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.. _appendixc:

MAVRIC Appendix C: Advanced Features
====================================

This appendix contains information on several advanced features that are
still under development or are non-standard use of the MAVRIC sequence.

Alternate normalization of the importance map and biased source
---------------------------------------------------------------

The importance map and biased source implemented in MAVRIC are only
functions of space and energy. The importance for a specific location
and energy represents the average over all directions. For applications
involving a collimated beam source, a space/energy importance map may
not be representative of the true importance of the particles as they
stream away from the source.

As an example, consider a 14.1 MeV active interrogation beam source 1
meter from a small spherical boat containing illicit nuclear material.
The objective is to compute the fission rate in the nuclear material. To
create the biasing parameters, an adjoint source is located within the
nuclear material and the resulting importance map is shown in
:numref:`fig4c1`. Note that in both the air and water, the importances
change with distance from the ship, but for the beam source, the
importance (to causing a fission in the nuclear material) anywhere along
the beam should be the same, since there is little chance a 14.1 MeV
neutron will interact with the air before striking the ship.

.. _fig4c1:

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.. figure:: figs/MAVRIC/fig41c.png
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  :align: center

  Importance map computed using standard CADIS.

The CADIS algorithm has done exactly what it was supposed to: it made a
space/energy importance map and normalized it such that the target
weight where the 14.1 MeV source particles are born is 1. The problem
with this is that the source particles will stream towards the ship and
strike the hull where the target weight is 0.092. Since source particles
have little chance of interacting in the air, the weight windows are not
used to split the particle as they travel towards the ship. When source
particles cross into the ship, they are split by a factor of 11 to match
the target weight. For this example, splitting each particle by a factor
of 11 once they strike the ship is not so bad, but for longer distances,
this will result in much larger splits. For a polyenergetic source, this
could lead to undersampling of the source and could result in higher
variances.

To remedy this problem when using beam sources, the normalization of the
importance map and biased source should not be done at the source
location but instead at the point where the source particles first
interact with the ship. The keyword shiftNormPos Δx Δy Δz will shift
the source normalization position by the amounts Δx, Δy, and Δz when the
biased source and importance map are developed. For the Monaco Monte
Carlo calculation, the source is returned to its normal position. The
source input for the above problem would then be

.. highlight:: scale

::

  read sources
      src 1
          title="14.1 DT neutrons - collimated"
          strength=1e30
          sphere 0 origin x=-195 y=0 z=0   (true source position)
          eDistributionID=1    (a mono-energetic 14.1 MeV distribution)
          direction 1.0 0.0 0.0
          dDistributionID=2    (a 2° beam )
          shiftNormPos 107.7 0.0 0.0    (just inside the hull)
      end src
  end sources

where the shift moves the source position from x = -195 to x = -87.3,
just inside the hull. The resulting target weights are shown in
:numref:`fig4c2` The source particles are born with weight 1 in a
location with a target weight 10.9. The particle weight is not checked
until the particle crosses into the hull, where the target weight is
1.0.

.. _fig4c2:
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.. figure:: figs/MAVRIC/fig4c2.png
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  :align: center

  Targets weights using the shiftNormPos keyword.

Other options to manipulate the importance map for special situations
include the mapMultiplier=\ *f*\  keyword (in the importanceMap block
or the biasing block), which will multiply every target weight by the
factor *f*, and the keyword noCheckAtBirth in the parameters block
will prevent the weight windows from being applied to source particles
when they are started. When used in the MAVRIC sequence, the
shiftNormPos capability automatically adds noCheckAtBirth to the
Monaco input that is created.

Importance maps with directional information
--------------------------------------------

In MAVRIC, the CADIS method is implemented in space and energy, but in
general, it could also include particle direction as well. This
formulation would be the following:



  True source:
    .. math:: q\left( \overrightarrow{r},E,\widehat{\Omega} \right)

  Desired response:
    .. math:: \sigma\left( \overrightarrow{r},E,\widehat{\Omega}\right)

  Adjoint flux using  :math:`q^{+}\left( \overrightarrow{r},E,\widehat{\Omega} \right) = \ \sigma \left( \overrightarrow{r},E,\widehat{\Omega} \right)`:
    .. math:: \psi^{+}\left( \overrightarrow{r},E,\widehat{\Omega} \right)

  Estimate of detector response
    .. math::
      :label: eq4c1

      R = \iiint_{}^{}{q\left( \overrightarrow{r},E,\widehat{\Omega} \right)\ \psi^{+}\left( \overrightarrow{r},E,\widehat{\Omega}\right)}d\text{Ω } dE \ dV

  Biased source:
    .. math::
      :label: eq4c2

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

  Target weight windows:
    .. math::
      :label: eq4c3

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

For a system using a deterministic method to compute the adjoint fluxes,
this completely general, space/energy/angle, approach presents many
difficulties in implementation, namely,

a. dealing with the amount of memory required for a
   :math:`\left( \overrightarrow{r},E,\widehat{\Omega} \right)`
   importance map in memory,

b. interpolating the importance for particle directions in between
   quadrature angles, and

c. expressing the biased source in a form suitable for a general MC code
   since the above biased source is, in general, not separable.

Approaches incorporating directional information
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Completely general space/energy/angle CADIS is most likely too difficult
to implement and may not be necessary for most applications. In most
real problems that involve directionally dependent source distributions,
the directional dependence is azimuthally symmetric about some reference
direction, :math:`\widehat{d}`. The angular distribution,
:math:`q_{i}\left( \widehat{\Omega} \right)`, can be expressed as the
product of the uniform azimuthal distribution and a polar distribution
about reference direction :math:`{\widehat{d}}_{i}` giving
:math:`\frac{1}{2\pi}q_{i}\left( \widehat{\Omega} \bullet {\widehat{d}}_{i} \right)`.
The geometric size of these sources tends to be small, allowing each
source distribution to be expressed as the product of two separable
distributions:
:math:`q_{i}\left( \overrightarrow{r},E,\widehat{\Omega} \right) \cong q_{i}\left( \overrightarrow{r},E \right)\ q_{i}\left( \widehat{\Omega} \right)`.

