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docs/source/methods/random_numbers.rst

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+.. _methods_random_numbers:
+
+========================
+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 :math:`[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.
+
+------------------------------
+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* :math:`\xi_0` and a sequence
+of random numbers can then be generated using the following recurrence relation:
+
+.. math::
+    :label: lcg
+
+    \xi_{i+1} = g \xi_i + c \mod M
+
+where :math:`g`, :math:`c`, and :math:`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, :math:`M` is chosen to be a power of two as this enables :math:`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
+:math:`g = 2806196910506780709`, :math:`c = 1`, and :math:`M = 2^{63}` (see
+`L'Ecuyer`_).
+
+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 :math:`N` random numbers and :math:`N` is large, the cost of
+sampling :math:`N` random numbers to get to that position may be prohibitively
+expensive. Fortunately, algorithms have been developed that allow us to skip
+ahead in :math:`O(\log_2 N)` operations instead of :math:`O(N)`. One algorithm
+to do so is described in a paper by Brown_. This algorithm relies on the following
+relationship:
+
+.. math::
+    :label: lcg-skipahead
+
+    \xi_{i+k} = g^k \xi_i + c \frac{g^k - 1}{g - 1} \mod M
+
+Note that equation :eq:`lcg-skipahead` has the same general form as equation :eq:`lcg`, so
+the idea is to determine the new multiplicative and additive constants in
+:math:`O(\log_2 N)` operations.
+
+.. only:: html
+
+   .. rubric:: References
+
+
+.. _L'Ecuyer: https://doi.org/10.1090/S0025-5718-99-00996-5
+.. _Brown: https://laws.lanl.gov/vhosts/mcnp.lanl.gov/pdf_files/anl-rn-arb-stride.pdf
+.. _linear congruential generator: https://en.wikipedia.org/wiki/Linear_congruential_generator