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.. _MAVRIC:

MAVRIC: Monaco with Automated Variance Reduction using Importance Calculations
==============================================================================

*D. E. Peplow and C. Celik*

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

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Monte Carlo particle transport calculations for deep penetration problems can require very long run
times in order to achieve an acceptable level of statistical uncertainty in the final answers.
Discrete-ordinates codes can be faster but have limitations relative to the discretization of space, energy,
and direction. Monte Carlo calculations can be modified (biased) to produce results with the same variance in
less time if an approximate answer or some other additional information is already known about the problem.
If importances can be assigned to different particles based on how much they will contribute to the final answer,
then more time can be spent on important particles, with less time devoted to unimportant particles. One of the best
ways to bias a Monte Carlo code for a particular tally is to form an importance map from the adjoint flux based on
that tally. Unfortunately, determining the exact adjoint flux could be just as difficult as computing the original
problem itself.  However, an approximate adjoint can still be very useful in biasing the Monte Carlo
solution :cite:`wagner_acceleration_1997`. Discrete ordinates can be used to quickly compute that approximate adjoint. Together, Monte Carlo and discrete ordinates can be used to find solutions to thick shielding problems in reasonable times.

The MAVRIC (Monaco with Automated Variance Reduction using Importance Calculations) sequence is based on the
CADIS (Consistent Adjoint Driven Importance Sampling) and FW-CADIS (Forward-Weighted CADIS)
methodologies :cite:`wagner_automated_1998` :cite:`wagner_automated_2002` :cite:`haghighat_monte_2003`
:cite:`wagner_forward-weighted_2007` MAVRIC automatically performs a three-dimensional, discrete-ordinates
calculation using Denovo to compute the adjoint flux as a function of position and energy. This adjoint flux
information is then used to construct an importance map (i.e., target weights for weight windows) and a biased
source distribution that work togetherparticles are born with a weight matching the target weight of the cell
into which they are born. The fixed-source Monte Carlo radiation transport Monaco :cite:`peplow_monte_2011`
then uses the importance map for biasing during particle transport, and it uses the biased source distribution
as its source. During transport, the particle weight is compared with the importance map after each particle
interaction and whenever a particle crosses into a new importance cell in the map.


For problems that do not require variance reduction to complete in a reasonable time,
execution of MAVRIC without the importance map calculation provides an easy way to run Monaco.
For problems that do require variance reduction to complete in a reasonable time, MAVRIC removes the burden of setting weight windows from the user and performs it automatically with a minimal amount of additional input. Note that the MAVRIC sequence can be used with the final Monaco calculation as either a multigroup (MG) or a continuous-energy (CE) calculation.

Monaco has a wide variety of tally options: it can calculate fluxes (by group) at a point in space,
over any geometrical region, or for a user-defined, three-dimensional, rectangular grid.
These tallies can also integrate the fluxes with either standard response functions from the cross
section library or user-defined response functions. All of these tallies are available in the MAVRIC sequence.

Although it was originally designed for CADIS, the MAVRIC sequence is also capable of
creating importance maps using both forward and adjoint deterministic estimates.
The FW-CADIS method :cite:`wagner_fw-cadis_2014` can be used for optimizing several tallies at once,
a mesh tally over a large region, or a mesh tally over the entire problem. Several other methods for
producing importance maps are also available in MAVRIC and are explored in :ref:`appendixc`.
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CADIS Methodology
-----------------

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MAVRIC is an implementation of CADIS (Consistent Adjoint Driven Importance Sampling) using the Denovo
SN and Monaco Monte Carlo functional modules. Source biasing and a mesh-based importance map, overlaying
the physical geometry, are the basic methods of variance reduction. To make the best use of an
importance map, the map must be made consistent with the source biasing. If the source biasing is inconsistent
with the weight windows that will be used during the transport process, then source particles will undergo Russian
roulette or splitting immediately, wasting computational time and negating the intent of the biasing.
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Overview of CADIS
~~~~~~~~~~~~~~~~~

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CADIS is well described in the literature, so only a
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brief overview is given here. Consider a class source-detector problem
described by a unit source with emission probability distribution
function :math:`q\left(\overrightarrow{r},E \right)` and a detector
response function :math:`\sigma_{d}\left(\overrightarrow{r},E \right)`.
To determine the total detector response, *R*, the forward scalar flux
:math:`\phi\left(\overrightarrow{r},E \right)` must be known. The
response is found by integrating the product of the detector response
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function and the flux over the detector volume :math:`V_{d}`:
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.. math::
  :label: mavric-1

  R = \int_{V_{d}}^{}{\int_{E}^{}{\sigma_{d}\left( \overrightarrow{r},E \right)}}\phi\left(\overrightarrow{r},E \right)\textit{dE dV.}


Alternatively, if the adjoint scalar flux,
:math:`\phi^{+}\left(\overrightarrow{r},E \right)`, is known from the
corresponding adjoint problem with adjoint source
:math:`q^{+}\left(\overrightarrow{r},E \right) = \sigma_{d}\left(\overrightarrow{r},E \right)`,
then the total detector response could be found by integrating the
product of the forward source and the adjoint flux over the source
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volume, :math:`V_{s}`:
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.. math::
  :label: mavric-2

  R = \int_{V_{s}}^{}{\int_{E}^{}{q\left(\overrightarrow{r},E \right)}}\phi^{+}\left( \overrightarrow{r},E \right)\textit{dE dV.}

Unfortunately, the exact adjoint flux may be just as difficult to
determine as the forward flux, but an approximation of the adjoint flux
can still be used to form an importance map and a biased source
distribution for use in the forward Monte Carlo calculation.

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Wagner :cite:`wagner_acceleration_1997` showed that if an estimate of the adjoint scalar flux
for the corresponding adjoint problem can be found, then an estimate
of the response *R* can be made using :eq:`mavric-2`. The adjoint source for the
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adjoint problem is typically separable and corresponds to the detector
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response and spatial area of the tally to be optimized:
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:math:`q^{+}\left(\overrightarrow{r},E \right) = \sigma_{d}\left(E \right)g\left( \overrightarrow{r} \right)`,
where :math:`\sigma_{d}\left( E \right)` is a flux-to-dose conversion
factor and :math:`g\left( \overrightarrow{r} \right)` is 1 in the tally
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volume and is 0 otherwise. Then, from the adjoint flux
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:math:`\phi^{+}\left( \overrightarrow{r},E \right)` and response
estimate *R*, a biased source distribution,
:math:`\widehat{q}\left( \overrightarrow{r},E \right)`, for source
sampling of the form


.. math::
  :label: mavric-3

  \widehat{q}\left(\overrightarrow{r},E \right) = \frac{1}{R}q\left(\overrightarrow{r},E\right)\phi^{+}\left( \overrightarrow{r},E \right)


and weight window target values,
:math:`\overline{w}\left( \overrightarrow{r},E \right)`, for particle
transport of the form


.. math::
  :label: mavric-4

  \overline{w}\left( \overrightarrow{r},E \right) = \frac{R}{\phi^{+}\left( \overrightarrow{r},E \right)}


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can be constructed, which minimizes the variance in the forward Monte
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Carlo calculation of *R*.

When a particle is sampled from the biased source distribution
:math:`\widehat{q}\left( \overrightarrow{r},E \right)`, to preserve a
fair game, its initial weight is set to


.. math::
  :label: mavric-5

  w_{0}\left(\overrightarrow{r},E \right) = \frac{q\left(\overrightarrow{r},E \right)}{\widehat{q}\left( \overrightarrow{r},E \right)} = \frac{R}{\phi^{+}\left( \overrightarrow{r},E \right)}\,


which exactly matches the target weight for that particles position and
energy. This is the consistent part of CADISsource particles are born
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with a weight matching the weight window of the region/energy into which they are
born. The source biasing and the weight windows work together.
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CADIS has been applied to many problemsincluding reactor ex-core
detectors, well-logging instruments, cask shielding studies, and
independent spent fuel storage facility modelsand has demonstrated very
significant speed-ups in calculation time compared to analog
simulations.

Multiple sources with CADIS
~~~~~~~~~~~~~~~~~~~~~~~~~~~

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For a typical Monte Carlo calculation with multiple sources---each with a
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probability distribution function
:math:`q_{i}\left( \overrightarrow{r},E \right)` and a strength
:math:`S_{i}`, giving a total source strength of
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:math:`S = \sum_{}^{}S_{i}`---the source is sampled in two steps. First,
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the specific source *i* is sampled with probability
:math:`p\left( i \right) = \ S_{i}/S`, and then the particle is sampled
from the specific source distribution
:math:`q_{i}\left( \overrightarrow{r},E \right)`.

