Monte Carlo Sampling Method

Dear  illaoi,

illaoi said:

I have an option to choose three sampling methods

1- Random

2- Latin HyperCube

3- Low Discrepancy Sequence

They also show slightly different mean and sigma. My question is which one is recommended in a sense to correlate better with measured Silicon?

To be honest illaoi, the question you are asking about the sampling method and its relevance to measured data are really different questions. The sampling method chosen is not related to the accuracy of its result relative to measured silicon data. The only way that comes to mind where the two are related is if the number of Monte Carlo simulations you chose is not sufficient to accurately estimate the variance and mean of the parameter of interest. I suspect the dominant sources of differences between Monte Carlo simulation results and measured results are the accuracy of the Monte Carlo parameters in your PDK, the simulation methodology (ie., how you choose to simulate a specific parameter) and improper post-processing of Monte Carlo simulation data.

The three methods you note for sampling methods are described at the On-line support site at URL:

https://support.cadence.com/apex/techpubDocViewerPage?xmlName=vvoug.xml&title=Virtuoso%20Variation%20Option%20User%20Guide%20--%20Statistical%20Sampling%20Methods%20-%20Statistical%20Sampling%20Methods&hash=pgfId-1066262&c_version=ICADVM20.1&path=vvoUG/vvoUGICADVM20.1/introVVO_Statistical_Sampling_Methods.html#pgfId-1066262

In short, the three provide different levels of convergence to the 'actual" statistical parameter. A faster convergence rate suggests that fewer simulations are required to estimate a statistical parameter. As an example, from a recent Monte-Carlo simulation I performed to study the variation in the output duty cycle, I plot the simulated standard deviation as a function of the number of Monte-Carlo simulations. To estimate a stable estimate of the standard deviation requires about 100 simulations.

illaoi said:

if you can comment on seed number selection since that also has slight effect on the simulation results.

This is expected, but will be less if a greater number of simulations are performed. Allowing the use of a specific seed provides the ability to repeat a Monte-Carlo simulation set with for example, an improved circuit topology and provide a direct comparison with the former results.

I hope this provides some help illaoi.

Shawn

In general there is no need to set the seed for a Monte Carlo simulation unless you specifically want to produce a different sequence. The randomness is the same (so the mean and standard deviation will tend towards the same result), but the actual sequence of random numbers will be different. There's a default seed that ADE uses, which means you should get consistent results if you run the same simulation twice, but you can perturb that by altering the seed - but that doesn't really tell you anything useful. I generally advise people to leave it alone.

As Shawn points out, setting the sampling method to something other than random is a way of reducing the number of samples needed to converge on the mean and standard deviation. If you think if rolling a dice, with "random" you can get the same number more than once and hence you are not learning anything new about the result - it's wasted effort. Latin Hypercube is a method which divides up the surface into a grid with the desired number of points and then re-rolls the dice if it ends up with a sample that's already in a grid point already covered. The downside of that is that you have to pre-decide how many samples you want and stopping early makes no sense (nor does adding more points afterwards because the grid is already decided). The Low Discrepancy Sequence is a deterministic generator (so you'll get the same sequence each time) which gradually gets denser as you add more samples. The benefit of LDS is that you need fewer samples than random to achieve similar mean/standard deviation and similar or fewer points than LHS and it works with auto-stop methods. So nearly always LDS is the mode to choose.

Here's a short video of a few slides from a presentation I gave a few years ago which illustrates the difference between the three - note that I step between 64, 256 and 1024 samples to show - and "sobol" is a low-discrepancy sequence algorithm (not exactly what is used in Spectre, but good enough to illustrate the principle):

Andrew

Dear Andrew,

This is a very nice graphical presentation to illustrate the differences between the three basic algorithms! Your insight and comments, as always, are much appreciated!

Shawn

Shawn and Andrew, thanks.