From PBDN

Introduction

This page provides an overview of what the different graphs mean and how to interpret them. A description of the computational experiments is also provided

Graphs

For each of the variance reduction techniques, there are graphs of pooled variance vs. number of pools. Pooled variance is an empirical estimate of the error in the solution. Pooled variance can be computed with different numbers of pools. The number of pools is not terribly important, we compare results for a particular pool-size and it shouldn't matter very much which one we choose. For example, the graph below shows pooled variance vs. number of pools for the birth-death model with antithetic variates (red) and with no variance reduction (blue, see Baseline below). The data points on the red line are well below the corresponding points on the blue line and, relatively, by about the same amount.

Error Bars

Each data point is the average of 20 independent simulations. The error-bars indicate how the 20 values were spread around the mean value.

Experiments

For most of the computational experiments, the number of trajectories was fixed at 120,000 and the stop-time at 100 seconds. When computing the pooled variance with stratified sampling or QMC, it is important that a group of trajectories isn't split over two or more pools, so the numbers of pools and sizes of groups were chosen with that constraint in mind and then used for all of the techniques to facilitate direct comparisons.

The "usual" numbers of pools were 10, 12, 15, 20, 24, 30, 40, 60, and 120. With 120,000 trajectories, the pool-sizes were then 12,000, 10,000, 8,000, 6,000, 5,000, 4,000, 3,000, 2,000, and 1,000 respectively.

The usual numbers of strata were 1 (no stratification), 2, 5, 10, 20, 50, 100, 200, 500, and 1,000.

Baseline

For each chemical model, the results of the computational experiments with a variance reduction technique applied are compared with a baseline with no variance reduction. In the baseline implementation, a conventional, standard pseudo-random number generator (PRNG) is used for generating the random number for reaction selection in SSA.

We know that the baseline can be speeded up by a factor of two easily. In an early implementation of the software, written before stratified sampling or RQMC were implemented, the baseline ran more than twice as fast as it did in a the later implementations. The reason for this is that stratified sampling and RQMC both require trajectories to be treated in groups. Adding this "grouping support" slows down the entire program, even when the baseline PRNG, which doesn't need grouping support, is used. We have to take this additional factor into account when comparing running times.

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