From 5d91f102b307642c917a48989f901852d5c7f66d Mon Sep 17 00:00:00 2001 From: John Tramm Date: Thu, 23 Jul 2026 02:36:56 +0000 Subject: [PATCH 1/9] Fix a source-sampling inconsistency in the linear source flux update The transport sweep evaluates each region's linear source against the accumulated centroid, but a batch's tracks average that source at their own track-length-weighted centroid, which fluctuates about the accumulated one as rays sample the region unevenly. The flux update added back only the flat source, so the difference -- the source gradient dotted with the per-batch centroid fluctuation -- was silently absorbed by the flux estimate as zero-mean noise with no flat-source counterpart. This broke, in linear source mode only, the defining property of a naive-volume update: that it adds no noise of its own beyond the angular fluxes it averages (its error being ratio bias alone). In near-cancellation regions (scattering ratio near one), where the reduced source approximately equals the flux, the absorbed noise can exceed the flux itself at modest hit counts and ignite self-sustaining negativity that should be structurally impossible. The update now adds back the source as actually integrated by the batch's tracks, weighted by the ratio of the batch volume to the volume in use. Under the batch (naive) volume the weight is one and the update becomes an exact per-batch track identity, so the flux inherits the sign of the angular fluxes; under the simulation-averaged volume the weight prevents the correlation between batch volume and batch centroid (sampled by the same rays) from becoming a systematic gradient-aligned bias in cut regions. Flat source solves are unaffected. On a fixed-source shielding reproducer (near-void hall with scattering ratio 0.99, naive estimator, 2,000 rays), the fix reduces negative flux bins from 5,788 (amplitudes up to 70% of the peak flux) to 21 (0.05% of peak). Linear source test golds are regenerated, and a new regression test pins the previously unstable naive-plus-linear regime. Co-Authored-By: Claude Fable 5 --- docs/source/methods/random_ray.rst | 46 +++ include/openmc/random_ray/source_region.h | 11 + src/random_ray/linear_source_domain.cpp | 44 ++- src/random_ray/source_region.cpp | 4 + .../results_true.dat | 2 +- .../linear/results_true.dat | 12 +- .../linear_xy/results_true.dat | 12 +- .../linear/results_true.dat | 12 +- .../linear_xy/results_true.dat | 168 ++++----- .../random_ray_linear/linear/results_true.dat | 338 +++++++++--------- .../linear_xy/results_true.dat | 338 +++++++++--------- .../__init__.py | 0 .../inputs_true.dat | 258 +++++++++++++ .../results_true.dat | 9 + .../test.py | 35 ++ .../random_ray_void/linear/results_true.dat | 12 +- .../hybrid/results_true.dat | 12 +- .../naive/results_true.dat | 12 +- .../simulation_averaged/results_true.dat | 12 +- 19 files changed, 871 insertions(+), 466 deletions(-) create mode 100644 tests/regression_tests/random_ray_linear_source_stability/__init__.py create mode 100644 tests/regression_tests/random_ray_linear_source_stability/inputs_true.dat create mode 100644 tests/regression_tests/random_ray_linear_source_stability/results_true.dat create mode 100644 tests/regression_tests/random_ray_linear_source_stability/test.py diff --git a/docs/source/methods/random_ray.rst b/docs/source/methods/random_ray.rst index 8bc2a0a1bf5..9410984acff 100644 --- a/docs/source/methods/random_ray.rst +++ b/docs/source/methods/random_ray.rst @@ -999,6 +999,52 @@ The contents of this section, alongside the equations for the flat source and scalar flux, Equations :eq:`source_update` and :eq:`phi_sim` respectively, completes the set of equations for LS. +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +Consistency of the Scalar Flux Estimate +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +One subtlety of the linear source scheme deserves explicit treatment. During +a batch, the transport sweep evaluates the linear source of region :math:`i` +against the accumulated centroid estimate :math:`\mathbf{C}_i`, so every +segment's attenuation calculation subtracts the local source value +:math:`\bar{Q}_i + \vec{Q}_i \cdot (\mathbf{m}_s - \mathbf{C}_i)` at that +segment's midpoint :math:`\mathbf{m}_s`. Because the chord integral of a +linear function is exactly its midpoint value times the chord length, +summing the segment balance equations over a batch's tracks shows that the +tracks integrate a mean emission of + +.. math:: + :label: batch_sampled_source + + \bar{Q}_i + \vec{Q}_i \cdot (\mathbf{c}_{i,b} - \mathbf{C}_i)\;, + +where :math:`\mathbf{c}_{i,b}` is the batch's track-length-weighted centroid, +which fluctuates about :math:`\mathbf{C}_i` from batch to batch as the rays +sample the region unevenly. If the scalar flux update adds back only +:math:`\bar{Q}_i`, the difference -- the source gradient dotted with the +per-batch centroid fluctuation -- is silently absorbed by the flux estimate +as a zero-mean noise term that has no counterpart in flat source mode (a +constant source has the same average over any set of tracks). When a region +is updated with its own batch (iteration) volume, this term is the only +noise the update itself contributes: adding back the full batch-sampled +source of Equation :eq:`batch_sampled_source` makes the update an exact +per-batch track identity, in which the scalar flux estimate equals the +track-length average of the angular flux and therefore inherits its sign. +Omitting the term instead breaks, in linear source mode only, the property +that a naive-volume update is exact given its rays -- and in +near-cancellation regions (scattering ratios near unity, as in the +air-filled regions of shielding problems), where the reduced source +approximately equals the scalar flux, the absorbed noise can exceed the flux +itself at modest per-batch hit counts and ignite self-sustaining negativity +through the scattering feedback. When a region is instead updated with the +simulation-averaged volume, the batch-sampled gradient term enters the +transport sum weighted by the ratio of the batch volume to the +simulation-averaged volume, and the update adds it back with the same +weight; the weight matters because the batch volume and batch centroid are +sampled by the same rays, and an unweighted term would convert their +correlation into a systematic, gradient-aligned bias in regions cut by +geometry. + .. _methods-shannon-entropy-random-ray: ----------------------------- diff --git a/include/openmc/random_ray/source_region.h b/include/openmc/random_ray/source_region.h index 1d2bbe1e8dc..3c784d893c3 100644 --- a/include/openmc/random_ray/source_region.h +++ b/include/openmc/random_ray/source_region.h @@ -456,6 +456,12 @@ class SourceRegionContainer { Position& centroid(int64_t sr) { return centroid_[sr]; } const Position centroid(int64_t sr) const { return centroid_[sr]; } + Position& centroid_offset(int64_t sr) { return centroid_offset_[sr]; } + const Position centroid_offset(int64_t sr) const + { + return centroid_offset_[sr]; + } + Position& centroid_iteration(int64_t sr) { return centroid_iteration_[sr]; } const Position centroid_iteration(int64_t sr) const { @@ -660,6 +666,11 @@ class SourceRegionContainer { vector centroid_; vector centroid_iteration_; vector centroid_t_; + // Offset of this batch's track-length-weighted centroid from the + // accumulated centroid the transport sweep evaluated the linear source + // against, used by the batch-consistent flux update (linear source solver + // only) + vector centroid_offset_; vector mom_matrix_; vector mom_matrix_t_; // A set of volume tally tasks. This more complicated data structure is diff --git a/src/random_ray/linear_source_domain.cpp b/src/random_ray/linear_source_domain.cpp index b4701ed1fa9..f7d7bc4aaf8 100644 --- a/src/random_ray/linear_source_domain.cpp +++ b/src/random_ray/linear_source_domain.cpp @@ -131,6 +131,22 @@ void LinearSourceDomain::normalize_scalar_flux_and_volumes( // update the simulation-averaged cell-wise volume estimates #pragma omp parallel for for (int64_t sr = 0; sr < n_source_regions(); sr++) { + // Offset of this batch's track centroid from the accumulated centroid + // the transport sweep evaluated the linear source against (the value + // held here before this batch is folded in). The flux update uses it to + // add back the source