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Random Ray Adjoint Mode (#3191)

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Co-authored-by: Paul Romano <paul.k.romano@gmail.com>
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45
docs/source/methods/random_ray.rst
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@ -542,7 +542,7 @@ in that cell for the iteration from Equation :eq:`phi_naive` to:
.. math::
:label: phi_missed_one
\phi_{i,g,n}^{missed} = \frac{Q_{i,g,n} }{\Sigma_{t,i,g}}
\phi_{i,g,n}^{missed} = \frac{Q_{i,g,n} }{\Sigma_{t,i,g}}
as the streaming operator has gone to zero. While this is obviously innacurate
as it ignores transport, for most problems where the region is only occasionally
@ -1060,6 +1060,49 @@ random ray and Monte Carlo, however.
develop the scattering source by way of inactive batches before beginning
active batches.
------------------------
Adjoint Flux Solver Mode
------------------------
The random ray solver in OpenMC can also be used to solve for the adjoint flux,
:math:`\psi^{\dagger}`. In combination with the regular (forward) flux solution,
the adjoint flux is useful for perturbation methods as well as for computing
weight windows for subsequent Monte Carlo simulations. The adjoint flux can be
thought of as the "backwards" flux, representing the flux where a particle is
born at an absoprtion point (and typical absorption energy), and then undergoes
transport with a transposed scattering matrix. That is, instead of sampling a
particle and seeing where it might go as in a standard forward solve, we will
sample an absorption location and see where the particle that was absorbed there
might have come from. Notably, for typical neutron absorption at low energy
levels, this means that adjoint flux particles are typically sampled at a low
energy and then upscatter (via a transposed scattering matrix) over their
lifetimes.
In OpenMC, the random ray adjoint solver is implemented simply by transposing
the scattering matrix, swapping :math:`\nu\Sigma_f` and :math:`\chi`, and then
running a normal transport solve. When no external fixed source is present, no
additional changes are needed in the transport process. However, if an external
fixed forward source is present in the simulation problem, then an additional
step is taken to compute the accompanying fixed adjoint source. In OpenMC, the
adjoint flux does *not* represent a response function for a particular detector
region. Rather, the adjoint flux is the global response, making it appropriate
for use with weight window generation schemes for global variance reduction.
Thus, if using a fixed source, the external source for the adjoint mode is
simply computed as being :math:`1 / \phi`, where :math:`\phi` is the forward
scalar flux that results from a normal forward solve (which OpenMC will run
first automatically when in adjoint mode). The adjoint external source will be
computed for each source region in the simulation mesh, independent of any
tallies. The adjoint external source is always flat, even when a linear
scattering and fission source shape is used. When in adjoint mode, all reported
results (e.g., tallies, eigenvalues, etc.) are derived from the adjoint flux,
even when the physical meaning is not necessarily obvious. These values are
still reported, though we emphasize that the primary use case for adjoint mode
is for producing adjoint flux tallies to support subsequent perturbation studies
and weight window generation.
Note that the adjoint :math:`k_{eff}` is statistically the same as the forward
:math:`k_{eff}`, despite the flux distributions taking different shapes.
---------------------------
Fundamental Sources of Bias
---------------------------