quasi-monte carlo methods fall 2012 by yaohang li, ph.d
TRANSCRIPT
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
is the number of points in interval [0,u)
– For d dimensions
• E: a sub-rectangle
• m(E): the volume of E
N
nnu
unNN ux
NxxDD
1),0[
101
** |)(1
|sup),...,(
|)(#
|sup),...,( 1** Em
N
EofxxxDD i
EnNN
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
*
1
1
0
)(|)()(1
| N
N
nn DfVdxxfxf
N
1* )(log][][][ NNcfVDfVf kN
Quasi-Monte Carlo Integration
• 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
n
iixf
ndxxf
1
1
0
)(1
)(
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
– radical-inverse function
• for an integer b>1, the van der Corput sequence in base b is {x0, x1, …} with xn=b(n) for all n>=0
0
)(j
jj bnan
0
)1()()(j
jjb bnan
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