.. _methods_variance_reduction:

==================
Variance Reduction
==================

.. _methods_variance_reduction_intro:

------------
Introduction
------------

Transport problems can sometimes involve a significant degree of attenuation
between the source and a detector (tally) region, which can result in a flux
differential of ten orders of magnitude (or more) throughout the simulation
domain. As Monte Carlo uncertainties tend to be inversely proportional to the
physical flux density, it can be extremely difficult to accurately resolve
tallies in locations that are optically far from the source. This issue is
particularly common in fixed source simulations, where some tally locations may
not experience a single scoring event, even after billions of analog histories.

Variance reduction techniques aim to either flatten the global uncertainty
distribution, such that all regions of phase space have a fairly similar
uncertainty, or to reduce the uncertainty in specific locations (such as a
detector). There are three strategies available in OpenMC for variance
reduction: weight windows generated via the MAGIC method or the FW-CADIS method,
and source biasing. Both weight windowing strategies work by developing a mesh
that can be utilized by subsequent Monte Carlo solves to split particles heading
towards areas of lower flux densities while terminating particles in higher flux
regions. In contrast, source biasing modifies source site sampling behavior to
preferentially track particles more likely to reach phase space regions of
interest.

------------
MAGIC Method
------------

The Method of Automatic Generation of Importances by Calculation, or `MAGIC
method <https://doi.org/10.1016/j.fusengdes.2011.01.059>`_, is an iterative
technique that uses spatial flux information :math:`\phi(r)` obtained from a
normal Monte Carlo solve to produce weight windows :math:`w(r)` that can be
utilized by a subsequent iteration of Monte Carlo. While the first generation of
weight windows produced may only help to reduce variance slightly, use of these
weights to generate another set of weight windows results in a progressively
improving iterative scheme.

Equation :eq:`magic` defines how the lower bound of weight windows
:math:`w_{\ell}(r)` are generated with MAGIC using forward flux information.
Here, we can see that the flux at location :math:`r` is normalized by the
maximum flux in any group at that location. We can also see that the weights are
divided by a factor of two, which accounts for the typical :math:`5\times`
factor separating the lower and upper weight window bounds in OpenMC.

.. math::
    :label: magic

    w_{\ell}(r) = \frac{\phi(r)}{2\,\text{max}(\phi(r))}

A major advantage of this technique is that it does not require any special
transport machinery; it simply uses multiple Monte Carlo simulations to
iteratively improve a set of weight windows (which are typically defined on a
mesh covering the simulation domain). The downside to this method is that as the
flux differential increases between areas near and far from the source, it
requires more outer Monte Carlo iterations, each of which can be expensive in
itself. Additionally, computation of weight windows based on regular (forward)
neutron flux tally information does not produce the most numerically effective
set of weight windows. Nonetheless, MAGIC remains a simple and effective
technique for generating weight windows.

--------
FW-CADIS
--------

As discussed in the previous section, computation of weight windows based on
regular (forward) neutron flux tally information does not produce the most
numerically efficient set of weight windows. It is highly preferable to generate
weight windows based on spatial adjoint flux :math:`\phi^{\dag}(r)`
information. The adjoint flux is essentially the "reverse" simulation problem,
where we sample a random point and assume this is where a particle was absorbed,
and then trace it backwards (upscattering in energy), until we sample the point
where it was born from.

The Forward-Weighted Consistent Adjoint Driven Importance Sampling method, or
`FW-CADIS method <https://doi.org/10.13182/NSE12-33>`_, produces weight windows
for global or local variance reduction given adjoint flux information throughout 
the entire domain. The weight window lower bound is defined in Equation
:eq:`fw_cadis`, and also involves a normalization step not shown here.

.. math::
    :label: fw_cadis

    w_{\ell}(r) = \frac{1}{2\phi^{\dag}(r)}

While the algorithm itself is quite simple, it requires estimates of the global
adjoint flux distribution, which is difficult to generate directly with Monte
Carlo transport. Thus, FW-CADIS typically uses an alternative solver (often
deterministic) that can be more readily adapted for generating adjoint flux
information, and which is often much cheaper than Monte Carlo given that a rough
solution is often sufficient for weight window generation.

The FW-CADIS implementation in OpenMC utilizes its own internal random ray
multigroup transport solver to generate the adjoint source distribution. No
coupling to any external transport is solver is necessary. The random ray solver
operates on the same geometry as the Monte Carlo solver, so no redefinition of
the simulation geometry is required. More details on how the adjoint flux is
computed are given in the :ref:`adjoint methods section <adjoint>`.

