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 an importance can be assigned to different particles based on how much they will contribute to the final answer, 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 together—particles 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 then uses the importance map for biasing during particle transport and 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.

While 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 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 Appendix C.

CADIS Methodology

-----------------

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. In order 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, source particles will undergo Russian roulette or splitting immediately, wasting computational time and negating the intent of the biasing.

Overview of CADIS

~~~~~~~~~~~~~~~~~

CADIS has been well described in the literature, so only a

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

function and the flux over the detector volume :math:`V_{d}`.

.. 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

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 where multiple tallies need to all be found with low statistical uncertainties. The extension of this idea is the mesh tally—where each voxel is a tally where 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. The drawback to this approach is the difficult question of “how close.” If the tallies are too far apart, 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 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, 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.

Forward-weighted CADIS

~~~~~~~~~~~~~~~~~~~~~~

In order to converge several tallies to the same relative uncertainty in

one simulation, the adjoint source corresponding to each of those

tallies needs to be weighted inversely by the expected tally value. In

order 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 needs to

have two parts. The adjoint source far from the reactor needs to have

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.

This concept can be extended to mesh tallies as well. Instead of using a

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

method: forward-weighted CADIS (FW-CADIS).**Error! Bookmark not

defined.** First, a forward S\ :sub:`N` calculation is performed to

estimate the expected tally results. A total adjoint source is

constructed where the adjoint source corresponding to each tally is

weighted inversely by those forward tally estimates. Then the standard

CADIS approach is used—an 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

q^{+}\left( \overrightarrow{r},E \right) = \frac{\sigma_{d}\left( E \right)\text{g}\left( \overrightarrow{r} \right)}{\int_{}^{}{\sigma_{d}\left( E \right)\text{ϕ}\left( \overrightarrow{r},E \right)}\textit{dE}}\ ,

where :math:`\phi\left( \overrightarrow{r},E \right)` is an estimate of

the forward flux and the energy integral is over the voxel at :math:`\overrightarrow{r}`.

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:

first, the number of tracks contributing to the tally and, second, the

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

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school={Pennsylvania State University},

author={Wagner, John C.},

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file={Full Text:/Users/john/Zotero/storage/VHWL9S7A/Wagner - 1997 - Acceleration of Monte Carlo shielding calculations.pdf:application/pdf}

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doi={10.13182/NT171-171},

abstract={Denovo is a new, three-dimensional, discrete ordinates (SN) transport code that uses state-of-the-art solution methods to obtain accurate solutions to the Boltzmann transport equation. Denovo uses the Koch-Baker-Alcouffe parallel sweep algorithm to obtain high parallel efficiency on O(100) processors on XYZ orthogonal meshes. As opposed to traditional SN codes that use source iteration, Denovo uses nonstationary Krylov methods to solve the within-group equations. Krylov methods are far more efficient than stationary schemes. Additionally, classic acceleration schemes (diffusion synthetic acceleration) do not suffer stability problems when used as a preconditioner to a Krylov solver. Denovo’s generic programming framework allows multiple spatial discretization schemes and solution methodologies. Denovo currently provides diamond-difference, theta-weighted diamond-difference, linear-discontinuous finite element, trilinear-discontinuous finite element, and step characteristics spatial differencing schemes. Also, users have the option of running traditional source iteration instead of Krylov iteration. Multigroup upscatter problems can be solved using Gauss-Seidel iteration with transport, two-grid acceleration. A parallel first-collision source is also available. Denovo solutions to the Kobayashi benchmarks are in excellent agreement with published results. Parallel performance shows excellent weak scaling up to 20000 cores and good scaling up to 40000 cores.},

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urldate={2020-06-26},

journal={Nuclear Technology},

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