as actually sampled by this batch's tracks (see + // set_flux_to_flux_plus_source). Regions with no prior accumulated + // volume used per-segment midpoints in place of a centroid, for which + // the batch offset is identically zero. + if (source_regions_.volume_t(sr) > 0.0 && + source_regions_.volume(sr) > 0.0) { + source_regions_.centroid_offset(sr) = + source_regions_.centroid_iteration(sr) * + (1.0 / source_regions_.volume(sr)) - + source_regions_.centroid(sr); + } else { + source_regions_.centroid_offset(sr) = {0.0, 0.0, 0.0}; + } source_regions_.centroid_t(sr) += source_regions_.centroid_iteration(sr); source_regions_.mom_matrix_t(sr) += source_regions_.mom_matrix(sr); source_regions_.volume_t(sr) += source_regions_.volume(sr); @@ -159,7 +175,33 @@ void LinearSourceDomain::set_flux_to_flux_plus_source( FlatSourceDomain::set_flux_to_flux_plus_source(sr, volume, g); } else { source_regions_.scalar_flux_new(sr, g) /= volume; - source_regions_.scalar_flux_new(sr, g) += source_regions_.source(sr, g); + // The source term added back includes the gradient part of the source + // that this batch's rays actually integrated. The sweep evaluates the + // linear source against the accumulated centroid, but the batch's tracks + // average it at their own track-length-weighted centroid: summing the + // exact chord integrals of a linear source over the batch's tracks gives + // a mean emission of source + gradient . centroid_offset, entering the + // transport sum with the batch's tracked volume -- hence the + // volume_naive / volume weight, which also prevents the correlation + // between the batch volume and batch centroid (both sampled by the same + // rays) from biasing the update when the two volumes differ. Adding back + // the flat source alone instead leaves the difference in the flux + // estimate as noise proportional to the gradient times the per-batch + // centroid fluctuation. In near-cancellation regions (scattering ratio + // near one), where the reduced source approximately equals the flux, + // that noise can exceed the flux itself at modest per-batch hit counts + // and ignite self-sustaining negativity through the scattering feedback. + // When the region is updated with its own batch (naive) volume the + // weight is one and the update becomes an exact per-batch track + // identity: the flux estimate is the track-length average of the angular + // flux, and so inherits its sign -- restoring to linear sources the + // property that a naive-volume update adds no noise of its own beyond + // the angular fluxes it averages. + source_regions_.scalar_flux_new(sr, g) += + source_regions_.source(sr, g) + + (source_regions_.volume_naive(sr) / volume) * + source_regions_.source_gradients(sr, g) + .dot(source_regions_.centroid_offset(sr)); } // If a source region is small, then the moments are likely noisy, so we zero // them. This is reasonable, given that small regions can get by with a flat diff --git a/src/random_ray/source_region.cpp b/src/random_ray/source_region.cpp index 78543c5ab53..265840308f8 100644 --- a/src/random_ray/source_region.cpp +++ b/src/random_ray/source_region.cpp @@ -93,6 +93,7 @@ void SourceRegionContainer::push_back(const SourceRegion& sr) centroid_.push_back(sr.centroid_); centroid_iteration_.push_back(sr.centroid_iteration_); centroid_t_.push_back(sr.centroid_t_); + centroid_offset_.push_back({0.0, 0.0, 0.0}); mom_matrix_.push_back(sr.mom_matrix_); mom_matrix_t_.push_back(sr.mom_matrix_t_); } @@ -146,6 +147,7 @@ void SourceRegionContainer::assign( centroid_.clear(); centroid_iteration_.clear(); centroid_t_.clear(); + centroid_offset_.clear(); mom_matrix_.clear(); mom_matrix_t_.clear(); } @@ -243,6 +245,8 @@ void SourceRegionContainer::adjoint_reset() std::fill(centroid_iteration_.begin(), centroid_iteration_.end(), Position {0.0, 0.0, 0.0}); std::fill(centroid_t_.begin(), centroid_t_.end(), Position {0.0, 0.0, 0.0}); + std::fill(centroid_offset_.begin(), centroid_offset_.end(), + Position {0.0, 0.0, 0.0}); std::fill(mom_matrix_.begin(), mom_matrix_.end(), MomentMatrix {0.0, 0.0, 0.0, 0.0, 0.0, 0.0}); std::fill(mom_matrix_t_.begin(), mom_matrix_t_.end(), diff --git