More information on the workflow is available in the :ref:`user guide
<variance_reduction>`, but generally production of weight windows with FW-CADIS
involves several stages (some of which are highly automated). These tasks
include generation of approximate multigroup cross section data for use by the
random ray solver, running of the random ray solver in normal (forward flux)
mode to generate a source for the adjoint solver, running of the random ray
solver in adjoint mode to generate adjoint flux tallies, and finally the
production of weight windows via the FW-CADIS method. As is discussed in the
user guide, most of these steps are automated together, making the additional
burden on the user fairly small.

The major advantage of this technique is that it typically produces much more
numerically efficient weight windows as compared to those generated with MAGIC,
sometimes with an order-of-magnitude improvement in the figure of merit
(Equation :eq:`variance_fom`), which accounts for both the variance and the
execution time. Another major advantage is that the cost of the random ray
solver is typically negligible compared to the cost of the subsequent Monte
Carlo solve itself, making it a very cheap method to deploy. The downside to
this method is that it introduces a second transport method into the mix (random
ray), such that there are more free input parameters for the user to know about
and adjust, potentially making the method more complex to use. However, as many
of the parameters have natural choices, much of this parameterization can be
handled automatically behind the scenes without the need for the user to be
aware of this.

.. math::
    :label: variance_fom

    \text{FOM} = \frac{1}{\text{Time} \times \sigma^2}

Finally, one unique capability of the FW-CADIS weight window generator is to 
produce weight windows for local variance reduction, given a list of the 
responses of interest. This is controlled by optionally specifying target 
tallies from the :class:`openmc.model.Model` to the 
:class:`openmc.WeightWindowGenerator`, as illustrated in the
:ref:`user guide<variance_reduction>`. If target tallies for local variance 
reduction are supplied, then the adjoint sources are only populated after the 
initial forward simulation in the source regions associated with those tallies. 
In other regions, the adjoint source term is instead set to zero. The Random  
Ray solver then determines the adjoint flux map used to generate FW-CADIS 
weight windows following the usual technique.

.. _methods_source_biasing:

--------------
Source Biasing
--------------

In contrast to the previous two methods that introduce population controls
during transport, source biasing modifies the sampling of the external source
distribution. The basic premise of the technique is that for each spatial,
angular, energy, or time distribution of a source, an additional distribution
can be specified provided that the two share a common support (set of points
where the distribution is nonzero). Samples are then drawn from this "bias"
distribution, which can be chosen to preferentially direct particles towards
phase space regions of interest. In order to avoid biasing the tally results,
however, a weight adjustment is applied to each sampled site as described below.

Assume that the unbiased probability density function of a random variable
:math:`X:x \rightarrow \mathbb{R}` is given by :math:`f(x)`, but that using the
biased distribution :math:`g(x)` will result in a greater number of particle
trajectories reaching some phase space region of interest. Then a sample
:math:`x_0` may be drawn from :math:`g(x)` while maintaining a fair game,
provided that its weight is adjusted as:

.. math::
    :label: source_bias

    w = w_0 \times \frac{f(x_0)}{g(x_0)}

where :math:`w_0` is the weight of an unbiased sample from :math:`f(x)`,
typically unity.

Returning now to Equation :eq:`source_bias`, the requirement for common support
becomes evident. If :math:`\mathrm{supp} (g)` fully contains but is not
identical to :math:`\mathrm{supp} (f)`, then some samples from :math:`g(x)` will
correspond to points where :math:`f(x) = 0`. Thus these source sites would be
assigned a starting weight of 0, meaning the particles would be killed
immediately upon transport, effectively wasting computation time. Conversely, if
:math:`\mathrm{supp} (g)` is fully contained by but not identical to
:math:`\mathrm{supp} (f)`, the contributions of some regions outside
:math:`\mathrm{supp} (g)` will not be counted towards the integral, potentially
biasing the tally. The weight assigned to such points would be undefined since
:math:`g(x) = \mathbf{0}` at these points.

When an independent source is sampled in OpenMC, the particle's coordinate in
each variable of phase space :math:`(\mathbf{r},\mathbf{\Omega},E,t)` is
successively drawn from an independent probability distribution. Multiple
variables can be biased, in which case the resultant weight :math:`w` applied to
the particle is the product of the weights assigned from all sampled
distributions: space, angle, energy, and time, as shown in Equation
:eq:`tot_wgt`.

.. math::
    :label: tot_wgt

    w = w_r \times w_{\Omega} \times w_E \times w_t

Finally, source biasing and weight windows serve different purposes. Source
biasing changes how particles are born, allowing the initial source sites to be
sampled preferentially from important regions of phase space (space, angle,
energy, and time) with an accompanying weight adjustment. Weight windows, by
contrast, apply population control during transport (splitting and Russian
roulette) to help particles reach and contribute in important regions as they
move through the system. Because particle transport proceeds as usual after a
biased source is sampled, particle attenuation in optically thick regions
outside the source volume will not be affected by source biasing; in such
scenarios, transport biasing techniques such as weight windows are often more
effective.