What is needed is a CADIS method that (1) can account for the importance
of a particle traveling in a certain direction; (2) can be cast as a
simple modification of the space/energy CADIS method using
:math:`\overline{w}\left( \overrightarrow{r},E \right)` and
:math:`\widehat{q}\left( \overrightarrow{r},E \right)`; and (3) is
simpler than the full space/angle/energy approach. This can be done
starting with the approximation that the angular component of the
adjoint flux
:math:`\psi^{+}\left( \overrightarrow{r},E,\widehat{\Omega} \right)` is
separable and symmetric about the average adjoint current direction
:math:`\widehat{n}\left( \overrightarrow{r},E \right)`, such that

.. math:: \psi^{+}\left( \overrightarrow{r},E,\widehat{\Omega} \right) \cong \phi^{+}\left( \overrightarrow{r},E \right)\ \frac{1}{2\pi}f\left( \widehat{\Omega} \bullet \widehat{n} \right)\text{\ .}

This is similar to the AVATAR approach :cite:`van_riper_avatar_1997` but with explicitly
including the azimuthal distribution so that the standard definition
:math:`\int_{}^{}{\phi^{+}\left( \overrightarrow{r},E \right)\ \frac{1}{2\pi}f\left( \widehat{\Omega} \bullet \widehat{n} \right)\ d\widehat{\Omega}} = \phi^{+}\left( \overrightarrow{r},E \right)`
applies. The probability distribution function
:math:`f\left( \mu \right)` describing the shape of the azimuthally
symmetric current at :math:`\left( \overrightarrow{r},E \right)` has the
form of

.. math:: f\left( \mu \right) = \frac{\lambda e^{\text{λμ}}}{2\ \mathrm{\sinh}\left( \lambda \right)}\ ,

with the single parameter
:math:`\lambda\left( \overrightarrow{r},E \right)` determined from
:math:`\overline{\mu}\left( \overrightarrow{r},E \right)`, the average
cosine of scatter.

From this, we can propose that weight window targets be developed that
are inversely proportional to the approximation of the adjoint angular
flux:

.. math::
  :label: eq4c4

  \overline{w}\left( \overrightarrow{r},E,\widehat{\Omega} \right) = \frac{2\pi\ k}{\phi^{+}\left( \overrightarrow{r},E \right) \ \ f\left( \widehat{\Omega} \bullet \widehat{n} \right)}\ ,

where :math:`k` is the constant of proportionality that will be adjusted
to make the importance map consistent with the biased source(s). Two
methods will be examined here, one without and one with biasing of the
source directional dependence.

For both of the methods, the S\ :sub:`N` code Denovo was modified to
report not only the adjoint scalar fluxes,
:math:`\phi^{+}\left( \overrightarrow{r},E \right)`, but also the
adjoint net currents in :math:`x`, :math:`y`, and :math:`z` directions:
:math:`J_{x}\left( \overrightarrow{r},E \right)`,
:math:`\ J_{y}\left( \overrightarrow{r},E \right)`, and
:math:`J_{z}\left( \overrightarrow{r},E \right)`. These currents are
used to find :math:`\widehat{n}\left( \overrightarrow{r},E \right)` and
:math:`\lambda\left( \overrightarrow{r},E \right)`. The following
methods have been developed so that the standard CADIS routines can be
used to compute space/energy quantities of the response per unit source
:math:`R`, the weight window target values
:math:`\overline{w}\left( \overrightarrow{r},E \right)`, and biased
source :math:`\widehat{q}\left( \overrightarrow{r},E \right)` with just
the adjoint scalar fluxes. These quantities are then modified by the
directional information.

Directionally dependent weight windows without directional source biasing
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

It is proposed that the biased source
:math:`\widehat{q}\left( \overrightarrow{r},E,\widehat{\Omega} \right)`
should be proportional to both the true source distribution and the
space/energy component of the adjoint flux:

.. math:: \widehat{q}\left( \overrightarrow{r},E,\widehat{\Omega} \right) = \frac{1}{R}\left\lbrack q\left( \overrightarrow{r},E \right)\ \frac{1}{2\pi}q\left( \widehat{\Omega} \bullet \widehat{d} \right) \right\rbrack\ \phi^{+}\left( \overrightarrow{r},E \right)\ ,

where the constant of proportionality, :math:`R`, is determined by
forcing
:math:`\widehat{q}\left( \overrightarrow{r},E,\widehat{\Omega} \right)`
to be a pdf. Since the angular component of the adjoint flux is not
included, the directional distribution of the biased source will be
exactly the same as the true source. Note that this approach would be
exact for cases where no directional biasing could be applied  beam
sources.