a/tests/regression_tests/random_ray_diagonal_stabilization/results_true.dat b/tests/regression_tests/random_ray_diagonal_stabilization/results_true.dat index 034d7f7c64d..503d8cd02ce 100644 --- a/tests/regression_tests/random_ray_diagonal_stabilization/results_true.dat +++ b/tests/regression_tests/random_ray_diagonal_stabilization/results_true.dat @@ -1,2 +1,2 @@ k-combined: -7.134473E-01 1.422763E-02 +7.074914E-01 1.463204E-02 diff --git a/tests/regression_tests/random_ray_fixed_source_linear/linear/results_true.dat b/tests/regression_tests/random_ray_fixed_source_linear/linear/results_true.dat index e90d6bfdcb8..f461cb08bd6 100644 --- a/tests/regression_tests/random_ray_fixed_source_linear/linear/results_true.dat +++ b/tests/regression_tests/random_ray_fixed_source_linear/linear/results_true.dat @@ -1,9 +1,9 @@ tally 1: -2.339086E+00 -2.747305E-01 +2.346966E+00 +2.759162E-01 tally 2: -1.089827E-01 -6.069324E-04 +1.093918E-01 +6.109599E-04 tally 3: -7.300831E-03 -2.715940E-06 +7.306078E-03 +2.715682E-06 diff --git a/tests/regression_tests/random_ray_fixed_source_linear/linear_xy/results_true.dat b/tests/regression_tests/random_ray_fixed_source_linear/linear_xy/results_true.dat index f4c8546aae4..8f3f5eca7e4 100644 --- a/tests/regression_tests/random_ray_fixed_source_linear/linear_xy/results_true.dat +++ b/tests/regression_tests/random_ray_fixed_source_linear/linear_xy/results_true.dat @@ -1,9 +1,9 @@ tally 1: -2.335703E+00 -2.742866E-01 +2.345165E+00 +2.754957E-01 tally 2: -1.081884E-01 -5.983316E-04 +1.084489E-01 +6.008484E-04 tally 3: -7.295389E-03 -2.711859E-06 +7.293108E-03 +2.707065E-06 diff --git a/tests/regression_tests/random_ray_fixed_source_mesh/linear/results_true.dat b/tests/regression_tests/random_ray_fixed_source_mesh/linear/results_true.dat index 23754a15734..48618bd866a 100644 --- a/tests/regression_tests/random_ray_fixed_source_mesh/linear/results_true.dat +++ b/tests/regression_tests/random_ray_fixed_source_mesh/linear/results_true.dat @@ -1,9 +1,9 @@ tally 1: -1.780361E+00 -2.137912E-01 +1.768878E+00 +2.095422E-01 tally 2: -8.230400E-02 -4.596391E-04 +8.105951E-02 +4.452456E-04 tally 3: -5.207797E-03 -1.834531E-06 +5.241401E-03 +1.858513E-06 diff --git a/tests/regression_tests/random_ray_fixed_source_subcritical/linear_xy/results_true.dat b/tests/regression_tests/random_ray_fixed_source_subcritical/linear_xy/results_true.dat index bf602d3e2d5..3d6df4554d2 100644 --- a/tests/regression_tests/random_ray_fixed_source_subcritical/linear_xy/results_true.dat +++ b/tests/regression_tests/random_ray_fixed_source_subcritical/linear_xy/results_true.dat @@ -1,168 +1,168 @@ tally 1: -1.583465E+02 -1.007029E+03 +1.652767E+02 +1.097494E+03 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -5.891526E+01 -1.392807E+02 +6.126912E+01 +1.506521E+02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -2.028338E+01 -1.649224E+01 +2.097207E+01 +1.762968E+01 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -2.453687E+01 -2.413498E+01 +2.539966E+01 +2.586305E+01 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -5.235813E+01 -1.099243E+02 +5.426858E+01 +1.181176E+02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -9.394406E+01 -3.543513E+02 +9.745708E+01 +3.815289E+02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -5.676284E+01 -1.303104E+02 +5.883282E+01 +1.400901E+02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -2.007858E+02 -1.616216E+03 +2.080105E+02 +1.734796E+03 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -7.319709E+01 -2.146301E+02 +7.562140E+01 +2.291019E+02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -2.566730E+01 -2.640697E+01 +2.636396E+01 +2.786048E+01 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -2.934703E+01 -3.453026E+01 +3.020939E+01 +3.659350E+01 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -5.737215E+01 -1.318580E+02 +5.924782E+01 +1.406529E+02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -9.495275E+01 -3.613373E+02 +9.827629E+01 +3.871903E+02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -5.799251E+01 -1.356178E+02 +5.994992E+01 +1.449707E+02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -1.066414E+02 -4.570291E+02 -1.254014E+01 -6.325641E+00 -3.052020E+01 -3.746918E+01 -4.977302E+01 -9.940548E+01 -2.345790E+00 -2.208444E-01 -5.709184E+00 -1.308146E+00 -1.953082E+01 -1.528292E+01 -1.998234E-01 -1.599792E-03 -4.863302E-01 -9.476180E-03 -2.307196E+01 -2.133059E+01 -2.432481E-01 -2.371074E-03 -5.920173E-01 -1.404478E-02 -5.299863E+01 -1.125576E+02 -1.959914E-01 -1.539191E-03 -4.770098E-01 -9.117441E-03 -1.186133E+02 -5.641060E+02 -5.781230E-02 -1.341759E-04 -1.430525E-01 -8.215327E-04 -7.995734E+01 -2.570099E+02 -3.452934E-01 -4.844059E-03 -9.604162E-01 -3.747587E-02 -1.556795E+02 -9.726232E+02 +1.111749E+02 +4.970219E+02 +1.307403E+01 +6.879737E+00 +3.181958E+01 +4.075131E+01 +5.169641E+01 +1.072779E+02 +2.436341E+00 +2.383121E-01 +5.929567E+00 +1.411614E+00 +2.018230E+01 +1.632218E+01 +2.064712E-01 +1.708276E-03 +5.025096E-01 +1.011877E-02 +2.386309E+01 +2.282486E+01 +2.515462E-01 +2.536294E-03 +6.122131E-01 +1.502344E-02 +5.489518E+01 +1.208074E+02 +2.029943E-01 +1.651813E-03 +4.940536E-01 +9.784560E-03 +1.231103E+02 +6.080634E+02 +6.002316E-02 +1.447403E-04 +1.485231E-01 +8.862168E-04 +8.296586E+01 +2.769320E+02 +3.583878E-01 +5.224525E-03 +9.968376E-01 +4.041933E-02 +1.621564E+02 +1.055824E+03 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -5.816726E+01 -1.357531E+02 +6.043092E+01 +1.465617E+02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -2.018480E+01 -1.633551E+01 +2.087833E+01 +1.747681E+01 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -2.453476E+01 -2.413615E+01 +2.540906E+01 +2.588814E+01 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -5.269380E+01 -1.113102E+02 +5.464001E+01 +1.197011E+02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -9.550216E+01 -3.658264E+02 +9.909886E+01 +3.940513E+02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -5.846320E+01 -1.379625E+02 +6.059452E+01 +1.482769E+02 0.000000E+00 0.000000E+00 0.000000E+00 diff --git a/tests/regression_tests/random_ray_linear/linear/results_true.dat b/tests/regression_tests/random_ray_linear/linear/results_true.dat index 77d41f3732c..e2b8f641a41 100644 --- a/tests/regression_tests/random_ray_linear/linear/results_true.dat +++ b/tests/regression_tests/random_ray_linear/linear/results_true.dat @@ -1,171 +1,171 @@ k-combined: -1.095967E+00 1.543581E-02 +9.796812E-01 2.615985E-02 tally 1: -5.425537E+00 -1.482137E+00 -1.974189E+00 -1.961888E-01 -4.804781E+00 -1.162101E+00 -3.866915E+00 -7.496163E-01 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3 3 3 3 +3 3 3 3 3 3 3 3 3 3 3 3 + + + + + + + + + + fixed source + 90 + 60 + 30 + + + 100.0 1.0 + + + universe + 1 + + + multi-group + + 500.0 + 100.0 + + + + 0.0 0.0 0.0 30.0 30.0 30.0 + + + + true + linear + naive + + + + + + + + 12 12 12 + 0.0 0.0 0.0 + 30.0 30.0 30.0 + + + + + 1 + + + 2 + + + 3 + + + 3 + flux + tracklength + + + 2 + flux + tracklength + + + 1 + flux + tracklength + + + diff --git a/tests/regression_tests/random_ray_linear_source_stability/results_true.dat b/tests/regression_tests/random_ray_linear_source_stability/results_true.dat new file mode 100644 index 00000000000..1401ddad731 --- /dev/null +++ b/tests/regression_tests/random_ray_linear_source_stability/results_true.dat @@ -0,0 +1,9 @@ +tally 1: +3.518081E+00 +4.139523E-01 +tally 2: +1.729928E-01 +1.032644E-03 +tally 3: +1.136650E-02 +4.448920E-06 diff --git a/tests/regression_tests/random_ray_linear_source_stability/test.py b/tests/regression_tests/random_ray_linear_source_stability/test.py new file mode 100644 index 00000000000..7eac708f301 --- /dev/null +++ b/tests/regression_tests/random_ray_linear_source_stability/test.py @@ -0,0 +1,35 @@ +import os + +import openmc +from openmc.examples import random_ray_three_region_cube + +from tests.testing_harness import TolerantPyAPITestHarness + + +class MGXSTestHarness(TolerantPyAPITestHarness): + def _cleanup(self): + super()._cleanup() + f = 'mgxs.h5' + if os.path.exists(f): + os.remove(f) + + +def test_random_ray_linear_source_stability(): + # A linear source run in the regime that stresses the gradient limiter + # and the batch-consistent flux update: the naive volume estimator, an + # overlay source-region mesh whose cells receive modest per-batch hit + # counts, and the example's