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The source sampling can be biased at both levels: from which source to sample
and how to sample each source. For example, the specific source can
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be sampled using some arbitrary distribution,
:math:`\widehat{p}\left( i \right)`, and then the individual sources can
be sampled using distributions
:math:`{\widehat{q}}_{i}\left( \overrightarrow{r},E \right)`. Particles
would then have a birth weight of


.. math::
  :label: mavric-6

  w_{0} \equiv \ \left(\frac{p\left( i \right)}{\widehat{p}\left( i \right)} \right)\left(\frac{q_{i}\left( \overrightarrow{r},E \right)}{{\widehat{q}}_{i}\left( \overrightarrow{r},E \right)} \right)\text{.}


For CADIS, a biased multiple source needs to be developed so that the
birth weights of sampled particles still match the target weights of the
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importance map. For a problem with multiple sources---each with a
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distribution :math:`q_{i}\left( \overrightarrow{r},E \right)` and a
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strength :math:`S_{i}`---the goal of the Monte Carlo calculation is to
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compute some response :math:`R` for a response function
:math:`\sigma_{d}\left( \overrightarrow{r},E \right)` at a given
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detector,
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.. math::
  :label: mavric-7

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  R = \ \int_{V}^{}{\int_{E}^{}{\sigma_{d}\left( \overrightarrow{r},E \right)\phi \left( \overrightarrow{r},E \right)\textit{dE dV.}}}
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Note that the flux :math:`\phi\left( \overrightarrow{r},E \right)` has
contributions from each source. The response, :math:`R_{i}`, from each
specific source (:math:`S_{i}` with
:math:`q_{i}\left( \overrightarrow{r},E \right)`) can be expressed using
just the flux from that source,
:math:`\phi_{i}\left( \overrightarrow{r},E \right)`, as


.. math::
  :label: mavric-8

  R_{i} = \ \int_{V}^{}{\int_{E}^{}{\sigma_{d}\left(\overrightarrow{r},E \right)\ \phi_{i}\left(\overrightarrow{r},E \right)\textit{dE dV .}}}


The total response is then found as :math:`R = \sum_{i}^{}R_{i}`.

For the adjoint problem, using the adjoint source of
:math:`q^{+}\left( \overrightarrow{r},E \right) = \sigma_{d}\left( \overrightarrow{r},E \right)`,
the response :math:`R` can also be calculated as


.. math::
  :label: mavric-9

  R = \ \int_{V}^{}{\int_{E}^{}{\left\lbrack \sum_{i}^{}{S_{i}q_{i}\left( \overrightarrow{r},E \right)} \right\rbrack\ \phi^{+}\left( \overrightarrow{r},E \right)\textit{dE dV}}},


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with the response contribution from each specific source being
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.. math::
  :label: mavric-10

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  R_{i} = \ \int_{V}^{}{\int_{E}^{}{\ {S_{i}q_{i}\left( \overrightarrow{r},E \right)\phi^{+}}\left( \overrightarrow{r}, E \right)\textit{dE dV.}}}
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The target weights
:math:`\overline{w}\left( \overrightarrow{r},E \right)` of the
importance map are found using


.. math::
  :label: mavric-11

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  \overline{w}\left( \overrightarrow{r},E \right) = \frac{R/S}{\phi^{+}\left( \overrightarrow{r},E \right)\ }.
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Each biased source
:math:`{\widehat{q}}_{i}\left( \overrightarrow{r},E \right)` pdf is
found using

.. math::
  :label: mavric-12

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  {\widehat{q}}_{i}\left(\overrightarrow{r},E \right) = \frac{S_{i}}{R_{i}}{q_{i}\left( \overrightarrow{r},E \right)\phi}^{+}\left( \overrightarrow{r},E \right)\ ,

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and the biased distribution used to select an individual source is
:math:`\widehat{p}\left( i \right) = \ R_{i}/\sum_{}^{}{R_{i} = R_{i}/R}`.

When using the biased distribution used to select an individual source,
:math:`\widehat{p}\left( i \right)`, and the biased source distribution,
:math:`{\widehat{q}}_{i}\left( \overrightarrow{r},E \right)`, the birth
weight of the sampled particle will be


.. math::
  :label: mavric-13

   \begin{matrix}
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      w_{0} & \equiv & \left( \frac{p\left( i \right)}{\widehat{p}\left( i \right)} \right)\left( \frac{q_{i}\left( \overrightarrow{r}, E \right)}{{\widehat{q}}_{i}\left(\overrightarrow{r},E \right)} \right) \\ & = & \ \left( \frac{\frac{S_{i}}{S}}{\frac{R_{i}}{R}} \right) \left( \frac{q_{i}\left( \overrightarrow{r},E \right)}{\frac{S_{i}}{R_{i}}{q_{i}\left( \overrightarrow{r},E \right)\phi^{+}}\left( \overrightarrow{r},E \right)} \right) \\
      & = & \frac{R/S}{{\phi}^{+}\left( \overrightarrow{r},E \right)\ }, \\
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  \end{matrix}


which matches the target weight,
:math:`\overline{w}\left( \overrightarrow{r},E \right)`.

Multiple tallies with CADIS
~~~~~~~~~~~~~~~~~~~~~~~~~~~

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The CADIS methodology works quite well for classic source/detector problems.
The statistical uncertainty of the tally that serves as the adjoint source is greatly reduced since the
Monte Carlo transport is optimized to spend more simulation time on those particles that contribute to the
tally, at the expense of tracking particles in other parts of phase space. However, more recently,
Monte Carlo has been applied to problems in which multiple tallies need to all be found with low statistical
uncertainties. The extension of this idea is the mesh tallywhere each voxel is a tally for which the user desires
low statistical uncertainties. For these problems, the user must accept a total simulation time that is controlled
by the tally with the slowest convergence and simulation results where the tallies have a wide range of relative
uncertainties.

The obvious way around this problem is to create a separate problem for each tally and use CADIS to optimize each.
Each simulation can then be run until the tally reaches the level of acceptable uncertainty.
For more than a few tallies, this approach becomes complicated and time-consuming for the user.
For large mesh tallies, this approach is not reasonable.

Another approach to treat several tallies, if they are in close proximity to each other,
or a mesh tally covering a small portion of the physical problem, is to use the CADIS methodology
with the adjoint source near the middle of the tallies to be optimized. Since particles in the
forward Monte Carlo simulation are optimized to reach the location of the adjoint source, all the
tallies surrounding that adjoint source should converge quickly. This approach requires the
difficult question of how close. If the tallies are too far apart, then certain energies or regions that are
needed for one tally may be of low importance for getting particles to the central adjoint source. This may
under-predict the flux or dose at the tally sites far from the adjoint source.

MAVRIC has the capability to have multiple adjoint sources with this problem in mind.
For several tallies that are far from each other, multiple adjoint sources could be used.
In the forward Monte Carlo, particles would be drawn to one of those adjoint sources.
The difficulty with this approach is that typically the tally that is closest to the true
physical source converges faster than the other tallies--showing that the closest adjoint source
seems to attract more particles than the others. Assigning more strength to the adjoint
source further from the true physical source helps to address this issue, but finding the correct strengths so
that all of the tallies converge to the same relative uncertainty in one simulation is an iterative process for the user.
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Forward-weighted CADIS
~~~~~~~~~~~~~~~~~~~~~~

