1. Optimization of Monte Carlo Integration
John Wilson
Western Kentucky University
Department of Physics & Astronomy
Applied Physics Institute 1 1 1 1 1 1

2. Introduction
Nuclear collision reaction cross sections are dependent upon the nuclear density distribution
Reaction cross section can be calculated numerically with Monte Carlo integration using coordinates randomly sampled from nuclear density distribution
Using cross section calculation to find nucleon distribution parameters

3. Reaction Cross Section
Experimental x-section: measured by scattering nuclei off of a carbon-12 target
Cross section calculation:
Calculated cross section fit to experimental data
Fitting parameter is parameter of distribution function of nucleons 3 3 3 3 3 3 3

4. Monte Carlo Integration
By definition:
where
MC is a numerical integration technique using random numbers
In order to achieve random numbers distributed according to the chosen nucleon distribution function, Markov Chain Monte Carlo (MCMC) is used.
Calculate f(x) using these numbers, find mean 4 4 4 4 4 4 4

5. Metropolis-Hastings
Drawing points from nucleon distribution allows fewer points than if sampling uniformly
Many ways to generate random coordinates distributed according to predetermined function
The Metropolis-Hastings algorithm was used
Does not require analytically solvable distributions (as is the case with the Box-Muller transform) 5 5 5 5 5 5 5

6. Metropolis-Hastings Algorithm
Start with initial point u from the target distribution
Sample a step from step distribution: N(0, σgen)
Candidate point u* = u + step
Given the candidate point u*, find ratio of the densities at u* and u
If α >1, the candidate point u* is accepted, and we take next step from there
If α ≤1 , α is compared to random number from uniform distribution U(0, 1). If α is greater, the step is accepted. Otherwise, don’t accept step, stay at u and attempt to step again 6 6 6 6 6 6 6

7. Algorithm Performance
Algorithm performance refers to how quickly set of convergence with target distribution is achieved
Convergence occurs quickly when optimal “mixing” of random walk occurs
Mixing characterized by autocorrelation of sets of random numbers, depends on step size, σgen 7 7 7 7 7 7 7

8. Mixing and Convergence
Target Distribution: N(0, 1.31)
μ =
σ = 1.377
μ = 0.388
σ = 1.07
μ =
σ = 1.58 8 8

9. Goal
Study how the accuracy of the integration depends on the parameters of MCMC
Parameters of MCMC determine correlation of data sets as well as data autocorrelation
Correlation arises when:
Same coordinates are used for target/projectile
Coordinates sampled from distribution function with dimensions that are not independent
Autocorrelation arises based on the step parameter of the MCMC random walk

10. Integrated Autocorrelation Time
Minimization of autocorrelation time τ1 gives optimal σgen
Lag-1 autocorrelation holds most information regarding optimization
Roberts, G. Rosenthal, J. Optimal Scaling for Various Metropolis-Hastings Algorithms Statistical Science, 2001 10 10 10 10 10 10 10

11. Autocorrelation Effect
Woods-Saxon
Normal 11

12. Calculated in optical approximation
Correlation Effect
Calculated in optical approximation 12 12 12 12

13. Conclusion
Error of cross section calculation in Glauber theory is dependent upon the autocorrelation of data sets and the correlation between data sets used in Monte Carlo integration
Care must be taken in environments where these traits are difficult to control (ie parallel number generation environments)
If these cross-section results are used to obtain other information (parameters of nuclear distributions) the errors could significantly change the final results 13

14. Acknowledgements
SESAPS attendance was supported in part by SESAPS Travel Grant