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Computational Physics - M. Jensen Episode 1 Part 9

Tham khảo tài liệu 'computational physics - m. jensen episode 1 part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 9.5. IMPROVED MONTE CARLO INTEGRATION 149 The second term on the rhs disappears since this is just the mean and employing the definition of Ơ2 we have 1 _ 22V2 . 9.59 resulting in 1- 21V2 9.60 and in the limit N oo we obtain 1 í A r . z exp --2 ự2Ĩ r ự V V 2 r ựjv -p 9.61 which is the normal distribution with variance Ơ2 2V where Ơ is the variance of the PDF p x and 1 is also the mean of the PDF p x . Thus the central limit theorem states that the PDF p z of the average of N random values corresponding to a PDF p x is a normal distribution whose mean is the mean value of the PDF p x and whose variance is the variance of the PDF p x divided by N the number of values used to compute . The theorem is satisfied by a large class of PDFs. Note however that for a finite A it is not always possible to find a closed expression for p tr . 9.5 Improved Monte Carlo integration In section 5.1 we presented a simple brute force approach to integration with the Monte Carlo method. There we sampled over a given number of points distributed uniformly in the interval 0.1 A . A . . . I f f dx K 2f xf . with the weights U ị 1 . Here we introduce two important topics which in most cases improve upon the above simple brute force approach with the uniform distribution p x 1 for X e 0 1 . With improvements we think of a smaller variance and the need for fewer Monte Carlo samples although each new Monte Carlo sample will most likely be more times consuming than corresponding ones of the brute force method. The first topic deals with change of variables and is linked to the cumulative function p x of a PDF p x . Obviously not all integration limits go from X 0 to X 1 rather in physics we are often confronted with integration domains like X e 0 oo or X e oo oo etc. Since all random number generators give numbers in the interval X e 0 1 we need a mapping from this integration interval to the explicit one under consideration. 150 CHAPTER 9. OUTLINE OF THE MONTE-CARLO STRATEGY The next topic .