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To converge several tallies to the same relative uncertainty in
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one simulation, the adjoint source corresponding to each of those
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tallies must be weighted inversely by the expected tally value. To calculate the
dose rate at two points--say one near a reactor
and one far from a reactor--in one simulation, then the total adjoint
source used to develop the weight windows and biased source must
have two parts. The adjoint source far from the reactor must have
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more strength than the adjoint source near the reactor by a factor equal
to the ratio of the expected near dose rate to the expected far dose
rate.

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This concept can be extended to mesh tallies, as well. Instead of using a
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uniform adjoint source strength over the entire mesh tally volume, each
voxel of the adjoint source should be weighted inversely by the expected
forward tally value for that voxel. Areas of low flux or low dose rate
would have more adjoint source strength than areas of high flux or high
dose rate.

An estimate of the expected tally results can be found by using a quick
discrete-ordinates calculation. This leads to an extension of the CADIS
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method: forward-weighted CADIS (FW-CADIS). First, a forward S\ :sub:`N` calculation is performed to
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estimate the expected tally results. A total adjoint source is
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constructed so that the adjoint source corresponding to each tally is
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weighted inversely by those forward tally estimates. Then the standard
CADIS approach is usedan importance map (target weight windows) and a
biased source are made using the adjoint flux computed from the adjoint
S\ :sub:`N` calculation.

For example, if the goal is to calculate a detector response function
:math:`\sigma_{d}\left( E \right)` (such as dose rate using
flux-to-dose-rate conversion factors) over a volume (defined by
:math:`g\left( \overrightarrow{r} \right)`) corresponding to mesh tally,
then instead of simply using
:math:`q^{+}\left( \overrightarrow{r},E \right) = \sigma_{d}\left( E \right)\ g(\overrightarrow{r})`,
the adjoint source would be


.. math::
  :label: mavric-14

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   q^{+}\left( \overrightarrow{r},E \right) = \frac{\sigma_{d}\left( E \right)\text{g}\left( \overrightarrow{r} \right)}{\int_{}^{}{\sigma_{d}\left( E \right)\phi\left( \overrightarrow{r},E \right)}\textit{dE}}\ ,
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where :math:`\phi\left( \overrightarrow{r},E \right)` is an estimate of
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the forward flux, and the energy integral is over the voxel at :math:`\overrightarrow{r}`.
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The adjoint source is nonzero only where the mesh tally is defined
(:math:`g\left( \overrightarrow{r} \right)`), and its strength is
inversely proportional to the forward estimate of dose rate.

The relative uncertainty of a tally is controlled by two components:
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(1) the number of tracks contributing to the tally and (2) the
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shape of the distribution of scores contributing to that tally. In the
Monte Carlo game, the number of simulated particles,
:math:`m\left( \overrightarrow{r},E \right)`, can be related to the true
physical particle density, :math:`n\left( \overrightarrow{r},E \right),`
by the average Monte Carlo weight of scoring particles,
:math:`\overline{w}\left( \overrightarrow{r},E \right)`, by


.. math::
  :label: mavric-15

  n\left( \overrightarrow{r},E \right) = \ \overline{w}\left( \overrightarrow{r},E \right)\text{m}\left( \overrightarrow{r},E \right).


In a typical Monte Carlo calculation, tallies are made by adding some
score, multiplied by the current particle weight, to an accumulator. To
calculate a similar quantity related to the Monte Carlo particle density
would be very close to calculating any other quantity but without
including the particle weight. The goal of FW-CADIS is to make the Monte
Carlo particle density, :math:`m\left( \overrightarrow{r},E \right)`,
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uniform over the tally areas, so an importance map must be developed
that represents the importance of achieving uniform Monte Carlo particle
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density. By attempting to keep the Monte Carlo particle density more
uniform, more uniform relative errors for the tallies should be
realized.

Two options for forward weighting are possible. For tallies over some
area where the entire group-wise flux is needed with low relative
uncertainties, the adjoint source should be weighted inversely by the
forward flux, :math:`\phi\left( \overrightarrow{r},E \right)`. The other
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option, for a tally in which only an energy-integrated quantity is desired,
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is to weight the adjoint inversely by that energy-integrated
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quantity,\ :math:`\int_{}^{}{\sigma_{d}\left( E \right)\phi\left( \overrightarrow{r},E \right)}\text{\ dE}`.
For a tally in which the total flux is desired, then the response in the
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adjoint source is simply :math:`\sigma_{d}\left( E \right) = 1`.

To optimize the forward Monte Carlo simulation for the calculation of
some quantity at multiple tally locations or across a mesh tally, the
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adjoint source must be weighted by the estimate of that quantity.
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For a tally defined by its spatial location
:math:`g\left( \overrightarrow{r} \right)` and its optional response
:math:`\sigma_{d}\left( E \right)`, the standard adjoint source would be
:math:`q^{+}\left( \overrightarrow{r},E \right) = \sigma_{d}\left( E \right)\text{g}\left( \overrightarrow{r} \right)`.
The forward-weighted adjoint source,
:math:`q^{+}\left( \overrightarrow{r},E \right)`, depending on what
quantity is to be optimized, is listed below.

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.. list-table::
  :header-rows: 1
  :widths: 50 100 200

  * - For the calculation of
    -
    - Adjoint source
  * - Energy and spatially dependent flux
    - :math:`\phi\left(\overrightarrow{r},E \right)`
    - .. math:: \frac{g\left( \overrightarrow{r}\right)}{\phi\left(\overrightarrow{r},E \right)}
  * - Spatially dependent total flux
    - :math:`\int_{}^{}{\phi\left( \overrightarrow{r},E \right)}\textit{dE}`
    - .. math:: \frac{g\left( \overrightarrow{r}\right)}{\int_{}^{}{\phi\left( \overrightarrow{r},E \right)}\textit{dE}}
  * - Spatially dependent total response
    - :math:`\int_{}^{}{\sigma_{d}\left( E \right)\phi    \left(\overrightarrow{r},E\right)}\textit{dE}`
    - .. math:: \frac{\sigma_{d}\left( E \right)\text{g}\left( \overrightarrow{r} \right)}{\int_{}^{}{\sigma_{d}\left( E \right)\phi    \left( \overrightarrow{r},E \right)}\textit{dE}}

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The bottom line of FW-CADIS is that in order to calculate a quantity at
multiple tally locations (or across a mesh tally) with more uniform
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relative uncertainties, an adjoint source must be developed for an
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objective function that keeps some non-physical quantityrelated to the
Monte Carlo particle density and similar in form to the desired
quantityconstant. FW-CADIS uses the solution of a forward
discrete-ordinates calculation to properly weight the adjoint source.
After that, the standard CADIS approach is used.

