4. Random Number Generation

In order to sample probability distributions, one must be able to produce random numbers. The standard technique to do this is to generate numbers on the interval \([0,1)\) from a deterministic sequence that has properties that make it appear to be random, e.g. being uniformly distributed and not exhibiting correlation between successive terms. Since the numbers produced this way are not truly “random” in a strict sense, they are typically referred to as pseudorandom numbers, and the techniques used to generate them are pseudorandom number generators (PRNGs). Numbers sampled on the unit interval can then be transformed for the purpose of sampling other continuous or discrete probability distributions.

4.1. Linear Congruential Generators

There are a great number of algorithms for generating random numbers. One of the simplest and commonly used algorithms is called a linear congruential generator. We start with a random number seed \(\xi_0\) and a sequence of random numbers can then be generated using the following recurrence relation:

(1)\[\xi_{i+1} = g \xi_i + c \mod M\]

where \(g\), \(c\), and \(M\) are constants. The choice of these constants will have a profound effect on the quality and performance of the generator, so they should not be chosen arbitrarily. As Donald Knuth stated in his seminal work The Art of Computer Programming, “random numbers should not be generated with a method chosen at random. Some theory should be used.” Typically, \(M\) is chosen to be a power of two as this enables \(x \mod M\) to be performed using the bitwise AND operator with a bit mask. The constants for the linear congruential generator used by default in OpenMC are \(g = 2806196910506780709\), \(c = 1\), and \(M = 2^{63}\) (see L’Ecuyer).

4.1.1. Skip-ahead Capability

One of the important capabilities for a random number generator is to be able to skip ahead in the sequence of random numbers. Without this capability, it would be very difficult to maintain reproducibility in a parallel calculation. If we want to skip ahead \(N\) random numbers and \(N\) is large, the cost of sampling \(N\) random numbers to get to that position may be prohibitively expensive. Fortunately, algorithms have been developed that allow us to skip ahead in \(O(\log_2 N)\) operations instead of \(O(N)\). One algorithm to do so is described in a paper by Brown. This algorithm relies on the following relationship:

(2)\[\xi_{i+k} = g^k \xi_i + c \frac{g^k - 1}{g - 1} \mod M\]

Note that equation (2) has the same general form as equation (1), so the idea is to determine the new multiplicative and additive constants in \(O(\log_2 N)\) operations.