17. Variance Reduction

Global and local variance reduction are possible in OpenMC through both weight windowing and source biasing techniques. OpenMC is capable of generating weight windows using either the MAGIC or FW-CADIS methods, the latter with an optional capability for local variance reduction. Both techniques will produce a weight_windows.h5 file that can be loaded and used later on. In this section, we first break down the steps required to generate and apply weight windows, then describe how source biasing may be applied.

17.1. Generating Global Weight Windows with MAGIC

As discussed in the methods section, MAGIC is an iterative method that uses flux tally information from a Monte Carlo simulation to produce weight windows for a user-defined mesh with the objective of global variance reduction. While generating the weight windows, OpenMC is capable of applying the weight windows generated from a previous batch while processing the next batch, allowing for progressive improvement in the weight window quality across iterations.

The typical way of generating weight windows is to define a mesh and then add an openmc.WeightWindowGenerator object to an openmc.Settings instance, as follows:

# Define weight window spatial mesh
ww_mesh = openmc.RegularMesh()
ww_mesh.dimension = (10, 10, 10)
ww_mesh.lower_left = (0.0, 0.0, 0.0)
ww_mesh.upper_right = (100.0, 100.0, 100.0)

# Create weight window object and adjust parameters
wwg = openmc.WeightWindowGenerator(
    method='magic',
    mesh=ww_mesh,
    max_realizations=settings.batches
)

# Add generator to Settings instance
settings.weight_window_generators = wwg

Notably, the max_realizations attribute is adjusted to the number of batches, such that all iterations are used to refine the weight window parameters.

With the WeightWindowGenerator instance added to the Settings, the rest of the problem can be defined as normal. When running, note that the second iteration and beyond may be several orders of magnitude slower than the first. As the weight windows are applied in each iteration, particles may be aggressively split, resulting in a large number of secondary (split) particles being generated per initial source particle. This is not necessarily a bad thing, as the split particles are much more efficient at exploring low flux regions of phase space as compared to initial particles. Thus, even though the reported “particles/second” metric of OpenMC may be much lower when generating (or just applying) weight windows as compared to analog MC, it typically leads to an overall improvement in the figure of merit accounting for the reduction in the variance.

Warning

The number of particles per batch may need to be adjusted downward significantly to result in reasonable runtimes when weight windows are being generated or used.

At the end of the simulation, a weight_windows.h5 file will be saved to disk for later use. Loading it in another subsequent simulation will be discussed in the “Using Weight Windows” section below.

17.2. Generating Global or Local Weight Windows with FW-CADIS and Random Ray

Weight window generation with FW-CADIS and random ray in OpenMC uses the same exact strategy as with MAGIC. Using FW-CADIS, however, also enables local variance reduction in fixed source problems through the targets attribute, which is described later in this section. To enable FW-CADIS, an openmc.WeightWindowGenerator object is added to the openmc.Settings object, and a weight_windows.h5 will be generated at the end of the simulation. The only procedural difference is that the code must be run in random ray mode. A full description of how to enable and setup random ray mode can be found in the Random Ray User Guide.

Note

It is a long term goal for OpenMC to be able to generate FW-CADIS weight windows with only a few tweaks to an existing continuous energy Monte Carlo input deck. However, at the present time, the workflow requires several steps to generate multigroup cross section data and to configure the random ray solver. A high level overview of the current workflow for generation of weight windows with FW-CADIS using random ray is given below.

Begin by making a deep copy of your continuous energy Python model and then convert the copy to be multigroup and use the random ray transport solver. The conversion process can largely be automated as described in more detail in the random ray quick start guide, summarized below:

# Define continuous energy model
ce_model = openmc.pwr_pin_cell() # example, replace with your model

# Make a copy to convert to multigroup and random ray
model = copy.deepcopy(ce_model)

# Convert model to multigroup (will auto-generate MGXS library if needed)
model.convert_to_multigroup()

# Convert model to random ray and initialize random ray parameters
# to reasonable defaults based on the specifics of the geometry
model.convert_to_random_ray()

# (Optional) Overlay source region decomposition mesh to improve fidelity of the
# random ray solver. Adjust 'n' for fidelity vs runtime.
n = 10
mesh = openmc.RegularMesh()
mesh.dimension = (n, n, n)
mesh.lower_left = model.geometry.bounding_box.lower_left
mesh.upper_right = model.geometry.bounding_box.upper_right
model.settings.random_ray['source_region_meshes'] = [(mesh, [model.geometry.root_universe])]

# (Optional) Improve fidelity of the random ray solver by enabling linear sources
model.settings.random_ray['source_shape'] = 'linear'

# (Optional) Increase the number of rays/batch, to reduce uncertainty
model.settings.particles = 500

If you need to improve the fidelity of the MGXS library, there is more information on generating multigroup cross sections via OpenMC in the random ray MGXS guide.

