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