For multiple sources (each with a probability distribution function
:math:`q_{i}\left( \overrightarrow{r},E \right)` and a strength
:math:`S_{i}`, giving a total source strength of
:math:`S = \sum_{}^{}S_{i}`), the user is required to provide one point
in phase space
:math:`\left( {\overrightarrow{r}}_{i},E_{i},{\widehat{\Omega}}_{i} \right)`
for each source :math:`i` that is representative of that entire source
where the biased source will match the target weight windows. For each
source, a vector
:math:`{\widehat{n}}_{i} = \widehat{n}\left( {\overrightarrow{r}}_{i},E_{i} \right)`
is computed using that point. For the general case of multiple sources,
the biased source sampling distribution, the biased source
distributions, and the weight windows are computed using



:math:`R_{i} =  \iint_{}^{}{q_{i}\left( \overrightarrow{r},E\right)\ \phi^{+}\left( \overrightarrow{r},E \right)} dE \ dr \ \ \ \ \ \ \ \ \  \text{(estimated response from source} \ i)`

:math:`\widehat{p}\left( i \right) = \frac{{S_{i}R}_{i}\ f\left( {\widehat{\Omega}}_{i} \bullet {\widehat{n}}_{i} \right)}{\sum_{}^{}{{S_{i}R}_{i}\ f\left( {\widehat{\Omega}}_{i} \bullet {\widehat{n}}_{i} \right)}} \ \ \ \ \ \ \ \text{(biased sampling of source} \ i)`

:math:`{\widehat{q}}_{i}\left(\overrightarrow{r},E,\widehat{\Omega} \right) \ =  \ \frac{1}{R_{i}}q_{i}\left( \overrightarrow{r},E \right)\ \phi^{+}\left( \overrightarrow{r},E \right)\ \frac{1}{2\pi}q_{i}\left( \widehat{\Omega} \bullet {\widehat{d}}_{i} \right) \ = \ {\widehat{q}}_{i}\left( \overrightarrow{r},E\right)\ \frac{1}{2\pi}q_{i}\left( \widehat{\Omega} \bullet {\widehat{d}}_{i} \right)`

:math:`\overline{w}\left( \overrightarrow{r},E,\widehat{\Omega} \right) \ \ \ =  \ \ \ \frac{\sum_{}^{}{{S_{i}R}_{i}\ f\left( {\widehat{\Omega}}_{i} \bullet {\widehat{n}}_{i} \right)}}{S\phi^{+}\left( \overrightarrow{r},E \right)}\frac{1}{\ f\left( \widehat{\Omega} \bullet \widehat{n} \right)} \ \ \ = \ \ \  \frac{\sum_{}^{}{{S_{i}R}_{i}\ f\left( {\widehat{\Omega}}_{i} \bullet {\widehat{n}}_{i} \right)}}{\sum_{}^{}{S_{i}R}_{i}}\overline{w}\left( \overrightarrow{r},E \right)\frac{1}{f\left( \widehat{\Omega} \bullet \widehat{n} \right)}`

Directionally dependent weight windows with directional source biasing
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Here it is proposed that the biased source be proportional to both the
true source distribution and the approximation of the adjoint angular
flux. With a small geometric source, it is also assumed that there is
one vector,
:math:`{\widehat{n}}_{0} = \widehat{n}\left( {\overrightarrow{r}}_{0},E_{0} \right),`
evaluated at a specific location and energy, which represents the
adjoint current direction over that source. The biased source then looks
like


.. math::

     \widehat{q}\left( \overrightarrow{r},E,\widehat{\Omega} \right) & = \frac{1}{\text{Rc}} q\left( \overrightarrow{r},E,\widehat{\Omega} \right) \ \psi^{+}\left( \overrightarrow{r},E,\widehat{\Omega} \right)

   & = \frac{1}{\text{Rc}}\left\lbrack q\left( \overrightarrow{r},E \right)\ \frac{1}{2\pi}q\left( \widehat{\Omega} \bullet {\widehat{d}}_{i} \right) \right\rbrack\ \left\lbrack \phi^{+}\left( \overrightarrow{r},E \right)\ \frac{1}{2\pi}\ f\left( \widehat{\Omega} \bullet {\widehat{n}}_{0} \right) \right\rbrack\ ,



where the constant :math:`\text{Rc}` is used to make :math:`\widehat{q}`
a pdf. Note that if either the original source directional distribution
:math:`q\left( \widehat{\Omega} \right)` or the adjoint angular flux
distribution at the source is isotropic, then
:math:`c = \frac{1}{4\pi}`.

For the general case of multiple sources, the biased source sampling
distribution, the biased source distributions and the weight windows are


.. math::

  R_{i} = \iint_{}^{}{q_{i}\left( \overrightarrow{r},E\right)\ \phi^{+}\left( \overrightarrow{r},E \right)}\text{dE}\ \text{dr}

.. math::

  c_{i} = \int_{}^{}{\frac{1}{2\pi}q_{i}\left( \widehat{\Omega} \bullet {\widehat{d}}_{i} \right)\ \frac{1}{2\pi}f\left( \widehat{\Omega} \bullet {\widehat{n}}_{i} \right)}d\widehat{\Omega}

.. math::

  \widehat{p}\left( i \right) = \frac{{S_{i}R}_{i}c_{i}}{\sum_{}^{}{{S_{i}R}_{i}c_{i}}}

.. math::
  {\widehat{q}}_{i}\left( \overrightarrow{r},E,\widehat{\Omega} \right) = \left\lbrack \frac{1}{R_{i}}\ q_{i}\left( \overrightarrow{r},E \right)\ \phi^{+}\left( \overrightarrow{r},E \right) \right\rbrack\ \left\lbrack \frac{1}{c_{i}}\ q_{i}\left( \widehat{\Omega} \right)\ f\left( \widehat{\Omega} \right) \right\rbrack = {\widehat{q}}_{i}\left( \overrightarrow{r},E \right)\ \frac{1}{c_{i}}\ \frac{1}{2\pi}q_{i}\left( \widehat{\Omega} \bullet {\widehat{d}}_{i} \right)\ \frac{1}{2\pi}f\left( \widehat{\Omega} \bullet {\widehat{n}}_{i} \right)

.. math::

  \overline{w}\left( \overrightarrow{r},E,\widehat{\Omega} \right) = \frac{\sum_{}^{}{{S_{i}R}_{i}c_{i}}}{S\phi^{+}\left( \overrightarrow{r},E \right)} \ \ \frac{2\pi}{\ f\left( \widehat{\Omega} \bullet \widehat{n} \right)} = \frac{\sum_{}^{}{{S_{i}R}_{i}c_{i}}}{\sum_{}^{}{S_{i}R}_{i}} \ \ \overline{w}\left( \overrightarrow{r},E \right)\ \frac{2\pi}{f\left( \widehat{\Omega} \bullet \widehat{n} \right)} \ .

