Simple arithmetic progression seems to be the answer.
But how do we add points one at a time while preserving uniformity?
Covering the Interval [0, 1)
Hermann Weyl 100 years ago: the sequence \(\alpha, 2\alpha, 3\alpha,\dots,n\alpha \pmod{1}\) is equidistributed if \(\alpha\) is any irrational number.
Proved by an argument about Riemann integrals that leads directly to the idea of Monte Carlo integration.
Irrationality is necessary; without it, only a finite set of fractions.
Irrationality is inconvenient for finite computing machines.
Covering the Interval [0, 1)
Johannes van der Corput, 1930s: Asked if any sequence could have discrepancy bounded by a constant for all N.
In other words, if the discrepancy is given by some function \(f(N)/N\), can \(f(N)\) remain below some constant \(C\) as \(N\) goes to infinity?
The answer is no. Proved in the 1940s by Tatyana Pavlovna van Aardenne-Ehrenfest.
Stronger bounds by K. F. Roth in the 1950s, and others.
Extensions to point sets in two or more dimensions.
A First Attempt at Quasi-MC
R. D. Richtmyer, Los Alamos, 1951.
Problem of neutron diffusion (same as first MC studies).
A Weyl sequence: \(n \alpha \pmod 1 \) for irrational \(\alpha\).
Outcome: No better than ordinary Monte Carlo.
But he published the negative result! (This is to be encouraged: You can claim priority even for your failures.)
Part 5
Making Quasirandom Numbers
Van der Corput Procedure
Quasirandomness Generators
Van der Corput algorithm is a bizarre mashup of operations on numbers and operation on numerals—the representation of numbers.
Other quasirandom generators are just as strange.
Maybe this sort of bit-twiddling is where the “randomness” comes from in quasirandom sequences.
I. M. Sobol Quasipattern
Wikipedia
Part 6
Convergence Rates
Convergence Rates
How many points do you have to sample to achieve a given accuracy?
For pseudorandom Monte Carlo, the expected sampling error is proportional to \(\sqrt{N}/N\).
This bound follows from the law of large numbers or the central limit theorem.
Convergence Rates
In quasi-Monte Carlo, the corresponding expected error goes as \((\log N)^{d}/N\), where \(d\) is the spatial dimension.
Trading \(\log N\) for \(\sqrt{N}\) is potentially a huge improvement: the square root of 1,000,000 is 1,000, but \(\log_{2}\) 1,000,000 is only 20.
But there’s that pesky exponent \(d\). For any fixed \(d\), \(\sqrt{N}/N\) will eventually exceed \((\log N)^{d}/N\), but it may take a very large value of \(N\) to reach the crossover.
The Curse of Dimensionality
In high dimensions, all the volume is in the corners.
Measuring that volume, effort grows as \(2^d\).
Another Problem: Error Estimates
Randomness in conventional MC yields an intrinsic error estimate: Just measure the variance of multiple trials.
Deterministic methods (quasi or lattice) have no concept of variance.
The Twilight of Quasi-MC
The situation from the 1950s into the 1990s:
Quasi-Monte Carlo has a theoretical advantage in low dimensions.
But most low-dimensional problems are easy for any numeric integration scheme.
For pseudorandom MC, the \(\sqrt{N}\) convergence rate is less than wonderful, but it does have the virtue of being independent of dimension.
As a result, quasi-MC was largely set aside by practitioners, although mathematical work continued.
Part 7
A New Dawn for Quasi-MC
Financial Engineering
Collateralized mortgage obligation (CMO): One of the clever inventions that brought us the great recession of 2008.
Gathers thousands of mortgage loans into a single, marketable security.
Whether you’re buying or selling, you need a way to assign a price to a specific CMO.
Pricing a CMO
Isn’t this a straightforward problem of calculating net present value?
A 30-year mortgage represents a stream of 360 monthly payments; you apply a discount rate to determine how much you ought to pay now for the promise of a dollar 30 years hence.
But most mortgages do not survive for a full 360 monthly payments. When a house is sold or refinanced, the balance is paid in advance. There’s also the possibility of default.
Monte Carlo on Wall Street
Formulate the problem as an integral in a space of 360 dimensions.
For evaluating such integrals Monte Carlo methods are favored not because they work well but because nothing else works at all.
The use of such techniques, borrowed from physics or computer science, became part of the great “quant” revolution in finance in subsequent years.
Quasi-Monte Carlo on Wall Street
In 1992 Goldman Sachs, the investment bank, approached Joseph Traub’s group at Columbia seeking advice on methods for evaluating CMOs.
Traub and several graduate students, including Spassimir Paskov, decided to try quasi-Monte Carlo, ignoring the received wisdom.
It worked. Convergence was faster by a factor between 2 and 5.
[Paskov 1994]
The Aftermath
Paskov earned his degree and went to work for a Swiss bank.
Traub, Paskov, and another colleague received a patent related to the work.
Use of quasi-MC spread rapidly through the world of investment banking.
Twelve years later, the economy tanked. (Causation?)
Meanwhile, the small community of quasi-fanatics went to work figuring out what it all means.
Part 8
Quasi Miracles?
Laws of Nature Still Enforced
The earlier results on convergence rates and accuracy were not wrong.
The curse of dimensionality has not been lifted.
Fully evaluating a 360-dimensional integral requires \(\approx 2^{360}\) samples, not \(10^{6}\).
\(\therefore\) either the quasi-MC technique is more powerful than we knew, or the problem is easier than it looks.
Effective Dimensions?
The Paskov-Traub result provoked a spate of quasi reassessments (e.g., [Caflisch 1998], [Owen 2002], [Sloan 2005].
The verdict: The 360-dimensional integral doesn’t really have 360 “effective” dimensions.
ANOVA approach: If almost all the variance lies in a few dimensions, the rest can be truncated.
Analogous to compressed-sensing.
Open Questions
How do we recognize that a problem has effective dimensionality lower than its formal dimensionality?
How common are such problems?
Will quasirandom methods work for other kinds of Monte Carlo studies (e.g., random walks, percolation, statistical mechanics, optimization)?