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
• Random Numbers
– Pseudorandom Numbers
• Monte Carlo Methods
– Quasirandom Numbers
• Uniformity
• Low-discrepancy
• Quasi-Monte Carlo Methods
– Mixed-random Numbers
• Hybrid-Monte Carlo Methods
Discrepancy
•Discrepancy
– For one dimension
DN*  DN* ( x1 ,..., xn )  sup |
0u 1
1
N
N

n 1
[ 0 ,u )
( xn )  u |
•  is the number of points in interval [0,u)
– For d dimensions
DN*  DN* ( x1 ,..., xn )  sup |
E
# ofxi  E
 m( E ) |
N
• 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
1
1 N
|  f ( xn )   f ( x)dx |  V ( f ) DN*
N n1
0
– V(f): bounded variation
• Criterion
 [ f ]  V [ f ]DN*  V [ f ]c(log N ) k N 1
– k is a dimension dependent constant
Quasi-Monte Carlo Integration
• Quasi-Monte Carlo Integration
– If x1, …, xn are from a quasirandom number sequence
1

0
1 n
f ( x)dx   f ( xi )
n i 1
– 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)kN-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

n   a j (n)b j
j 0
– radical-inverse function

b (n)   a j (n)b ( j 1)
j 0
• for an integer b>1, the van der Corput sequence in base b is {x0,
x1, …} with xn=b(n) for all n>=0
Halton Sequence
• Halton Sequence
– s dimensional van der Corput sequence
• xn=(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
What I want you to do?
• Review Slides
• Review basic probability/statistics concepts
• Select your presentation topic