FW-CADIS Weight Window Generation with Random Ray (#3273)

Co-authored-by: Olek <45364492+yardasol@users.noreply.github.com>
Co-authored-by: Paul Romano <paul.k.romano@gmail.com>

docs/source/methods/index.rst

@@ -20,4 +20,5 @@ Theory and Methodology
     energy_deposition
     parallelization
     cmfd
+    variance_reduction
     random_ray

docs/source/methods/random_ray.rst

@@ -1060,6 +1060,8 @@ random ray and Monte Carlo, however.
   develop the scattering source by way of inactive batches before beginning
   active batches.
 
+.. _adjoint:
+
 ------------------------
 Adjoint Flux Solver Mode
 ------------------------

docs/source/methods/variance_reduction.rst

@@ -0,0 +1,134 @@
+.. _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 two strategies available in OpenMC for variance reduction:
+the Monte Carlo MAGIC method and the FW-CADIS method. Both strategies work by
+developing a weight window 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---all while maintaining a fair
+game.
+
+------------
+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 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}