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MAVRIC Implementation of CADIS
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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

With MAVRIC, as with other shielding codes, the user defines the problem as a set of
physical modelsthe material compositions, the geometry, the source, and the detectors
(locations and response functions)as well as some mathematical parameters on how to solve
the problem (number of histories, etc.). For the variance reduction portion of MAVRIC, the
only additional inputs required are (1) the mesh planes to use in the discrete-ordinates
calculation(s) and (2) the adjoint source description--basically the location and the response
of each tally to optimize in the forward Monte Carlo calculation. MAVRIC uses this information
to construct a Denovo adjoint problem. (The adjoint source is weighted by a Denovo forward flux
or response estimate for FW-CADIS applications.)  MAVRIC then uses the CADIS methodology: it combines
the adjoint flux from the Denovo calculation with the source description and creates the importance map
(weight window targets) and the mesh-based biased source. Monaco is then run using the CADIS biased source
distribution and the weight window targets.
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Denovo
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^^^^^^
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Denovo is a parallel three-dimensional SN code that is used to generate adjoint (and, for FW-CADIS, forward)
scalar fluxes for the CADIS methods in MAVRIC. For use in MAVRIC/CADIS, it is highly desirable that the SN code be fast,
positive, and robust. The phase-space shape of the forward and adjoint fluxes, as opposed to a highly accurate solution,
is the most important quality for Monte Carlo weight-window generation. Accordingly,
Denovo provides a step-characteristics spatial differencing option that produces positive scalar fluxes as
long as the source (volume plus in-scatter) is positive. Denovo uses an orthogonal, nonuniform mesh that is
ideal for CADIS applications because of the speed and robustness of calculations on this mesh type.
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Denovo uses the highly robust GMRES (Generalized Minimum Residual) Krylov method to solve the SN equations in each group. GMRES has been shown to be more robust and efficient than traditional source (fixed-point) iteration. The in-group discrete SN equations are defined as


.. math::
  :label: mavric-16

  \mathbf{L}\psi = \mathbf{\text{MS}}\phi + q

where **L** is the differential transport operator, **M** is the
moment-to-discrete operator, **S** is the matrix of scattering
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cross section moments, *q* is the external and in-scatter source,
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:math:`\phi` is the vector of angular flux moments, and :math:`\psi` is
the vector of angular fluxes at discrete angles. Applying the operator
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**D**, where :math:`\phi = \mathbf{D}\psi`, and rearranging terms, casts
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the in-group equations in the form of a traditional linear system,
:math:`\mathbf{A}x = b`,

 .. math::
  :label: mavric-17

  \left( \mathbf{I} - \mathbf{D}\mathbf{L}^{- 1}\mathbf{\text{MS}} \right) = \mathbf{D}\mathbf{L}^{- 1}q .

The operation :math:`\mathbf{L}^{- 1}\nu`, where :math:`\nu` is an
iteration vector, is performed using a traditional wave-front solve
(transport sweep). The parallel implementation of the Denovo wave-front
solver uses the well-known Koch-Baker-Alcouffe (KBA) algorithm, which is
a two-dimensional blockspatial decomposition of a three-dimensional
orthogonal mesh :cite:`baker_sn_1998`. The Trilinos package is used for the GMRES
implementation :cite:`willenbring_trilinos_2003` Denovo stores the mesh-based scalar fluxes in a
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double precision binary file (\*.dff) called a *Denovo flux file*. Past
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versions of SCALE/Denovo used the TORT :cite:`rhoades_tort_1997` \*.varscl file format
(DOORS package :cite:`rhoades_doors_1998`), but this was limited to single precision. Since
the rest of the MAVRIC sequence has not yet been parallelized, Denovo is
currently used only in serial mode within MAVRIC.

Monaco
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^^^^^^
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The forward Monte Carlo transport is performed using Monaco, a
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fixed-source shielding code that uses the SCALE General Geometry
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Package (SGGP, the same as used by the criticality code KENO-VI) and the
standard SCALE material information processor. Monaco can use either MG
or CE cross section libraries. Monaco was originally based on the MORSE
Monte Carlo code but has been extensively modified to modernize the
coding, incorporate more flexibility in terms of sources/tallies, and
read a user-friendly block/keyword style input.

Much of the input to MAVRIC is the same as Monaco. More details can be
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found in the Monaco chapter of the SCALE manual (SECTIONREFERENCE).
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Running MAVRIC
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^^^^^^^^^^^^^^
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The objective of a SCALE sequence is to execute several codes, passing
the output from one to the input of the next, in order to perform some
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analysis--tasks that users typically had to do in the past. MAVRIC does
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this for difficult shielding problems by running approximate
discrete-ordinates calculations, constructing an importance map and
biased source for one or more tallies that the user wants to optimize in
the Monte Carlo calculation, and then using those in a forward Monaco
Monte Carlo calculation. MAVRIC also prepares the forward and adjoint
cross sections when needed. The steps of a MAVRIC sequence are listed in
:numref:`Mavric-sequence`. The user can instruct MAVRIC to run this whole sequence of
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steps or just some subset of the steps to verify the
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intermediate steps or to reuse previously calculated quantities in a new
analyses.

The MAVRIC sequence can be stopped after key points by using the
parm= *parameter*  operator on the =mavric command line, which is
the first line of the input file. The various parameters are listed in
Table :numref:`mavric-param`. These parameters allow the user to perform checks and make
changes to the importance map calculation before the actual Monte Carlo
calculation in Monaco.

MAVRIC also allows the sequence to start at several different points. If
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an importance map and biased source have already been computed, they can then
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be used directly. If the adjoint scalar fluxes are known, they can
quickly be used to create the importance map and biased source and then
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begin the forward Monte Carlo calculation. All of the different combinations of
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starting MAVRIC with some previously calculated quantities are listed in
the following section detailing the input options.

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When using MG cross section libraries that do not have flux-to-dose-rate
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conversion factors, use parm=nodose to prevent the cross section
processing codes from trying to move these values into the working
library.

MAVRIC creates many files that use the base problem name from the output
file. For an output file called c:\path1\path2\\\ *outputName*.out or
/home/path1/path2/ *outputName*.inp, spaces in the output name will
cause trouble and should not be used.

.. list-table:: Steps in the MAVRIC sequence
   :name: Mavric-sequence
   :widths: 100 100
   :header-rows: 0
   :align: center

   * - **Cross section calculation**
     - XSProc is used to calculate the forward cross sections for Monaco
   * - **Forward Denovo (optional)**
     -
   * -  Cross section calculation
     - XSProc is used to calculate the forward cross sections for Denovo
   * -  Forward flux calculation
     - Denovo calculates the estimate of the forward flux
   * - **Adjoint Denovo (optional)**
     -
   * -  Cross section calculation
     - XSProc is used to calculate the adjoint cross sections for Denovo
   * -  Adjoint flux calculation
     - Denovo calculates the estimate of the adjoint flux
   * - **CADIS (optional)**
     - The scalar flux file from Denovo is then used to create the biased source distribution and transport weight windows
   * - **Monte Carlo calculation**
     - Monaco uses the biased source distribution and transport weight windows to calculate the various tallies

.. list-table:: Parameters for the MAVRIC command line (parm=…”)
   :name: mavric-param
   :widths: 50 50
   :header-rows: 1
   :align: center

   * - Parameter
     - MAVRIC will stop after
   * - check
     - input checking
   * - forinp
     - Forward Denovo input construction (makes ``xkba_b.inp`` in the tmp area)
   * - forward
     - The forward Denovo calculation
   * - adjinp
     - Adjoint Denovo input construction (makes ``xkba_b.inp`` in the tmp area)
   * - adjoint
     - The adjoint Denovo calculation
   * - impmap
     - Calculation of importance map and biased source

MAVRIC input
------------

The input file for MAVRIC consists of three lines of text (=mavric
command line with optional parameters, the problem title, and SCALE
cross section library name) and then several blocks, with each block
starting with read xxxx and ending with end xxxx. There are three
required blocks and nine optional blocks. Material and geometry blocks
must be listed first and in the specified order. Other blocks may be
listed in any order.

