Optimization of Monte Carlo Integration

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

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

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

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)

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

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

8. Mixing and Convergence Target Distribution: N(0, 1.31) μ = -0.291 σ = 1.58 μ = 0.388 σ = 1.07 μ = -0.116 σ = 1.377

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

11. Autocorrelation Effect Woods-Saxon Normal

12. Correlation Effect Calculated in optical approximation

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

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