More details on the development of these methods and their application
for several problems have been presented :cite:`peplow_hybrid_2010,peplow_consistent_2012`.

Using space/energy/angle CADIS in MAVRIC
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The two angular CADIS methods that use the AVATAR-type approximation of
adjoint flux are specified in MAVRIC with the angularBiasing= keyword
in the importanceMap block. Values for this keyword are 1 or 2.

Space/Energy/Angle CADIS without directional biasing (for beam sources)
 This method uses one specific location,
:math:`{\overrightarrow{r}}_{0}`, energy, :math:`E_{0}`, and direction,
:math:`{\widehat{\Omega}}_{0},` which is the reference direction of the
source :math:`\widehat{d}`, where the weight of the biased source
matches the weight window.

Space/Energy/Angle CADIS with directional biasing (for general sources)
 This method uses one specific energy, :math:`E_{0}`, to determine the
adjoint current vector :math:`{\widehat{n}}_{0}` and the
:math:`\lambda_{0}` parameter for the biased angular distribution for
each source.

With each method, the user must specify at what energy the importance
map and the biased sources should be made consistent. The particle type
must also be specified. This is done with the keywords angBiasParType=
(1 for neutron or 2 for photon) and angBiasEnergy= (with a value in
eV), also in the importanceMap block.

Note that all sources should have a direction :math:`\widehat{d}` set,
using direction *u v w*\  within each source, even if the angular
distribution for a given source is isotropic. The direction is used for
source biasing and for aligning the weight windows and biased sources.
Also note that for either angular biasing method, the Denovo S\ :sub:`N`
calculation must use a Legendre order greater than 0.

With angular biasing, a mesh angular information (\*.mai) file is
produced which can be visualized with the MeshFileViewer. This file
contains the space/energy-dependent
:math:`\lambda\left( \overrightarrow{r},E \right)` values and components
of the average adjoint current direction
:math:`\widehat{n}\left( \overrightarrow{r},E \right) = \left\langle n_{x},n_{y},n_{z} \right\rangle`.
An existing mesh angular information (\*.mai) file can be used in a
separate MAVRIC problem by using the meshAngInfoFile= keyword in the
biasing block.

