From PBDN

Introduction

Quasi-Monte Carlo (QMC) methods are considerably more difficult to implement, and explain, than the other techniques.

The general idea is that, rather than attempting to condition randomly generated numbers in some way, we begin with a deterministic sequence which we know provides very uniform coverage of the interval from 0 to 1. In randomized QMC (RQMC), the sequence is randomized in such a way that the numbers look random, but (as a group), don't lose the desirable property of covering the interval (0 to 1) in a regular way.

The particular technique applied here is array randomized quasi-Monte Carlo. To implement this method in software is very complex, compared to the other techniques, and it executes very slowly in comparison with them.

Results

Birth-Death Model

Pooled variance plot for birth-death model

For the birth-death model, the result is at least as good as for antithetic variates, perhaps a little better. Although this means that, for a given error, it is only necessary to compute about one third as many trajectories, computing them is very slow indeed. For this problem, antithetic variates, giving comparable variance reduction, performs much better.

Metabolite-Enzyme Model

Pooled variance plot for metabolite A in metabolite-enzyme model

Pooled variance plot for metabolite B in metabolite-enzyme model

For the metabolite-enzyme model, the result is similar to what was found for stratified sampling. We get a consistent, if small, decrease in pooled variance for both of the metabolite species, and see little or no effect on the two enzyme species.

Pooled variance plot for enzyme A in metabolite-enzyme model

Pooled variance plot for enzyme B in metabolite-enzyme model

Gains and Losses

The RQMC implementation takes 5 to 7 times as long to compute a trajectory as the simpler baseline PRNG under the same conditions. We also know that the baseline PRNG can be made to run over twice as fast with some simplifications which cannot be applied to the RQMC implementation. Although we know that our RQMC implementation is not as fast as it could be, we don't believe that it can be made to run 10 to 14 times faster.

Because RQMC takes so long to compute each trajectory, it doesn't compare favourably with the baseline PRNG, or even with stratified sampling, in terms of time saved.

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