Blocks (must be in this order):

-  Composition  (required) SCALE standard composition, list of materials used in the problem

-  Celldata  SCALE resonance self-shielding

-  Geometry  (required) SCALE general geometry description

-  Array  optional addition to the above geometry description

-  Volume  optional calculation or listing of region volumes

-  Plot  create 2D slices of the SGGP geometry

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Other Blocks (in any order, following the blocks listed above):
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-  Definitions  defines locations, response functions, and grid geometries used by other blocks

-  Sources  (required) description of the particle source spatial, energy, and directional distributions

-  Tallies  description of what to calculate: point detector tallies, region tallies, or mesh tallies

-  Parameters  how to perform the simulation (random number seed, how many histories, etc.)

-  Biasing  data for reducing the variance of the simulation

-  ImportanceMap  instructions for creating an importance map based on a discrete-ordinates calculation

The material blocks (Composition and Celldata) and the physical model
blocks (Geometry, Array, Volume, and Plot) follow the standard SCALE
format. See the other SCALE references as noted in the following
sections for details. The Biasing block and ImportanceMap block cannot
both be used.

For the other six blocks, scalar variables are set by keyword=value,
fixed-length arrays are set with keyword value\ :sub:`1` ...
value\ :sub:`N`\ , variable-length arrays are set with keyword
value\ :sub:`1` ... value\ :sub:`N` end, and some text and filenames
are read in as quoted strings. Single keywords to set options are also
used in some instances. The indention, comment lines, and
upper/lowercase shown in this document are not required they are used
in the examples only for clarity. Except for strings in quotes (like
filenames), SCALE is case insensitive.

After all input blocks are listed, a single line with end data should be listed.
A final end should also be listed, to signify the end of all MAVRIC input.
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Nine of the blocks are the same input blocks as those used by the functional module Monaco,
with a few extra keywords only for use with MAVRIC. These extra keywords are highlighted here, but
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without relisting all of the standard Monaco keywords for those blocks.
See :numref:`input-format` for an overview of MAVRIC input file structure.

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Composition block
~~~~~~~~~~~~~~~~~

Material information input follows the standard SCALE format for
material input. Basic materials known to the SCALE library may be used
as well as completely user-defined materials (using isotopes with known
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cross sections). Input instructions are located in the XSProc chapter (SECTIONREFERENCE) in
the SCALE manual. The Standard Composition Library chapter (SECTIONREFERENCE) lists the
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different cross section libraries and the names of standard materials.
An example is as follows:

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.. code:: scale
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   read composition

       uo2 1 0.2 293.0 92234 0.0055 92235 3.5 92238 96.4945 end

       orconcrete 2 1.0 293.0 end

       ss304 3 1.0 293.0 end

   end composition

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Details on the cell data block are also included in the XSProc chapter (SECTIONREFERENCE).
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When using different libraries for the importance map production (listed
at the top of the input) and the final Monte Carlo calculation (listed
in the parameters block, if different), make sure that the materials are
present in both libraries.


.. list-table:: Overall input format
   :widths: 30 30
   :header-rows: 1
   :align: center
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   :name: input-format
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   * - input file
     - Comment
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   * - .. code:: scale
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         =mavric
         Some title for this problem
         v7-27n19g
         read composition
            ...
         end composition
         read celldata
            ...
         end celldata
         read geometry
            ...
         end geometry
         read array
            ...
         end array
         read volume
            ...
         end volume
         read plot
            ...
         end plot
         read definitions
            ...
         end definitions
         read sources
            ...
         end sources
         read tallies
            ...
         end tallies
         read parameters
            ...
         end parameters
         read biasing
            ...
         end biasing
         read importanceMap
            ...
         end importanceMap
         end data
         end
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     - .. code:: rest
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          name of sequence
          title
          cross section library name
          SCALE material compositions
              [required block]

          SCALE resonance self-shielding
              [optional block]

          SCALE SGGP geometry
              [required block]

          SCALE SGGP arrays
              [optional block]

          SCALE SGGP volume calc
              [optional block]

          SGGP Plots
              [optional block]

          Definitions
              [possibly required]

          Sources definition
              [required block]

          Tally specifications
              [optional block]

          Monte Carlo parameters
              [optional block]

          Biasing information
              [optional block]

          Importance map
              [optional block]

          end of all blocks
          end of MAVRIC input

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SGGP geometry blocks
~~~~~~~~~~~~~~~~~~~~

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MAVRIC uses the functional module Monaco for the forward Monte Carlo calculation.
Monaco tracks particles through the physical geometry described by the SGGP input
blocks, as well as through the mesh importance map and any mesh tallies, which are
defined in the global coordinates and overlay the physical geometry. Because Monaco
must track through all of these geometries at the same time, users should not use the
reflective boundary capability in the SGGP geometry.
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For more details on each SGGP geometry block, see the following sections of the KENO-VI chapter (SECTIONREFERENCE) of the SCALE Manual.
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    Geometry  *Geometry Data*

    Array  *Array Data*

    Volume  *Volume Data*

    Plot  *Plot Data*

Other blocks shared with Monaco
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The definitions, sources, tallies, and biasing blocks are all the same
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SCALE Manual.

   Definitions  *Definitions Block*

   Sources  *Sources Block*

   Tallies  *Tallies Block*

   Biasing  *Biasing Block*

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The parameters block includes several keywords that are not included in
Monaco (see the *Parameter Block* section of the Monaco chapter (SECTIONREFERENCE)) which
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are used when the cross section library used in the importance
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calculations differs from the library used in the final forward
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Monaco Monte Carlo calculation. The library listed at the beginning of
the MAVRIC input file will be used for the importance calculations
(forward and adjoint Denovo calculation, formation of the importance
map, and biased sources). To use a different MG library in the final
Monaco simulation, use the keyword library= with the cross section
library name in quotes. A cross section library for Monaco will be made
using csas-mg. If there are any extra parameters to use (parm= in the
=csas-mg line of the csas-mg input), they can be passed along using
the keyword parmString= with the extra information in quotes. For
example, the following input file would use a coarse-group library for
the importance calculations and a fine-group library for the final
Monaco, each with CENTRM processing.

