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 on account that its formal improvement within the Forties. each one of these tutorial efforts were directed both on the mathematical foundation of the method or at its sensible software as embodied within the numerous huge, formal laptop codes to be had for appearing Monte Carlo shipping calculations. This ebook makes an attempt to fill what seems to be a niche 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 sensible radiation delivery difficulties. this is often performed by utilizing the computer because the simple educating software. This publication assumes the reader has an information of necessary calculus, neutron delivery idea, and Fortran programming. It additionally assumes the reader has to be had a computer with a Fortran compiler. Any workstation of average dimension could be enough to breed the examples or resolve the routines contained herein. The authors think it is necessary for the reader to execute those examples and routines, and via doing to be able to develop into comprehensive at getting ready acceptable software program for fixing radiation shipping difficulties utilizing Monte Carlo. The step from the software program defined during this e-book to using construction Monte Carlo codes may be straight forward.
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Extra info for A Monte Carlo Primer: A Practical Approach to Radiation Transport
The standard deviation thus improves by one over the square root of the number of samples. For large n the estimates of the mean frequently approach a Normal or Gaussian distribution. [2;. l is the mean and 0' is the standard deviation of the distribution. Here Y is the probability of finding a sample value from a normally distributed popUlation within X of the mean. 73% of the time the estimate will be within three standard deviations of the correct value. 5 Introduction to Variance Reduction Techniques As indicated above, the purpose of a Monte Carlo calculation is usually to obtain an estimate of the expected value of a random variable.
L)var = (sumsq(i)/dj - (sumf(i)/dj)**2) totals=totals+sumf (i) /dj; tvarl=tvar1+var; tvar2=tvar2+var/dj 200 CCNI'INUE WRITE (8, 16)totals,DSQRT(tvarl),DSQRT(tvar2) 16 ~T(' sum:', flO . 6,' stdevof distr:', flO. 6) var=O . OdO ! 6,' stdev of distr: ',fl0 . 7,' stdevof sum:',eI4 . 6) REIURN END For many random variables of interest, the range over which a Monte Carlo estimate varies within a stratum can be less than that for the total sample space. Since the variance of the variable depends upon this range, and reducing the range reduces the variance, stratification generally reduces the variance.
100))nstrata=100 CALL stats delta=(b-a)/nstrata ! delta x value for each stratun delta2=(b-a) ! delta x for IIDStratified calculatioo 00 100 j=l,nsamples dK=fltrn () ! lber in (0, 1) n=linstrata*dK ! 1late the strata associated with dK dK=a+(b-a)*dK; score=delta*f(dK) ! calculate score at dK CALL statlp ! store statistics for score score=delta2*f (dK) ! calculate score for IIDStratified case n=nstrata+1 ! store IIDStratified scores in last strata CALL statlp ! store statistics for IIDStratified case 100 CCNl'INUE n=nstrata CALL statend !