Example problem
~~~~~~~~~~~~~~~

Consider the Ueki shielding problem used as sample problems in the
Monaco and MAVRIC manuals. The goal is to calculate the neutron dose on
one side of a shield from a partially collimated :sup:`252`\ Cf source
on the other side of the shield. Both of the angular approaches
discussed above can be compared to analog and standard space/energy
CADIS calculations. For the analog calculations, no importanceMap block
is used. For the other cases, the importance map blocks are shown below.

.. list-table::
  :align: center

  * - ::

        CADIS

    - ::

        Angular CADIS 1 -
        without a biased source
        angular dist.

    - ::

        Angular CADIS 2 -
        with a biased source
        angular dist.

  * - ::

        read importanceMap
            adjointSource 1
                locationID=1
                responseID=5
            end adjointSource
            gridGeometryID=7
            macromaterial
              mmTolerance=0.01
            end macromaterial



        end importanceMap

    - ::

        read importanceMap
            adjointSource 1
                locationID=1
                responseID=5
            end adjointSource
            gridGeometryID=7
            macromaterial
              mmTolerance=0.01
            end macromaterial

            angularbiasing=1
            angBiasParType=1
            angBiasEnergy=2.0e6
        end importanceMap

    - ::

        read importanceMap
            adjointSource 1
                locationID=1
                responseID=5
            end adjointSource
            gridGeometryID=7
            macromaterial
              mmTolerance=0.01
            end macromaterial

            angularbiasing=2
            angBiasParType=1
            angBiasEnergy=2.0e6
        end importanceMap

Note that the energy at which to tie the importance map to the biased
source, 2 MeV, is about the average energy of the source energy
distribution. The figure-of-merit (FOM) of the calculation could change
as this parameter is varied.

Results, shown in :numref:`tab4c1`, demonstrate that the two directional
approaches improved the FOM for this problem by more than a factor of 2.
A larger boost is seen in method 2 where biasing is applied to the
source directional distribution. In this case, the biased source
distribution was an exponential distribution with a power of 2.45,
significantly sampling more source neutrons in the direction of the
shield than into the paraffin collimator block.

Note that the space/energy/angle CADIS methods require more memory to
hold the importance information. Improvements to the CADIS calculation
can also be made with a judicious choice of standard source direction
biasing, without an increase in memory requirements. A simple
distribution (where :math:`\mu = 0.924` represents the edge of the cone
cutout and :math:`\mu = 0.974` is the center half of the shield
dimension) added that to the source

::

    distribution 2
        abscissa -1.0     0.924    .974   1.0 end
        truepdf      0.962     0.025   .013   end
        biasedpdf    0.500     0.250  0.250   end
    end distribution

    src 1
        neutrons  strength=4.05E+07
        cuboid  0.01 0.01  0 0  0 0
        eDistributionID=1
        direction 1.0 0.0 0.0
        dDistributionID=2
    end src

results in an FOM improvement of nearly 3 over the standard CADIS without the overhead of the angular CADIS methods.


.. list-table:: Results of the Ueki Shielding Problem (35 cm graphite shield)
  :align: center
  :name: tab4c1

  * - Calculation Method
    - Adj. S\ :sub:`N` (min)
    - MC (min)
    - dose rate (rem/hr)
    - relative
      uncert
    - MC FOM
      (/min)
  * - Analog
    - 0.0
    - 152.7
    - 3.998E-03
    - 0.0101
    - 64.7
  * - CADIS
    - 0.2
    - 9.9
    - 3.998E-03
    - 0.0081
    - 1550
  * - Directional CADIS 1,
      no src bias
    - 0.2
    - 10.3
    - 4.035E-03
    - 0.0054
    - 3390
  * - Directional CADIS 2,
      with src bias
    - 0.2
    - 10.0
    - 4.012E-03
    - 0.0049
    - 4190
  * - CADIS, standard src
      dir. bias
    - 0.2
    - 10.0
    - 3.998E-03
    - 0.0047
    - 4550

A series of problems was used to compare the space/energy/angle CADIS to
the standard space/energy CADIS. Most of the problems saw an improvement
of a factor of about 2 or 3. Some problems did not improve at all, and
some photon problems actually performed worse. In that case (a photon
litho-density gauge), it could be that the angular approximation for
importance, an exponential function in :math:`\mu`, cannot adequately
describe the true importance.

Since the space/energy/angle CADIS methods are more difficult to explain
(more theory, adjustable parameters set by the user), require more
memory than standard CADIS, and may not offer any improvement over
standard space/energy CADIS, they have not been made part of the main
MAVRIC manual and have been left as an advanced/developing feature.
These methods were not removed from the MAVRIC code since they may be
helpful to future problems.

University of Michigan methods for global variance reduction
------------------------------------------------------------

The use of hybrid deterministic/Monte Carlo methods, particularly for
global variance reduction, has been an active area of research by the
transport team at the University of Michigan for a long time. One of the
first approaches studied was a way to develop Monte Carlo weight window
target values that were proportional to deterministically estimated
values of the forward flux :cite:`cooper_automated_2001`. For global problems using isotropic
weight windows, this reference argues that in order to get uniform
relative uncertainties in the Monte Carlo calculation, the weight
windows should be set such that the number density of Monte Carlo
particles, :math:`m\left( \overrightarrow{r} \right)`, is constant. The
physical particle density, :math:`n\left( \overrightarrow{r} \right)`,
is related by the average weight,
:math:`\overline{w}\left( \overrightarrow{r} \right)`, to the Monte
Carlo particle density by

.. math:: n\left( \overrightarrow{r} \right) = \overline{w}\left( \overrightarrow{r} \right)\ m\left( \overrightarrow{r} \right) \ .

To make :math:`m\left( \overrightarrow{r} \right)` constant over the
geometry, the weight window targets
:math:`\overline{w}\left( \overrightarrow{r} \right)` need to be
proportional to the physical particle density. Cooper and Larsen
calculate the weight window targets from an estimate of the forward
scalar flux :math:`\phi\left( \overrightarrow{r} \right)` to be

.. math:: \overline{w}\left( \overrightarrow{r} \right) = \frac{\phi\left( \overrightarrow{r} \right)}{\mathrm{\max}\left( \phi\left( \overrightarrow{r} \right) \right)} \ .