Add in a WeightWindowGenerator in a similar manner as for MAGIC generation with Monte Carlo and set the method attribute set to "fw_cadis":

# Create weight window object and adjust parameters, using the same mesh
# we used for source region decomposition
wwg = openmc.WeightWindowGenerator(
    method='fw_cadis',
    mesh=mesh
)

# Add generator to openmc.settings object
settings.weight_window_generators = wwg

Warning

If using FW-CADIS weight window generation, ensure that the selected weight window mesh does not subdivide any source regions in the problem. This can be ensured by using the same mesh for both source region subdivision (i.e., assigning to model.settings.random_ray['source_region_meshes']) and for weight window generation.

(Optional) If local variance reduction is desired in a fixed-source problem, populate the targets attribute with an openmc.Tallies instance or an iterable of tally IDs indicating the tallies of interest for variance reduction:

# Build a new example and WWG for local variance reduction
from openmc.examples import random_ray_three_region_cube_with_detectors
new_model = random_ray_three_region_cube_with_detectors()

ww_mesh = openmc.RegularMesh()
n = 7
width = 35.0
ww_mesh.dimension = (n, n, n)
ww_mesh.lower_left = (0.0, 0.0, 0.0)
ww_mesh.upper_right = (width, width, width)

wwg = openmc.WeightWindowGenerator(
    method="fw_cadis",
    mesh=ww_mesh,
    max_realizations=new_model.settings.batches
)
new_model.settings.weight_window_generators = wwg
new_model.settings.random_ray['volume_estimator'] = 'naive'

# Get the tallies of interest
target_tallies = openmc.Tallies()

for tally in list(new_model.tallies):
    if tally.name in {"Detector 1 Tally", "Detector 2 Tally"}:
        target_tallies.append(tally)

# Add to WeightWindowGenerator
wwg.targets = target_tallies

Warning

The tallies designated as FW-CADIS targets to the WeightWindowGenerator must be present under the tallies attribute of the Model as well in order to be recognized as valid local variance reduction targets. This check is performed when the openmc.model.Model.export_to_model_xml() or openmc.model.Model.export_to_xml() functions are called, meaning that the standalone openmc.Settings.export_to_xml() and openmc.Tallies.export_to_xml() methods should not be used with FW-CADIS local variance reduction.

When running your multigroup random ray input deck, OpenMC will automatically run a forward solve followed by an adjoint solve, with a weight_windows.h5 file generated at the end. The weight_windows.h5 file will contain FW-CADIS generated weight windows. This file can be used in identical manner as one generated with MAGIC, as described below.

17.3. Using Weight Windows

To use a weight_windows.h5 weight window file with OpenMC’s Monte Carlo solver, the Python input just needs to load the h5 file:

settings.weight_window_checkpoints = {'collision': True, 'surface': True}
settings.survival_biasing = False
settings.weight_windows_file = "weight_windows.h5"
settings.weight_windows_on = True

The WeightWindowGenerator instance is not needed to load an existing weight_windows.h5 file. Inclusion of a WeightWindowGenerator instance will cause OpenMC to generate new weight windows and thus overwrite the existing weight_windows.h5 file. Weight window mesh information is embedded into the weight window file, so the mesh does not need to be redefined. Monte Carlo solves that load a weight window file as above will utilize weight windows to reduce the variance of the simulation.

17.4. Source Biasing

In fixed source problems, source biasing provides a means to reduce the variance on global or localized responses, depending on the biasing scheme. In either case, the premise of the method is to sample source sites from a biased distribution that directs a larger fraction of the simulated histories towards phase space regions of interest than would be found there under analog sampling. In order to preserve an unbiased estimate of the tally mean, the weight of these with analog sampling, divided by the probability assigned by the biased distribution. While the assignment of statistical weights is outlined in the methods section, this section demonstrates the implementation of source biasing to problems in OpenMC.

