on A. ) The standard deviation of J is taken as a measure of the error to be expected in taking a sample value of J, as above, as an estimate of I. The error thus decreases very slowly as N ( and the expense of the calculation ) increases and may be unacceptably large even for quite high values of N. As a result, a great deal of effort has gone into devising more sophisticated forms of Monte-Carlo procedures (see, e.g., [1] ) in order to replace J by estimators of lower variance. Each of these methods 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.
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تاریخ انتشار 2010