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.. code:: scale
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    =mavric parm=centrm
    v7-27n19g
    

    read parameters

        library=v7-200n47g parmString=centrm

        

    end parameters

    

    end data

    end


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To use a CE cross section in the final Monaco step, use the keyword ceLibrary= with the cross section
library name in quotes. When the library= or ceLibrary= keywords are used, they should precede the neutron, photon,
noNeutron, and noPhoton keywords. :numref:`extra-keywords` summarizes all of the keywords in the MAVRIC parameter block.
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When using two different cross section libraries, be sure that the responses and distributions are
defined in ways that do not depend on the cross section library. For example, any response that is
just a list of n values (corresponding to a cross section library of n groups) needs to have the
group energies specifically listed so that it can be evaluated properly on the other group structure.
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.. list-table:: Extra keywords for the parameters block
  :align: center
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  :name: extra-keywords

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  * - .. image:: figs/MAVRIC/table4.4.png
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Importance map block
~~~~~~~~~~~~~~~~~~~~

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The importance map block is the heart and soul of MAVRIC. This block lists the parameters for creating an
importance map and biased source from one (adjoint) or two (forward, followed by adjoint) Denovo
discrete-ordinates calculations. Without an importance map block, MAVRIC can be used to run Monaco
and use its conventional types of variance reduction. If both the importance map and biasing blocks
are specified, then only the importance map block will be used. The various ways to use the importance map block
are explained in the subsections below. Keywords for this block are summarized at the end of this section, in
:numref:`keywords-importance`.
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Constructing a mesh for the S\ :sub:`N` calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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All uses of the importance map block that run the
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discrete-ordinates code require the use of a grid geometry that overlays
the physical geometry. Grid geometries are defined in the definitions
block of the MAVRIC input. The extent and level of detail needed in a
grid geometry are discussed in the following paragraphs.

When using S\ :sub:`N` methods alone for solving radiation transport in
shielding problems, a good rule of thumb is to use mesh cell sizes on
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the order of a meanfree path of the particle. In complex shielding
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problems, this could lead to an extremely large number of mesh cells,
especially when considering the size of the meanfree path of the lowest
energy neutrons and photons in common shielding materials.

In MAVRIC, the goal is to use the S\ :sub:`N` calculation for a quick
approximate solution. Accuracy is not paramountjust getting an idea of
the overall shape of the true importance map will help accelerate the
convergence of the forward Monte Carlo calculation. The more accurate
the importance map, the better the forward Monte Carlo acceleration will
be. At some point there is a time trade-off when the computational time
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for calculating the importance map followed by the time to perform the Monte Carlo
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calculation exceeds that of a standard analog Monte Carlo calculation.
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Large numbers of mesh cells that result from using very small mesh sizes
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for S\ :sub:`N` calculations also use a great deal of computer memory.

Because the deterministic solution(s) for CADIS and FW-CADIS can have
moderate fidelity and still provide variance reduction parameters that
substantially accelerate the Monte Carlo solution, mesh cell sizes in
MAVRIC applications can be larger than what most S\ :sub:`N` practioners
would typically use. The use of relatively coarse mesh reduces memory
requirements and the run time of the deterministic solution(s). Some
general guidelines to keep in mind when creating a mesh for the
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importance map/biased source are as follows:
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-  The true source regions should be included in the mesh with mesh
   planes at their boundaries.

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-  Place point or very small sources in the center of a mesh cell, not on the mesh planes.
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-  Any region of the geometry where particles could eventually
   contribute to the tallies (the important areas) should be included
   in the mesh.

-  Point adjoint sources (corresponding to point detector locations) in
   standard CADIS calculations do not have to be included inside the
   mesh. For FW-CADIS, they must be in the mesh and should be located at
   a mesh cell center, not on any of the mesh planes.

-  Volumetric adjoint sources should be included in the mesh with mesh
   planes at their boundaries.

-  Mesh planes should be placed at significant material boundaries.

-  Neighboring cell sizes should not be drastically different.

-  Smaller cell sizes should be used where the adjoint flux is changing
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   rapidly, such as toward the surfaces of adjoint sources and
   shields (rather than in their interiors).
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Another aspect to keep in mind is that the source in the forward Monaco
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Monte Carlo calculation will be a biased mesh-based source. Source
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particles will be selected by first sampling which mesh cell to use and
then sampling a position uniformly within that mesh cell that meets the
user criteria of unit=, region=, or mixture= if specified. The
mesh should have enough resolution that the mesh source will be an
accurate representation of the true source.

The geometry for the Denovo calculation is specified using the keyword
gridGeometryID= and the identification number of a grid geometry that
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was defined in the definitions block. The material assigned to each voxel of the mesh is determined by
testing the center point in the SGGP geometry (unless the macro-material option is usedsee below).


.. _macromaterials:
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Macromaterials for S\ :sub:`N` geometries
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Part of the advantage of the CADIS method is that the adjoint
discrete-ordinates calculation only needs to be approximate in order to
form a reasonable importance map and biased source. This usually means
that the mesh used is much coarser than the mesh that would be used if
the problem were to be solved only with a discrete-ordinates code. This
coarse mesh may miss significant details (especially curves) in the
geometry and produce a less-than-optimal importance map.

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To get more accurate solutions from a coarse-mesh
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discrete-ordinates calculation, Denovo can represent the material in
each voxel of the mesh as a volume-weighted mixture of the real
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materials, called *macromaterials*, in the problem. When constructing the
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Denovo input, the Denovo EigenValue Calculation (DEVC, see section SECTIONREFERENCE)
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sequence can estimate the volume fraction occupied by using each real
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material in each voxel by a sampling method. The user can specify
parameters for how to sample the geometry. Note that finer sampling
makes more accurate estimates of the material fraction but requires more
setup time to create the Denovo input. Users should understand how the
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macromaterials are sampled and should consider this when constructing a mesh
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grid. This is especially important for geometries that contain arrays.
Careful consideration should be given when overlaying a mesh on a
geometry that contains arrays of arrays.

Because the list of macromaterials could become large, the user can also
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specify a tolerance for how close two different macromaterials can be in order to
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be considered the same, thereby reducing the total number of
macromaterials. The macromaterial tolerance, ``mmTolerance=``, is used for
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creating a different macromaterial from the those already created by
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looking at the infinity norm between two macromaterials.
The number of macromaterials does not appreciably impact Denovo run time
or memory requirements.

Two different sampling methods are availablepoint testing :cite:`ibrahim_improving_2009` with
the keyword ``mmPointTest`` and ray tracing :cite:`johnson_fast_2013` with the keyword
``mmRayTest``.

Ray Tracing
'''''''''''

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This method estimates the volume of different materials in the Denovo mesh grid elements by
tracing rays through the SGGP geometry and computing the average track lengths through each material.
Rays are traced in all three dimensions to better estimate the volume fractions of materials within each voxel.
The mmSubCell parameter controls how many rays will be traced in each voxel in each dimension. For example, if mmSubCell= n,
then when tracing rays in the z dimension, each column of voxels uses a set of n×n rays
starting uniformly spaced in the x  and y  dimensions. With rays being cast from all three orthogonal directions,
a total of 3n2 rays are used to sample each voxel. One can think of subcells as an equally spaced sub-mesh with a
single ray positioned at each center. The number of subcells in each direction, and hence the number of rays, can
be explicitly given with mmSubCells ny nz nx nz nx ny end keyword for rays parallel to the x axis, y axis, and z axis.
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:numref:`ray-positions` shows different subcell configurations (in two dimensions) for a given voxel.

