1. Author: B. C. Bromley Presented by: Shuaiyuan Zhou 10-30-2009 Quasi-random Number Generators for Parallel Monte Carlo Algorithms

2.  Why & How  Sobol’ sequence & subsequences  Parallel Algorithm  Performance issue

3. The will and the way Enable parallel systems to take full advantage of the benefits of quasi-random Monte Carlo algorithms Rapid sequence production. as quick as any pseudorandom generator Fast convergence rate. much faster than pseudorandom methods An algorithm of generating quasi-random numbers in parallel systems, using a leapfrog scheme

4. Main contribution “A recursion relation which allows an element in a Sobol’ sequence to be quickly calculated from a previous, but not necessarily adjacent, element without determining all of the intervening members of the sequence”. Enables parallel Monte Carlo algorithms to have each node of a parallel processor step through interleaved subsequences with the same computational load as if it were calculating the original sequence, without any internode communication.

5. Sobol’ sequence

6. Recursive computation

7. Implementing in parallel In parallel applications, Breaking up the spatial domain of integration in the M- dimensional cube among processing nodes Distributing the sequence of sample points among processing nodes Leapfrog technique: each node skips along the sequence, jumping over those sample points which are handled by other nodes.

8. Modification to recursion

9. Algorithm

10. Algorithm – Cont.

11. Performance issue The efficiency of the leapfrog algorithm is highest when the number of nodes is an integral power of 2. Tests with several parallel supercomputers demonstrate that as many as 10 6 integration points (up to 6 dimensions) can be generated per second per node in the optimal case. When the number of nodes is P, the computational load of generating interleaved sequences is, at worst, proportional to the number of nonzero bits in P.

12. Thank you!