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Exploring Quasi-Monte Carlo Methods: A Comprehensive Review on Numerical Integration Techniques

This review delves into Quasi-Monte Carlo methods, highlighting their advantages over traditional Monte Carlo techniques for numerical integration. It covers key topics such as random choices from finite sets, continuous distributions, and methods for generating quasi-random numbers like the van der Corput and Halton sequences. The document also discusses the error bounds associated with these methods, the Koksma-Hlawake Inequality, and the implications of discrepancy in various dimensions. A thorough understanding of these methods is essential for efficient computational applications across diverse fields.

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Exploring Quasi-Monte Carlo Methods: A Comprehensive Review on Numerical Integration Techniques

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  1. Quasi-Monte Carlo MethodsFall 2012 By Yaohang Li, Ph.D.

  2. 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

  3. 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

  4. 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

  5. A Picture is Worth a Thousand Words

  6. 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

  7. 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

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

  9. 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

  10. 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

  11. 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

  12. 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

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

  14. What I want you to do? • Review Slides • Review basic probability/statistics concepts • Select your presentation topic

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