.. _ray-positions:

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.. figure:: figs/MAVRIC/fig4.1.01_rayTrace6.png
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  :width: 500
  :align: center

  Ray positions within a voxel with different mmSubCells parameters.

Ray tracing is a more robust method compared to the simple point testing
method used in previous versions of SCALE/MAVRIC; however, it requires
more memory than point testing. Ray tracing gives more accurate
estimates of volume fractions because track lengths across a voxel give
more information than a series of test points. Ray tracing is also much
faster than point testing because the particle tracking routines are
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optimized to quickly determine lists of materials and distance along
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a given ray.

Ray tracing operates on the grid geometry supplied by the user and
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shoots rays in all three directions, starting from the lower bounds of
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the mesh grid. An example of an arbitrary assembly geometry is shown in
:numref:`geom-model`. A ray consists of a number of steps that each correspond
to crossing a material boundary along the path of the ray. Ratios of
each step’s length to the voxel length in the ray’s direction determine
the material volume fraction of that step in that voxel, and summation
of the same material volume fractions gives the material volume fraction
of that material in that voxel. Ray tracing through a single voxel that
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contains a fuel pin is illustrated in :numref:`ray-vox`.
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.. _geom-model:

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.. figure:: figs/MAVRIC/fig4.1.02_kenoDenovo.png
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  :width: 600
  :align: center

  Geometry model (left) and the Denovo representation (right) of an assembly using macromaterials determined by ray tracing.
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.. _ray-vox:

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.. figure:: figs/MAVRIC/fig4.1.03_rayTrace.png
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  :width: 300
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  :align: center

  Ray tracing (in two dimensions) through a voxel.

The final constructed macromaterials for this model are also shown in
:numref:`geom-model`. Voxels that contain only a single material are assigned
the original material number in the constructed macromaterials. For the
voxels that contain a fuel pin with three different materials, the
result is a new macromaterial consisting of the volume weighted
fractions of each original material.

After the rays are shot in all three directions, the material volume
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fractions are updated, and macromaterials are created by using these
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material volume fractions. Material volume fraction calculations for a
single voxel, as shown in :numref:`ray-vox`, are given by

.. math::
  :label: mavric-18

   F_{m} = \ \sum_{d = x,y,z}^{}{\sum_{r = 1}^{N_{r}}{\sum_{s = 1}^{N_{s}}\left\{ \begin{matrix}
   \frac{L_{d,r,s}}{L_{d}},\ \ \ & m_{s} = m \\
   0,\ \ \ & \mathrm{\text{otherwise}} \\
   \end{matrix} \right.\ }} \ \ \ \ \ \ \ and \ \ \ \ \ \ \ \ \ V_{m} = \frac{F_{m}}{\sum_{n = 1}^{N_{m}}F_{n}}\ ,

where *F\ m* = sampled fraction of material *m* in the voxel,

*d* = direction of the rays (*x, y, z*),

*r* = ray number,

:math:`N_r` = total number of rays in the voxel for direction of *d*,

*s* = step number,

:math:`N_s` = total number of steps for ray r in the voxel for direction of
*d*,

:math:`L_{d,r,s}` = length of the steps s for ray r in the voxel for direction
of *d*,

:math:`L_d` = length of the voxel along direction of *d*,

:math:`m_s` = material of step *s*,

*m* = material number,

:math:`N_m` = total number of materials in the voxel, and

:math:`V_m` = volume fraction of material m in the voxel.

Point Testing
'''''''''''''

The recursive bisection method is utilized in point testing and uses a
series of point tests to determine the macromaterial fractions. For a
given voxel, the material at the center is compared to the material at
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the eight corners. If they are all the same, then the entire volume is
considered to be made of that material. If they are different, then the volume is
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divided into two in each dimension. Each subvolume is tested, and the
method is then applied to the subvolumes that are not of a single
material. When the ratio of the volume of the tested region to the
original voxel becomes less than a user-specified tolerance (in the
range of 10-1 to 10-4), then further subdivision and testing are
stopped. This is illustrated in :numref:`rec-macro`.



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.. _rec-macro:
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.. figure:: figs/MAVRIC/rec-macro.png
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  :width: 99 %
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.. centered:: *Fig. 4 Successive steps in the recursive macromaterial method*
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In point testing, the keyword mmTolerance=f is interpreted to be where *f* is the smallest
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fraction of the voxel volume that can be achieved by bisection method and hence the limiting
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factor for dividing the voxel. This same tolerance *f* is also used to limit the number of macromaterials.
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Before a new macromaterial is created, if one already exists where the fraction of each actual
material matches to within the given tolerance, then the existing material will be used. If
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using only a single point at the center of each voxel, then use mmTolerance=1.
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The mmSubCell keyword is not used in point testing.

Example
'''''''

:numref:`cask-geom` shows an example of a cask geometry with two types of spent fuel (yellows),
steel (blue), resin (green), and other metals (gray). When the Denovo geometry is set up by
testing only the center of each mesh cell, the curved surfaces are not well represented (upper right).
By applying the ray-tracing method and defining a new material made of partial fractions of the original materials,
an improved Denovo model can be made. In the lower left of the figure, the Denovo
model was constructed using one ray (in each dimension) per voxel and a tolerance of 0.1.
This gives 20 new materials that are a mixture of the original 13 actual materials and void.
With mmSubCells=3 and an mmTolerance=0.01, 139 macromaterials are created.

A macromaterial table listing the fractions of each macromaterial is saved to a file called “outputName.mmt”,
where outputName is the name the user chose for his or her output file. This file can be used by the Mesh File
Viewer to display the macromaterials as mixtures of the actual materials, as seen in the lower row of :numref:`cask-geom`.
See the Mesh File Viewer help pages for more information on how to use colormap files and macromaterial tables.


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.. _cask-geom:
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.. figure:: figs/MAVRIC/cask-geom.png
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  Cask geometry model (upper left) and the Denovo representation using cell center testing (upper right). Representations using macromaterials determined by ray tracing are shown for mmSubCell=1/mmTolerance=0.1 (lower left) and mmSubCell=3/mmTolerance=0.01 (lower right).*
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Optimizing source/detector problems
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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For standard source/detector problems in which one tally is to be optimized
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in the forward Monte Carlo calculation, an adjoint source based on that
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tally must be constructed. An adjoint source requires a unique and
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positive identification number, a physical location, and an energy
spectrum. The adjoint source location can be specified either by (1) a
point location (“locationID=” keyword) or (2) a volume described by a
box (“boundingBox” array). A bounding box is specified by maximum and
minimum extent in each dimension—\ :math:`x_{max}` :math:`x_{min}` :math:`y_{max}` :math:`y_{min}` :math:`z_{max}`
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:math:`z_{min}`—in global coordinates. The boundingBox should not be degenerate
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(should have volume>0) but can be optionally limited to areas matching a
given unit number (“unit=”), a given region number (“region=”), or a
given material mixture number (“mixture=”). A mixture and a region
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cannot both be specified, since that would either be redundant or
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mutually exclusive. The energy spectrum of an adjoint source is a
response function (“responseID=”) listing one of the ID numbers of the
responses defined in the definitions block. An optional weight can be
assigned to each adjoint source using the “weight=” keyword. If not
given, the default weight is 1.0.

For example, to optimize a region tally, the user would construct an
adjoint source located in the same place as the tally, with an adjoint
source spectrum equal to the response function that the tally is
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computing. Note that the grid geometry 1 and response function 3 must
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already be defined in the definitions block.

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.. code:: scale
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  read importanceMap
     gridGeometryID=1
     adjointSource 24
         boundingBox 12.0 10.0  5.0 -5.0  10.0 -10.0
         unit=1 region=5
         responseID=3
     end adjointSource
  end importanceMap

For optimizing a point detector for the calculation of total photon flux,
the importance map block would look like the following:

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.. code:: scale
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  read importanceMap
     adjointSource 21
         locationID=4
         responseID=1
     end adjointSource
     gridGeometryID=1
  end importanceMap

