Monte Carlo and quasi-Monte Carlo Integration

John D. Cook

M. D. Anderson Cancer Center

July 24, 2002

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

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 havea fourth derivative.

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

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.

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.

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

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.

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.

Proportionality factors

Error bound in classical methods depends on maximum of derivatives.

MC error proportional to variance of function, E[f2] – E[f]2

Contrasting proportionality

Classical methods improve with smooth integrands

Monte Carlo doesn’t depend on differentiability at all, but improves with overall “flatness”.

Good MC, bad trapezoid

1.5 2 2.5 3

0.2

0.4

0.6

0.8

1

Good trapeziod, bad MC

-3 -2 -1 1 2 3

2

4

6

8

Simple Monte Carlo

If xi is a sequence of independent samples from a uniform random variable

Importance Sampling

Suppose X is a random variable with PDF and xi is a sequence of independent samples from X.

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.

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.

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.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Sobol’ Sequence Excel’s PRNG

120 Point Comparison

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.

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.

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

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.

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

Lattice Example

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

Software written

QMC integration implemented for generic sequence generator

Generators implemented: Sobol’, Halton, Hammersley

Randomized QMC

Lattice rules

Randomized lattice rules

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.

Future development

Variance reduction. Good transformations make any technique work better.

Need for lots of experiments.

Contact

http://www.JohnDCook.com