Source biasing in OpenMC is accomplished by applying a distribution to the bias attribute of one or more of the univariate or independent multivariate distributions which make up an IndependentSource instance as follows:

# First create the biased distribution
biased_dist = openmc.stats.PowerLaw(a=0, b=3, n=3)

# Construct a new distribution with the bias applied
dist = openmc.stats.PowerLaw(a=0, b=3, n=2, bias=biased_dist)

# The bias attribute can also be set on an existing "analog" distribution:
sphere_dist = openmc.stats.spherical_uniform(r_outer=3)
sphere_dist.r.bias = biased_dist

Univariate distributions may be sampled via the Python API, returning the sample(s) along with the associated weight(s):

sample_vec, wgt_vec = dist.sample(n_samples=100)

Here, if the distribution is unbiased, the weight of each sample will be unity. Finally, IndependentSource instances can be constructed with biased distributions:

# Create a source with a biased spatial distribution
source = openmc.IndependentSource(space=sphere_dist)

During the simulation, source sites are then sampled using the biased distributions where available and given starting statistical weights corresponding to the cumulative product of the weights assigned by each distribution in the source object. Hence multiple source variables (e.g., direction and energy) may be biased and the resulting source sites will have their weights adjusted accordingly.

Note

Combining source biasing with weight windows can be a powerful variance reduction technique if each is constructed appropriately for the response of interest. For example, if a source biasing scheme is devised for variance reduction of a specific localized response, the user may be able to specify their own weight window structure that results in more efficient transport than if weight windows were generated by either of OpenMC’s automatic weight window generators, which are intended for global variance reduction.

Biased distributions that could result in degenerate weight mappings are not recommended; this is most commonly seen when biasing the \(\phi\)-coordinate of spherical or cylindrical independent multivariate distributions. In such cases degenerate behavior will be observed at the pole about which \(\phi\) is measured, with all values of \(\phi\) (hence many possible statistical weights) mapping to the same point for \(r=0\) or \(\mu=0\), and large weight gradients in the vicinity. In most cases requiring a spherical independent source, it would be preferable to reorient the reference vector of the distribution such that biasing could be applied to the \(\mu\)-coordinate instead.

When biasing a distribution, care should also be taken to ensure that both the unbiased and biased distribution share a common support—that is, every region of phase space mapped to a nonzero probability density by the unbiased distribution should likewise map to nonzero probability under the biased distribution, and vice versa. In OpenMC, this places restrictions on the set of compatible distributions that may be used to bias sampling of each distribution type. The following table summarizes the method for each distribution in OpenMC that permits biased sampling.

**Distributions that support biased sampling**
Discrete Univariate PDFs	Biasing Method
`openmc.stats.Discrete`	Apply a vector of alternative probabilities to the `bias` attribute

Continuous Univariate PDFs	Biasing Method
`openmc.stats.Uniform`, `openmc.stats.PowerLaw`, `openmc.stats.Maxwell`, `openmc.stats.Watt`, `openmc.stats.Normal`, `openmc.stats.Tabular`	Apply a second, unbiased continous univariate PDF to the `bias` attribute, ensuring that the `support` attribute of each distribution is the same

Mixed Univariate PDFs	Biasing Method
`openmc.stats.Mixture`	May be constructed from multiple biased univariate distributions, or a second, unbiased continous univariate PDF may be applied to the `bias` attribute

Discrete Multivariate PDFs	Biasing Method
`openmc.stats.PointCloud`, `openmc.stats.MeshSpatial`	Apply a vector of the new relative probabilities of each point or mesh element under biased sampling to the `bias` attribute

Continuous Multivariate PDFs	Biasing Method
`openmc.stats.CartesianIndependent`, `openmc.stats.CylindricalIndependent`, `openmc.stats.SphericalIndependent`, `openmc.stats.PolarAzimuthal`	Construct from biased univariate distributions for `x`, `y`, `z`, etc.
`openmc.stats.Isotropic`	Apply an unbiased `openmc.stats.PolarAzimuthal` to the `bias` attribute