Two approaches have since been developed :cite:`becker_application_2009`, based on what global
information the user desires from the simulation: global flux weight
windows, for obtaining every energy group at every location, and global
response weight windows, for obtaining an energy-integrated response at
every spatial location. Both of these methods are designed for
calculating the global solution  everywhere in the geometry of the
problem  with nearly uniform statistics.

Note that none of the University of Michigan methods discussed here
included the development of biased sources. These methods have all been
extended to include a consistent biased source by ORNL during their
implementation in the MAVRIC sequence of SCALE. The methods have also
been extended by ORNL for multiple sources.

Weight windows using only forward estimates of flux
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Global flux weight windows
^^^^^^^^^^^^^^^^^^^^^^^^^^

This method keeps the Monte Carlo particle distribution uniform in space
and energy. Note that this is a space/energy version of the original
space-only Coopers Method. The target weight windows,
:math:`\overline{w}\left( \overrightarrow{r},E \right)`, should be
proportional to the estimate of the forward scalar flux,
:math:`\phi\left( \overrightarrow{r},E \right)`.

.. math:: \overline{w}\left( \overrightarrow{r},E \right) = c\ \phi\left( \overrightarrow{r},E \right) \ .

A biased source distribution,
:math:`\widehat{q}\left( \overrightarrow{r},E \right)`, that is
consistent with the target weight windows can be found from the true
source distribution, :math:`q\left( \overrightarrow{r},E \right)`, and
the forward flux to be

.. math:: \widehat{q}\left( \overrightarrow{r},E \right) = \ \frac{1}{c}\ \frac{q\left( \overrightarrow{r},E \right)}{\phi\left( \overrightarrow{r},E \right)}\ ,

where the constant *c* can be determined so that the biased source
distribution is a probability distribution function,

.. math:: c = \int_{V}^{}{\int_{E}^{}{\ \frac{q\left( \overrightarrow{r},E \right)}{\phi\left( \overrightarrow{r},E \right)}\ \text{dE}\ d\overrightarrow{r}}} \ .

Thus, using the estimate of forward flux, we can first compute *c* and
then form a consistent set of weight window target values and a biased
source distribution.

Global response weight windows
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

For an energy-integrated response (such as dose) desired at all spatial
locations, this method keeps the Monte Carlo particle distribution
proportional to the integral of the product of the response function,
:math:`\sigma_{d}\left( \overrightarrow{r},E \right)`, and the estimate
of the forward flux, :math:`\phi\left( \overrightarrow{r},E \right)`.
The energy-integrated response,
:math:`D\left( \overrightarrow{r} \right)`, is found from the estimate
of the forward flux as

.. math:: D\left( \overrightarrow{r} \right) = \int_{E}^{}{\sigma_{d}\left( \overrightarrow{r},E \right)\ \phi\left( \overrightarrow{r},E \right) DE } .

The target weight windows,
:math:`\overline{w}\left( \overrightarrow{r},E \right)`, should be

.. math:: \overline{w}\left( \overrightarrow{r},E \right) = c\ \frac{D\left( \overrightarrow{r} \right)}{\sigma_{d}\left( \overrightarrow{r},E \right)} \ .

A biased source distribution,
:math:`\widehat{q}\left( \overrightarrow{r},E \right)`, that is
consistent with the target weight windows can be found from the true
source distribution, :math:`q\left( \overrightarrow{r},E \right)`, and
the forward dose estimate to be

.. math:: \widehat{q}\left( \overrightarrow{r},E \right) = \ \frac{1}{c}\ \frac{q\left( \overrightarrow{r},E \right)\ \sigma_{d}\left( \overrightarrow{r},E \right)}{D\left( \overrightarrow{r} \right)}

using a proportionality constant of

.. math:: c = \int_{V}^{}{\frac{1}{D\left( \overrightarrow{r} \right)}\int_{E}^{}{q\left( \overrightarrow{r},E \right)\ \sigma_{d}\left( \overrightarrow{r},E \right)\ \text{dE}\ d\overrightarrow{r}}}\text{\ .}

Implementation in MAVRIC
^^^^^^^^^^^^^^^^^^^^^^^^

The global flux weight windows (GFWW) approach and the global response
weight windows (GRWW) are both triggered by specifying an importance map
block without any adjoint sources. For the GRWW approach, the response
of interest is listed in the importance map block. If none is listed,
GFWW is used.

For problems with multiple sources (each with probability distribution
function :math:`q_{i}\left( \overrightarrow{r},E \right)` and strength
:math:`S_{i}`, giving a total source strength of
:math:`S = \sum_{}^{}S_{i}`), the GWW methods require a biased mesh
source for each. This was implemented in MAVRIC in a manner similar to
the multiple source CADIS routines. Each biased source,
:math:`{\widehat{q}}_{i}\left( \overrightarrow{r},E \right)`, is
developed as


.. math::

   \widehat{q}_{i}\left( \vec{r},E \right) =
   \begin{cases}

      \frac{1}{c_{i}} \frac{q_{i}\left( \vec{r},E \right)}{\phi\left( \vec{r},E \right)}   \ \ \ \ \ & \text{global flux weight windows} \\
      \frac{1}{c_{i}} \frac{q_{i}\left( \vec{r},E \right) \sigma_{d} \left( \vec{r},E \right)}{D \left( \vec{r}\right)} \ \ \ & \text{global response weight windows}

   \end{cases}

where :math:`c_{i}` is a normalization constant. The weight windows are
then set to

.. math::

  \overline{w}\left( \overrightarrow{r},E \right) = \left\{ \begin{matrix}
  \frac{\sum_{}^{}c_{i}}{\sum_{}^{}S_{i}}\ \phi\left( \overrightarrow{r},E \right) & \text{global flux weight windows} \\
  \frac{\sum_{}^{}c_{i}}{\sum_{}^{}S_{i}}\ \frac{D\left( \overrightarrow{r} \right)}{\sigma_{d}\left( \overrightarrow{r},E \right)} & \text{global response weight windows} \\
  \end{matrix} \right.\ \ \ \ .

In the final Monte Carlo, the specific source *i* is sampled with
probability :math:`p\left( i \right) = \ S_{i}/S`, and then the particle
is sampled from the biased mesh source
:math:`{\widehat{q}}_{i}\left( \overrightarrow{r},E \right)`. Unlike the
CADIS method for multiple sources, there is no way to develop a biased
probability distribution for which source to sample without knowing the
contribution to the global estimate from each source separately.

For multiple source problems where the expected contribution from each
source is very different from the true strengths of those sources, it
may be more efficient to run a series of problems with one source each
for different amounts of time. The resulting mesh tallies can then be
added together using the mesh tally adder (part of the MAVRIC
utilities).

Methods using forward and adjoint estimates
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Becker :cite:`becker_hybrid_2009` proposed three methods for developing weight windows based