optically thin scattering-dominated interior. + openmc.reset_auto_ids() + model = random_ray_three_region_cube() + model.settings.random_ray['source_shape'] = 'linear' + model.settings.random_ray['volume_estimator'] = 'naive' + mesh = openmc.RegularMesh() + mesh.lower_left = (0.0, 0.0, 0.0) + mesh.upper_right = (30.0, 30.0, 30.0) + mesh.dimension = (12, 12, 12) + model.settings.random_ray['source_region_meshes'] = [ + (mesh, [model.geometry.root_universe])] + model.settings.inactive = 30 + model.settings.batches = 60 + harness = MGXSTestHarness('statepoint.60.h5', model) + harness.main() diff --git a/tests/regression_tests/random_ray_void/linear/results_true.dat b/tests/regression_tests/random_ray_void/linear/results_true.dat index befa507a59b..da7e9cdb29a 100644 --- a/tests/regression_tests/random_ray_void/linear/results_true.dat +++ b/tests/regression_tests/random_ray_void/linear/results_true.dat @@ -1,9 +1,9 @@ tally 1: -2.356818E+00 -2.782542E-01 +2.356752E+00 +2.782387E-01 tally 2: -1.082843E-01 -6.028734E-04 +1.082833E-01 +6.029190E-04 tally 3: -7.302705E-03 -2.718076E-06 +7.308057E-03 +2.717825E-06 diff --git a/tests/regression_tests/random_ray_volume_estimator_linear/hybrid/results_true.dat b/tests/regression_tests/random_ray_volume_estimator_linear/hybrid/results_true.dat index e90d6bfdcb8..f461cb08bd6 100644 --- a/tests/regression_tests/random_ray_volume_estimator_linear/hybrid/results_true.dat +++ b/tests/regression_tests/random_ray_volume_estimator_linear/hybrid/results_true.dat @@ -1,9 +1,9 @@ tally 1: -2.339086E+00 -2.747305E-01 +2.346966E+00 +2.759162E-01 tally 2: -1.089827E-01 -6.069324E-04 +1.093918E-01 +6.109599E-04 tally 3: -7.300831E-03 -2.715940E-06 +7.306078E-03 +2.715682E-06 diff --git a/tests/regression_tests/random_ray_volume_estimator_linear/naive/results_true.dat b/tests/regression_tests/random_ray_volume_estimator_linear/naive/results_true.dat index 5258ffd9c84..b2ab28b5691 100644 --- a/tests/regression_tests/random_ray_volume_estimator_linear/naive/results_true.dat +++ b/tests/regression_tests/random_ray_volume_estimator_linear/naive/results_true.dat @@ -1,9 +1,9 @@ tally 1: -2.339567E+00 -2.748423E-01 +2.347355E+00 +2.760113E-01 tally 2: -1.085878E-01 -6.024509E-04 +1.091443E-01 +6.086736E-04 tally 3: -7.299803E-03 -2.741867E-06 +7.303490E-03 +2.739532E-06 diff --git a/tests/regression_tests/random_ray_volume_estimator_linear/simulation_averaged/results_true.dat b/tests/regression_tests/random_ray_volume_estimator_linear/simulation_averaged/results_true.dat index 1e8aa9fb75f..9a75655c304 100644 --- a/tests/regression_tests/random_ray_volume_estimator_linear/simulation_averaged/results_true.dat +++ b/tests/regression_tests/random_ray_volume_estimator_linear/simulation_averaged/results_true.dat @@ -1,9 +1,9 @@ tally 1: -2.670850E+02 -4.432939E+05 +2.670984E+02 +4.433011E+05 tally 2: -1.116994E-01 -6.491358E-04 +1.122814E-01 +6.552230E-04 tally 3: -7.564527E-03 -2.947794E-06 +7.578232E-03 +2.957122E-06 From 6f8bba5023145d7affd1af190479cfba0b06eec4 Mon Sep 17 00:00:00 2001 From: John Tramm Date: Thu, 3 Sep 2026 15:47:40 +0000 Subject: [PATCH 2/9] Apply the linear source consistency term only with the batch volume Restrict the batch-sampled gradient term to regions updating with their own batch (naive) volume, where adding it back makes the update an exact per-batch track identity that inherits the sign of the angular fluxes it averages. Regions updating with the simulation-averaged volume keep the original update form. The omitted term cannot bias their accumulated results, since the accumulated centroid is built from the same batch centroids and the deviations sum to exactly zero, and the original form samples the residual between the angular flux and the linear source model, which carries less noise than the direct form wherever the model tracks the field. The estimator matrix agrees: the term is transformative for naive-volume updates and only adds variance under simulation-averaged ones (e.g., 0.5% average pin power error on C5G7 with the hybrid estimator at a 3.7% miss rate). The simulation-averaged reference results are untouched by this commit, confirming the original arithmetic