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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
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Quasi-Monte Carlo MethodsFall 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
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
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
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
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
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