By Stephen A. Dupree, Stanley K. Fraley
The mathematical means of Monte Carlo, as utilized to the delivery of sub-atomic debris, has been defined in different stories and books due to the fact that its formal improvement within the Forties. every one of these educational efforts were directed both on the mathematical foundation of the process or at its functional software as embodied within the a number of huge, formal laptop codes on hand for appearing Monte Carlo delivery calculations. This booklet makes an attempt to fill what seems to be a spot during this Monte Carlo literature among the maths and the software program. therefore, whereas the mathematical foundation for Monte Carlo delivery is roofed in a few element, emphasis is put on the appliance of the strategy to the answer of useful radiation delivery difficulties. this can be performed through the use of the computer because the uncomplicated instructing device. This booklet assumes the reader has an information of indispensable calculus, neutron shipping conception, and Fortran programming. It additionally assumes the reader has to be had a laptop with a Fortran compiler. Any computing device of moderate dimension can be sufficient to breed the examples or remedy the routines contained herein. The authors think it will be significant for the reader to execute those examples and workouts, and through doing to be able to turn into entire at getting ready applicable software program for fixing radiation delivery difficulties utilizing Monte Carlo. The step from the software program defined during this booklet to using creation Monte Carlo codes will be straightforward.
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Extra resources for A Monte Carlo Primer: A Practical Approach to Radiation Transport
This point determines the value of a random variable for purposes of the calculation. • The random variable is either the result of the calculation or is in one-to-one correspondence with the result. 8 forms the basis for a significant portion of the Monte Carlo technique. With this equation, random numbers distributed over the interval 2. , the random number ~ is used to select F(x), which assures a fair game. 9 (or a functionally equivalent inverse) is prohibitive, other methods of solving for x must be used.
Revised 1956. 1 Probability Theory and Statistics To understand the mathematical basis of Monte Carlo calculations, to develop means of increasing the efficiency of such calculations, and to estimate the statistical uncertainty in the results obtained, it is necessary to have some understanding of statistics and probability theory. A brief description of the terminology that will be used in the remainder of this book is also necessary. The intent of this chapter is to introduce the mathematical concepts that underlie the Monte Carlo method and provide a basis for further development of selected topics.
F For the purpose ofthis proof, a measurable set A of real numbers is any set that is a union of intervals of real numbers. 70) Then E(V1) = fV1(x)fl(x)dx= fVI(x) f(x) dx=.!. , E(V \) > E(V) - then E(V 2) < E(V), and vice versa. 76) 2. 77) R-A a J[V - E(V\ )]2 f(x)dx + (1- a) J[V - E(V2 )]2 f(x)dx R-A A In an unbiased sampling scheme with n samples for these two strata, nP(A) or na samples will be used for stratum A, and nP(R~A) or n(1-a) will be used for stratum R~A. 79) + (1:a) J[V -E(V2)]2f2(x)dx R- A =.!.
A Monte Carlo Primer: A Practical Approach to Radiation Transport by Stephen A. Dupree, Stanley K. Fraley