on both forward and adjoint deterministic solutions. These three methods
correspond to the portion of the phase space over which uniform relative
uncertainties are desired: a small detector region, a region
comprising a significant portion of the entire problem, and the global
problem. In this discussion, only a brief outline of each method,
focusing on its implementation into MAVRIC, will be given.

Source/detector problems
^^^^^^^^^^^^^^^^^^^^^^^^

For a small detector of volume :math:`V_{D}`\ where we want to optimize
the MC calculation of the detector response

.. math:: R = \int_{V_{D}}^{}{\int_{0}^{\infty}{\sigma\left( \overrightarrow{r},E \right)\ \phi\left( \overrightarrow{r},E \right)}\text{dE}\ \text{dV}}

or optimize for the energy dependent flux at the detector, the following
is used:



    forward flux estimate
      .. math::
        :label: eq4c5

        \phi\left( \overrightarrow{r},E \right)

    adjoint source for flux
      .. math:: q^{+}\left( \overrightarrow{r},E \right) = \frac{1}{\phi\left( \overrightarrow{r},E \right)}

    or for response
      .. math:: q^{+}\left( \overrightarrow{r},E \right) = \sigma\left( \overrightarrow{r},E \right)

    adjoint flux
      .. math:: \phi^{+}\left( \overrightarrow{r},E \right)

    contributon flux
      .. math:: \phi^{c}\left( \overrightarrow{r},E \right) = \phi\left( \overrightarrow{r},E \right)\ \phi^{+}\left( \overrightarrow{r},E \right)

    normalization constant
      .. math:: C_{\mathrm{\text{norm}}} = \frac{V_{D}}{\int_{V_{D}}^{}{\int_{0}^{\infty}{\phi^{c}\left( \overrightarrow{r},E \right)} dE} \ dV}

    space-only contributon flux
      .. math:: {\widetilde{\phi}}^{c}\left( \overrightarrow{r} \right) = \ C_{\mathrm{\text{norm}}}\int_{0}^{\infty}{\phi^{c}\left( \overrightarrow{r},E \right)} dE

    spatial parameter
      .. math:: \alpha\left( \overrightarrow{r} \right) = \left\lbrack 1 + exp\left( \frac{{\widetilde{\phi}}_{\mathrm{\max} \in V_{D}}^{c}}{{\widetilde{\phi}}^{c}\left( \overrightarrow{r} \right)} - \frac{{\widetilde{\phi}}^{c}\left( \overrightarrow{r} \right)}{{\widetilde{\phi}}_{\mathrm{\max} \in V_{D}}^{c}} \right) \right\rbrack^{- 1}

    spatial parameter
      .. math:: B\left( \overrightarrow{r} \right) = \ \alpha\left( \overrightarrow{r} \right){\widetilde{\phi}}^{c}\left( \overrightarrow{r} \right) + \ 1 - \ \alpha\left( \overrightarrow{r} \right)

    weight windows
      .. math::
        :label: eq4c6

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

Source-region problems
^^^^^^^^^^^^^^^^^^^^^^

For a detector of volume :math:`V_{D}` and surface area :math:`A_{D}`
(smaller than the entire problem) where we want to optimize the MC
calculation of the detector response

.. math:: R\left( \overrightarrow{r} \right) = \int_{0}^{\infty}{\sigma\left( \overrightarrow{r},E \right)\ \phi\left( \overrightarrow{r},E \right)} dE \ \ \ \ \ \ \ \ \ \overrightarrow{r} \in V_{D}

or optimize for the energy dependent flux in the region, the following
is used:


    forward flux estimate
      .. math::
        :label: eq4c7

        \phi\left( \overrightarrow{r},E \right)

    adjoint source for flux
      .. math:: q^{+}\left( \overrightarrow{r},E \right) = \frac{1}{\phi\left( \overrightarrow{r},E \right)}

    adjoint source for response
      .. math:: q^{+}\left( \overrightarrow{r},E \right) = \frac{\sigma\left( \overrightarrow{r},E \right)}{\int_{0}^{\infty}{\sigma\left( \overrightarrow{r},E \right)\ \phi\left( \overrightarrow{r},E \right)} dE}

    adjoint flux estimate
      .. math:: \phi^{+}\left( \overrightarrow{r},E \right)

    contributon flux
      .. math:: \phi^{c}\left( \overrightarrow{r},E \right) = \phi\left( \overrightarrow{r},E \right)\ \phi^{+}\left( \overrightarrow{r},E \right)

    normalization constant
      .. math:: C_{\mathrm{\text{norm}}} = \frac{A_{D}}{\int_{A_{D}}^{}{\int_{0}^{\infty}{\phi^{c}\left( \overrightarrow{r},E \right)} dE} \ dA}

    space-only contributon flux
      .. math:: {\widetilde{\phi}}^{c}\left( \overrightarrow{r} \right) = \ C_{\mathrm{\text{norm}}}\int_{0}^{\infty}{\phi^{c}\left( \overrightarrow{r},E \right)} dE

    spatial parameter
      .. math:: \alpha\left( \overrightarrow{r} \right) = \left\lbrack 1 + exp\left( \frac{{\widetilde{\phi}}_{\mathrm{\max} \in V_{D}}^{c}}{{\widetilde{\phi}}^{c}\left( \overrightarrow{r} \right)} - \frac{{\widetilde{\phi}}^{c}\left( \overrightarrow{r} \right)}{{\widetilde{\phi}}_{\mathrm{\max} \in V_{D}}^{c}} \right) \right\rbrack^{- 1}

    spatial parameter
      .. math::
        :label: eq4c8

         B\left( \vec{r} \right) =
         \begin{cases}

            \tilde{\phi^{c}}\left(\vec{r}\right) & \vec{r} \in V_{D}

            \alpha\left(\vec{r}\right)\tilde{\phi^{c}}\left(\vec{r}\right) + 1 - \alpha\left(\vec{r}\right) & \text{otherwise}

         \end{cases}

    weight windows
      .. math:: \overline{w}\left( \overrightarrow{r},E \right) = \frac{B\left( \overrightarrow{r} \right)}{\phi^{+}\left( \overrightarrow{r},E \right)}

Note that :math:`A_{D}` does not include surfaces of :math:`V_{D}` which
are on the boundary of the problem.

Global response problem
^^^^^^^^^^^^^^^^^^^^^^^

For optimizing the Monte Carlo calculation of a detector response
everywhere in phase space

.. math:: R(\overrightarrow{r}) = \int_{0}^{\infty}{\sigma\left( \overrightarrow{r},E \right)\ \phi\left( \overrightarrow{r},E \right)}\text{dE}

or optimizing for the energy-dependent flux everywhere, the following is
used:



    forward flux estimate
      .. math::
        :label: eq4c9

        \phi\left( \overrightarrow{r},E \right)

    adjoint source for flux
      .. math:: q^{+}\left( \overrightarrow{r},E \right) = \frac{1}{\phi\left( \overrightarrow{r},E \right)}

    adjoint source for response
      .. math:: q^{+}\left( \overrightarrow{r},E \right) = \frac{\sigma\left( \overrightarrow{r},E \right)}{\int_{0}^{\infty}{\sigma\left( \overrightarrow{r},E \right)\ \phi\left( \overrightarrow{r},E \right)} dE }

    adjoint flux estimate
      .. math:: \phi^{+}\left( \overrightarrow{r},E \right)

    contributon flux
      .. math:: \phi^{c}\left( \overrightarrow{r},E \right) = \phi\left( \overrightarrow{r},E \right)\ \phi^{+}\left( \overrightarrow{r},E \right)

    space-only contributon flux
      .. math:: \phi^{c}\left( \overrightarrow{r} \right) = \int_{0}^{\infty}{\phi^{c}\left( \overrightarrow{r},E \right)} dE

    spatial parameter
      .. math:: B\left( \overrightarrow{r} \right) = \phi^{c}\left( \overrightarrow{r} \right)

    weight windows
      .. math:: \overline{w}\left( \overrightarrow{r},E \right) = \frac{B\left( \overrightarrow{r} \right)}{\phi^{+}\left( \overrightarrow{r},E \right)}

Implementation in MAVRIC
~~~~~~~~~~~~~~~~~~~~~~~~