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where location 4 is the same location used by the point detector. To calculate total photon flux, response function 1 must be defined in the definitions block similar to this response:
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.. code:: scale
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  read definitions
      response 1
           values 27r0.0 19r1. end
      end response

  end definitions


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This response is used for computing total photon flux for the 27 neutron/19 photon group coupled cross section library or like this response
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.. code:: scale
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  read definitions
      response 1
           photon
           bounds 1000.0 2.0e7 end
           values  1.0   1.0   end
      end response

  end definitions

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which is independent of the cross section library.
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Multiple adjoint sources
^^^^^^^^^^^^^^^^^^^^^^^^

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If there are several tallies in very close proximity and/or several different responses being calculated by the tallies, multiple adjoint sources can be used.
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.. code:: scale
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  read importanceMap
     gridGeometryID=1
     adjointSource 1
         locationID=4  responseID=20
     end adjointSource
     adjointSource 2
         locationID=5  responseID=21
         weight=2.0
     end adjointSource
  end importanceMap

Note that adjoint sources using point locations can be mixed with volumetric adjoint sources (using bounding boxes).

Options for Denovo :math:`S_n` calculations
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

While the default values for various calculational parameters and settings used by Denovo for
the MAVRIC sequence should cover most applications, they can be changed if desired.
The two most basic parameters are the quadrature set used for the discrete ordinates and
the order of the Legendre polynomials used in describing the angular scattering.
The default quadrature order that MAVRIC uses is a level symmetric :math:`S_8` set, and the
default scattering order is :math:`P_3` (or the maximum number of coefficients contained in the
cross-section library if less than 3). :math:`S_8`/ :math:`P_3` is an adequate choice for many applications,
but the user is free to changes these. For complex ducts or transport over large distances at small angles,
:math:`S_{12}` may be required. :math:`S_4`/ :math:`P_1` or even :math:`S_2`/ :math:`P_0` would be useful in doing a very cursory run to confirm that the
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problem was input correctly, but this would likely be inadequate for weight window generation in a problem
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that is complex enough to require advanced variance reduction.

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These and other Denovo options are applied to both
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the forward and the adjoint calculations that are required from the
inputs given in the importance map block.

In problems with small sources or media that are not highly scattering,
discrete ordinates can suffer from "ray effects" :cite:`lathrop_ray_1968,lathrop_remedies_1971`
where artifacts of the discrete quadrature directions can be seen in the
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computed fluxes. Denovo has a
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first-collision capability to help alleviate ray effects. This method
computes the uncollided flux in each mesh cell from a point source. The
uncollided fluxes are then used as a distributed source in the main
discrete-ordinates solution. At the end of the main calculation, the
uncollided fluxes are added to the fluxes computed with the first
collision source, forming the total flux. While this helps reduce ray
effects in many problems, the first-collision capability can take a
significant amount of time to compute on a mesh with many cells or for
many point sources.

Adjoint sources that use point locations will automatically use the
Denovo first-collision capability. Volumetric adjoint sources (that use
a boundingBox) will be treated without the first-collision capability.
The keywords “firstCollision” and “noFirstCollision” will be ignored by
MAVRIC for adjoint calculations. Keywords for Denovo options in the
importance map block are summarized at the end of this section, in
:numref:`denovo-op`.

Starting with an existing adjoint flux file
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

An importance map can be made from an existing Denovo flux file by using
the keyword “adjointFluxes=” with the appropriate file name in quotes.
The file must be a binary file using the \*.dff file format, and the
number of groups must match the number of groups in the MAVRIC cross
section library (i.e., the library entered on the third line of the
MAVRIC input file). Instead of performing an adjoint calculation, the
fluxes read from the file will be used to create both the mesh-based
importance map and the biased mesh source.

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  read importanceMap
      adjointFluxes=”c:\mydocu~1\previousRun.adjoint.dff”
      gridGeometry=7
  end importanceMap

If the “adjointFluxes=” keyword is used and any adjoint sources are defined, an error will result. If a forward flux file is supplied for forward-weighting the adjoint source (see below), then an adjoint flux file cannot be specified.

The grid geometry is not required when using a pre-existing flux file. If grid geometry is not supplied, one will be created from the mesh planes that are contained in the Denovo flux file (which were used to compute the fluxes in that file).

Forward-weighting the adjoint source
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

To optimize a mesh tally or multiple region tallies/point detector
tallies over a large region, instead of a uniform weighting of the
adjoint source, a weighting based on the inverse of the forward response
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can be performed. This requires an extra discrete-ordinates calculation but
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can help the forward Monte Carlo calculation compute the mesh tally or
group of tallies with more uniform statistical uncertainties.

The same grid geometry will be used in both the forward calculation and
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the adjoint calculation, so the user must ensure that the mesh
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covers all of the forward sources and all of the adjoint sources, even
if they are point sources.

To use forward-weighted CADIS, specify either of the keywords –
“respWeighting” or “fluxWeighting”. For either, MAVRIC will run Denovo
to create an estimate of the forward flux,
:math:`\phi\left( \overrightarrow{r},E \right)`. For response weighting
(“respWeighting”), each adjoint source is inversely weighted by the
integral of the product of the response function used in that adjoint
source and the estimate of the forward flux. For an adjoint source
described by the geometric function :math:`g(\overrightarrow{r})` and
the response function :math:`\sigma_{d}\left( E \right)` (note that
:math:`\sigma_{d}\left( E \right) = 1` for computing total fluxes), the
forward-weighted adjoint source becomes

.. math::
  :label: mavric-19


   q_{i}^{+}\left( \overrightarrow{r},E \right) = \frac{\sigma_{d}\left( E \right)g(\overrightarrow{r})}{\int_{}^{}{\sigma_{d}\left( E \right)\ \phi\left( \overrightarrow{r},E \right)}\ \text{dE}} \ \ .


Response weighting will calculate more uniform relative uncertainties of
the integral quantities of the tallies in the final Monte Carlo
calculation.

To optimize the calculation of the entire group-wise flux with more
uniform relative uncertainties in each group, the adjoint source should
be weighted inversely by the forward flux,
:math:`\phi\left( \overrightarrow{r},E \right),` using the
“fluxWeighting” keyword. For an adjoint source described by the
geometric function :math:`g(\overrightarrow{r})` and the response
function :math:`\sigma_{d}\left( E \right) = 1`, the forward-weighted
adjoint source becomes

.. math::
 :label: mavric-20

 q_{i}^{+}\left( \overrightarrow{r},E \right) = \frac{\sigma_{d}\left( E \right)g(\overrightarrow{r})}{\phi\left( \overrightarrow{r},E \right)}\ .


For example, consider a problem with a single source and two detectors,
one near the source that measures flux and one far from the source that
measures some response. In a standard Monte Carlo calculation, it is
expected that since more Monte Carlo particles cross the near detector,
it will have a much lower relative uncertainty than the far detector.
Standard CADIS could be used to optimize the calculation of each in
separate simulations:

.. list-table::

  * - To optimize the flux in the near detector

    - To optimize the response in the far detector

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  * - .. code:: scale
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        read importanceMap
            gridGeometryID=1
            adjointSource 1
                boundingBox x1 x2 y1 y2 z1 z2
                responseID=1
            end adjointSource
        end importanceMap

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    - .. code:: scale
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        read importanceMap
            gridGeometryID=1
            adjointSource 2
                boundingBox u1 u2 v1 v2 w1 w2
                responseID=6
            end adjointSource
        end importanceMap

where response 1 was defined as :math:`\sigma_{1}\left( E \right) = 1`
and response 6 was defined as :math:`\sigma_{6}\left( E \right) =`
flux-to-response conversion factors. The two options for
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forward weighting allow the tallies for both detectors to be calculated
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in the same MAVRIC simulation. Using “fluxWeighting”, the importance map
and biased source will be made to help distribute Monte Carlo particles
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evenly through each energy group and every voxel in both detectors,
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making the relative uncertainties close to uniform. With
“respWeighting”, the importance map and biased source will optimize the
total integrated response of each tally.

.. list-table::

  * - To optimize :math:`\phi\left( \overrightarrow{r},E \right)` in each detector

    - To optimize a total response :math:`\int_{}^{}{\sigma_{d}\left ( E \right) \phi \left( \overrightarrow{r},E \right)} dE` (either total flux or total dose)

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  * - .. code:: scale
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