Quasi-Monte Carlo Methods Fall 2012 By Yaohang Li, Ph.D.

Review Last Class –Numerical Distribution Random Choices from a finite set General methods for continuous distributions –inverse function method –acceptance-rejection method Distributions –Normal distribution »Polar method –Exponential distribution Shuffling This Class –Quasi-Monte Carlo Next Class –Markov Chain Monte Carlo

Random Numbers –Pseudorandom Numbers Monte Carlo Methods –Quasirandom Numbers Uniformity Low-discrepancy Quasi-Monte Carlo Methods –Mixed-random Numbers Hybrid-Monte Carlo Methods

Discrepancy –For one dimension  is the number of points in interval [0,u) –For d dimensions E: a sub-rectangle m(E): the volume of E

A Picture is Worth a Thousand Words

Quasi-Monte Carlo Motivation –Convergence Monte Carlo methods: O(N -1/2 ) quasi-Monte Carlo methods: O(N -1 ) –Integration error bound Koksma-Hlwaka Inequality Theorem –V(f): bounded variation Criterion –k is a dimension dependent constant

Quasi-Monte Carlo Integration –If x1, …, xn are from a quasirandom number sequence –Compared with Crude Monte Carlo Only difference is the underlying random numbers –Crude Monte Carlo »pseudorandom numbers –Quasi-Monte Carlo »quasirandom numbers

Discrepancy of Pseudorandom Numbers and Quasirandom Numbers Discrepancy of Pseudorandom Numbers –O(N -1/2 ) Discrepancy of Quasirandom Numbers –O(N -1 )

Analysis of Quasi-Monte Carlo Convergence Rate –O(N -1 ) Actual Convergence Rate –O((logN) k N -1 ) k is a constant related to dimension –when dimension is large (>48) the (logN) k factor becomes large the advantage of quasi-Monte Carlo disappears

Quasi-random Numbers van der Corput sequence –digit expansion –radical-inverse function for an integer b>1, the van der Corput sequence in base b is {x0, x1, …} with x n =  b (n) for all n>=0

Halton Sequence –s dimensional van der Corput sequence x n =(  b1 (n),  b2 (n),…,  bs (n)) –b1, b2, … bs are relatively prime bases Scrambled Halton Sequence –Use permutations of digits in the digit expansion of each van der Corput sequence –Improve the randomness of the Halton sequence

Discussion In low diemensions (s<30 or 40), quasi-Monte Carlo methods in numerical integrations are better than usual Monte Carlo methods Quasi-Monte Carlo method is deterministic method –Monte Carlo methods are statistic methods There are serially efficient implementation of quasirandom number sequences –Halton –Sobol –Faure –Niederreiter quasi-Monte Carlo can now efficiently used in integration –Still in research in other areas

Summary Quasirandom Numbers –Discrepancy –Implementation van der Corput Halton Quasi-Monte Carlo –Integration –Convergence rate –Comparison with Crude Monte Carlo

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