QuasiMonte Carlo Methods Spring 2005

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Quasi-Monte Carlo MethodsSpring 2005

• By Yaohang Li, Ph.D.
• Department of Computer Science
• North Carolina AT State University
• yaohang_at_ncat.edu

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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
• Paper Presentation List
• Next Class
• Test of Random Numbers (William Mirugi)

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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

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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

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A Picture is Worth a Thousand Words
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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
• d is a dimension dependent constant

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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

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Discrepancy of Pseudorandom Numbers and
Quasirandom Numbers

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

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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 (gt48)
• the (logN)k factor becomes large
• the advantage of quasi-Monte Carlo disapears

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Quasi-random Numbers

• van der Corput sequence
• digit expansion
• radical-inverse function
• for an integer bgt1, the van der Corput sequence
  in base b is x0, x1, with xn?b(n) for all
  ngt0

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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

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Discussion

• In low diemensions (slt30 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

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Summary

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

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What I want you to do?

• Review Slides
• Read the UNIX handbook if you are not familiar
  with UNIX
• Review basic probability/statistics concepts
• Work on your Assignment 2 and 3
• Select your presentation topic