Like CADIS and FW-CADIS, the Denovo S\ :sub:`N` code is used to
calculate the forward flux estimate,
:math:`\phi\left( \overrightarrow{r},E \right)`, and the estimate of the
adjoint flux, :math:`\phi^{+}\left( \overrightarrow{r},E \right)`, for
all of the Michigan weight window methods.

None of the above discussions of the University of Michigan methods
include information on how the weight window target values were adjusted
to match the source sampling. When implemented into MAVRIC, each of the
above problem types will compute a biased source,
:math:`\widehat{q}\left( \overrightarrow{r},E \right)`, along with the
target weight, :math:`\overline{w}\left( \overrightarrow{r},E \right)`,
that are produced. For a problem with a single source of strength
:math:`S` and distribution :math:`q\left( \overrightarrow{r},E \right)`,
the biased source
distribution\ :math:`\ \widehat{q}\left( \overrightarrow{r},E \right)`
is found by using

.. math:: \widehat{q}\left( \overrightarrow{r},E \right) = \frac{q\left( \overrightarrow{r},E \right)}{\overline{w}\left( \overrightarrow{r},E \right)}\ .

The weight windows are multiplied by a factor of :math:`R/S`, where
:math:`R` is defined as

.. math:: R = \iint_{}^{}{\widehat{q}\left( \overrightarrow{r},E \right)}\phi^{+}\left( \overrightarrow{r},E \right)\ dE \ dV .

Sampled source particles will then be born with a weight that matches
the weight window of the phase space where they are born.

For multiple sources, each with strength :math:`S_{i}` and distribution
:math:`q_{i}\left( \overrightarrow{r},E \right)`, each biased source
distribution\ :math:`\ {\widehat{q}}_{i}\left( \overrightarrow{r},E \right)`
is found by using

.. math:: {\widehat{q}}_{i}\left( \overrightarrow{r},E \right) = \frac{q_{i}\left( \overrightarrow{r},E \right)}{\overline{w}\left( \overrightarrow{r},E \right)}

and the response from each source being

.. math:: R_{i} = \iint_{}^{}{{\widehat{q}}_{i}\left( \overrightarrow{r},E \right)}dE \ dV \ .

The individual sources are sampled with a biased probability of
:math:`\widehat{p}\left( i \right) = \ R_{i}/\sum_{}^{}R_{i}`. The
weight windows are then multiplied by a factor of

.. math:: \frac{\sum_{i}^{}R_{i}}{\sum_{i}^{}S_{i}}

to match the source birth weights.

To use one of the Becker methods in MAVRIC, the keyword beckerMethod=
is used with values of 1, 2, or 3 for the source/detector,
source/region, or global method. Adjoint sources are described just like
standard MAVRIC CADIS and FW-CADIS problems. To switch between
optimizing flux in every group or optimizing a response, the keywords
fluxWeighting and respWeighting are used. Just like FWCADIS, the
response listed in each adjoint source is the response that is
optimized. Note that even when starting a calculation with known forward
and adjoint flux files, the adjoint source(s) still need to be listed
since they are used in the final normalization of the weight windows.

Example problems
^^^^^^^^^^^^^^^^

The first Becker method (source/detector) is demonstrated using the Ueki
shielding problem (used as sample problems in the Monaco and MAVRIC
manuals and above in the space/energy/angle CADIS example). The goal is
to calculate the neutron dose on one side of a shield from a partially
collimated :sup:`252`\ Cf source on the other side of the shield. For
the analog calculations, no importanceMap block is used. For the other
cases, the importance map blocks are shown below.

.. list-table::
  :align: center
  :width: 80

  * - ::

        CADIS

    - ::

        Becker 1  flux optimization

    - ::

        Becker 2  response optimization

  * - ::

        read importanceMap
            adjointSource 1
                locationID=1
                responseID=5
            end adjointSource
            gridGeometryID=7
            macromaterial
              mmTolerance=0.01
            end macromaterial



        end importanceMap

    - ::

        read importanceMap
            adjointSource 1
                locationID=1
                responseID=5
            end adjointSource
            gridGeometryID=7
            macromaterial
              mmTolerance=0.01
            end macromaterial

            beckerMethod=1
            fluxWeighting
        end importanceMap

    - ::

        read importanceMap
            adjointSource 1
                locationID=1
                responseID=5
            end adjointSource
            gridGeometryID=7
            macromaterial
              mmTolerance=0.01
            end macromaterial

            beckerMethod=1
            respWeighting
        end importanceMap