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Introduction to numerical integration using Monte Carlo and quasi-Monte Carlo techniques

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- 1. Monte Carlo and quasi-Monte Carlo Integration John D. Cook M. D. Anderson Cancer Center July 24, 2002
- 2. Trapezoid rule in one dimension Error bound proportional to product of Step size squared Second derivative of integrand N = number of function evaluations Step size h = N-1 Error proportional to N-2
- 3. Simpson’s rule in one dimensions Error bound proportional to product of Step size to the fourth power Fourth derivative of integrand Step size h = N-1 Error proportional to N-4 All bets are off if integrand doesn’t have a fourth derivative.
- 4. Product rules In two dimensions, trapezoid error proportional to N-1 In d dimensions, trapezoid error proportional to N-2/d. If 1-dimensional rule has error N-p, n-dimensional product has error N-p/d
- 5. Dimension in a nutshell Assume the number of integration points N is fixed, as well as the order of the integration rule p. Moving from 1 dimension to d dimensions divides the number of correct figures by d.
- 6. Monte Carlo to the rescue Error proportional to N-1/2, independent of dimension! Convergence is slow, but doesn’t get worse as dimension increases. Quadruple points to double accuracy.
- 7. How many figures can you get with a million integration points? Dimension Trapezoid Monte Carlo 1 12 3 2 6 3 3 4 3 4 3 3 6 2 3 12 1 3
- 8. Fine print Error estimate means something different for product rules than for MC. Proportionality factors other than number of points very important. Different factors improve performance of the two methods.
- 9. Interpreting error bounds Trapezoid rule has deterministic error bounds: if you know an upper bound on the second derivative, you can bracket the error. Monte Carlo error is probabilistic. Roughly a 2/3 chance of integral being within one standard deviation.
- 10. Proportionality factors Error bound in classical methods depends on maximum of derivatives. MC error proportional to variance of function, E[f2] – E[f]2
- 11. Contrasting proportionality Classical methods improve with smooth integrands Monte Carlo doesn’t depend on differentiability at all, but improves with overall “flatness”.
- 12. Good MC, bad trapezoid 1 0.8 0.6 0.4 0.2 1.5 2 2.5 3
- 13. Good trapeziod, bad MC 8 6 4 2 -3 -2 -1 1 2 3
- 14. Simple Monte Carlo If xi is a sequence of independent samples from a uniform random variable
- 15. Importance Sampling Suppose X is a random variable with PDF and xi is a sequence of independent samples from X.
- 16. Variance reduction (example) If an integrand f is well approximated by a PDF that is easy to sample from, use the equation and apply importance sampling. Variance of the integrand will be small, and so convergence will be fast.
- 17. MC Good news / Bad news MC doesn’t get any worse when the integrand is not smooth. MC doesn’t get any better when the integrand is smooth. MC converges like N-1/2 in the worst case. MC converges like N-1/2 in the best case.
- 18. Quasi-random vs. Pseudo-random Both are deterministic. Pseudo-random numbers mimic the statistical properties of truly random numbers. Quasi-random numbers mimic the space-filling properties of random numbers, and improves on them.
- 19. 120 Point Comparison 1 .0 1 .0 0 .8 0 .8 0 .6 0 .6 0 .4 0 .4 0 .2 0 .2 0 .0 0 .0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 Sobol’ Sequence Excel’s PRNG
- 20. Quasi-random pros and cons The asymptotic convergence rate is more like N-1 than N-1/2. Actually, it’s more like log(N)dN-1. These bounds are very pessimistic in practice. QMC always beats MC eventually. Whether “eventually” is good enough depends on the problem and the particular QMC sequence.
- 21. MC-QMC compromise Randomized QMC Evaluate integral using a number of randomly shifted QMC series. Return average of estimates as integral. Return standard deviation of estimates as error estimate. Maybe better than MC or QMC! Can view as a variance reduction technique.
- 22. Some quasi-random sequences Halton – bit reversal in relatively prime bases Hammersly – finite sequence with one uniform component Sobol’ – common in practice, based on primitive polynomials over binary field
- 23. Sequence recommendations Experiment! Hammersley probably best for low dimensions if you know up front how many you’ll need. Must go through entire cycle or coverage will be uneven in one coordinate. Halton probably best for low dimensions. Sobol’ probably best for high dimensions.
- 24. Lattice Rules Nothing remotely random about them “Low discrepancy” Periodic functions on a unit cube There are standard transformations to reduce other integrals to this form
- 25. Lattice Example
- 26. Advantages and disadvantages Lattices work very well for smooth integrands Don’t work so well for discontinuous integrands Have good projections on to coordinate axes Finite sequences Good error posterior estimates Some a priori estimates, sometimes pessimistic
- 27. Software written QMC integration implemented for generic sequence generator Generators implemented: Sobol’, Halton, Hammersley Randomized QMC Lattice rules Randomized lattice rules
- 28. Randomization approaches Randomized lattice uses specified lattice size, randomize until error goal met RQMC uses specified number of randomizations, generate QMC until error goal met Lattice rules require this approach: they’re finite, and new ones found manually. QMC sequences can be expensive to compute (Halton, not Sobol) so compute once and reuse.
- 29. Future development Variance reduction. Good transformations make any technique work better. Need for lots of experiments.
- 30. Contact http://www.JohnDCook.com

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