survives bit for bit. The adaptive family shifts slightly: a transiently hit-starved region carries its fitted gradients for one batch before the flat fallback catches up, and now receives the consistent update in that batch. Co-Authored-By: Claude Fable 5 Claude-Session: https://claude.ai/code/session_01P9rxXRzNh13xQBGzePHEeK --- docs/source/methods/random_ray.rst | 49 +- src/random_ray/linear_source_domain.cpp | 54 +- src/random_ray/source_region.cpp | 4 +- .../results_true.dat | 2 +- .../linear/results_true.dat | 12 +- .../linear_xy/results_true.dat | 12 +- .../linear/results_true.dat | 12 +- .../linear_xy/results_true.dat | 168 +- .../random_ray_linear/linear/results_true.dat | 338 +- .../linear_xy/results_true.dat | 338 +- .../random_ray_void/linear/results_true.dat | 12 +- .../hybrid/results_true.dat | 12 +- .../simulation_averaged/results_true.dat | 12 +- .../linear/results_true.dat | 9058 ++++++++--------- 14 files changed, 5043 insertions(+), 5040 deletions(-) diff --git a/docs/source/methods/random_ray.rst b/docs/source/methods/random_ray.rst index 9410984acff..a97ed5c1f59 100644 --- a/docs/source/methods/random_ray.rst +++ b/docs/source/methods/random_ray.rst @@ -1021,29 +1021,32 @@ tracks integrate a mean emission of where :math:`\mathbf{c}_{i,b}` is the batch's track-length-weighted centroid, which fluctuates about :math:`\mathbf{C}_i` from batch to batch as the rays sample the region unevenly. If the scalar flux update adds back only -:math:`\bar{Q}_i`, the difference -- the source gradient dotted with the -per-batch centroid fluctuation -- is silently absorbed by the flux estimate -as a zero-mean noise term that has no counterpart in flat source mode (a -constant source has the same average over any set of tracks). When a region -is updated with its own batch (iteration) volume, this term is the only -noise the update itself contributes: adding back the full batch-sampled -source of Equation :eq:`batch_sampled_source` makes the update an exact -per-batch track identity, in which the scalar flux estimate equals the -track-length average of the angular flux and therefore inherits its sign. -Omitting the term instead breaks, in linear source mode only, the property -that a naive-volume update is exact given its rays -- and in -near-cancellation regions (scattering ratios near unity, as in the -air-filled regions of shielding problems), where the reduced source -approximately equals the scalar flux, the absorbed noise can exceed the flux -itself at modest per-batch hit counts and ignite self-sustaining negativity -through the scattering feedback. When a region is instead updated with the -simulation-averaged volume, the batch-sampled gradient term enters the -transport sum weighted by the ratio of the batch volume to the -simulation-averaged volume, and the update adds it back with the same -weight; the weight matters because the batch volume and batch centroid are -sampled by the same rays, and an unweighted term would convert their -correlation into a systematic, gradient-aligned bias in regions cut by -geometry. +:math:`\bar{Q}_i`, the difference (the source gradient dotted with the +per-batch centroid fluctuation) is silently absorbed by the flux estimate +as a zero-mean noise term that has no counterpart in flat source mode, as a +constant source has the same average over any set of tracks. + +When a region is updated with its own batch (iteration) volume, this term +is the only noise the update itself contributes. Adding back the full +batch-sampled source of Equation :eq:`batch_sampled_source` therefore makes +the update an exact per-batch track identity, in which the scalar flux +estimate equals the track-length average of the angular flux and inherits +its sign. Omitting the term breaks that exactness, and in near-cancellation +regions (scattering ratios near unity, as in the air-filled regions of +shielding problems) the absorbed noise can exceed the flux itself at modest +per-batch hit counts and ignite self-sustaining negativity. OpenMC +therefore includes the term whenever a region updates with its batch +volume. + +When a region is instead updated with the simulation-averaged volume, the +update omits the term and keeps its original form. This cannot bias the +accumulated result, because the accumulated centroid is built from the same +batch centroids and the omitted deviations sum to exactly zero over the +simulation. The original form also has a variance advantage there: it +samples the residual between the angular flux and the linear source model, +whose fluctuation is smaller than that of the flux itself wherever the +model tracks the field. Each volume treatment is thus paired with the +update form that is exact or optimal for it. .. _methods-shannon-entropy-random-ray: diff --git a/src/random_ray/linear_source_domain.cpp b/src/random_ray/linear_source_domain.cpp index f7d7bc4aaf8..e125313755d 100644 --- a/src/random_ray/linear_source_domain.cpp +++ b/src/random_ray/linear_source_domain.cpp @@ -175,33 +175,33 @@ void LinearSourceDomain::set_flux_to_flux_plus_source( FlatSourceDomain::set_flux_to_flux_plus_source(sr, volume, g); } else { source_regions_.scalar_flux_new(sr, g) /= volume; - // The source term added back includes the gradient part of the source - // that this batch's rays actually integrated. The sweep evaluates the - // linear source against the accumulated centroid, but the batch's tracks - // average it at their own track-length-weighted centroid: summing the - // exact chord integrals of a linear source over the batch's tracks gives - // a mean emission of source + gradient . centroid_offset, entering the - // transport sum with the batch's tracked volume -- hence the - // volume_naive / volume weight, which also prevents the correlation - // between the batch volume and batch centroid (both sampled by the same - // rays) from biasing the update when the two volumes differ. Adding back - // the flat source alone instead leaves the difference in the flux - // estimate as noise proportional to the gradient times the per-batch - // centroid fluctuation. In near-cancellation regions (scattering ratio - // near one), where the reduced source approximately equals the flux, - // that noise can exceed the flux itself at modest per-batch hit counts - // and ignite self-sustaining negativity through the scattering feedback. - // When the region is updated with its own batch (naive) volume the - // weight is one and the update becomes an exact per-batch track - // identity: the flux estimate is the track-length average of the angular - // flux, and so inherits its sign -- restoring to linear sources the - // property that a naive-volume update adds no noise of its own beyond - // the angular fluxes it averages. - source_regions_.scalar_flux_new(sr, g) += - source_regions_.source(sr, g) + - (source_regions_.volume_naive(sr) / volume) * - source_regions_.source_gradients(sr, g) - .dot(source_regions_.centroid_offset(sr)); + source_regions_.scalar_flux_new(sr, g) += source_regions_.source(sr, g); + // A region updated with its own batch (naive) volume also adds back the + // gradient part of the source that the batch's rays actually + // integrated. The sweep evaluates the linear source against the + // accumulated centroid, but the batch's tracks average it at their own + // centroid, so the mean emission the tracks sampled is the flat source + // plus the gradient dotted with the batch centroid offset. Including + // that term makes the naive-volume update an exact per-batch track + // identity, in which the flux estimate is the track-length average of + // the angular flux and so inherits its sign. Omitting it leaves + // gradient-scale noise in the flux with no flat-source counterpart, + // which in near-cancellation regions (scattering ratio near one) can + // exceed the flux itself and ignite self-sustaining negativity. + // + // A region updated with the simulation-averaged volume keeps the + // original form with no added term. The omitted term cannot bias the + // accumulated result, since the accumulated centroid is built from the + // same batch centroids and the deviations sum to exactly zero. The + // original form samples the residual between the angular flux and the + // linear source model, whose noise is smaller than the direct form's + // wherever the model tracks the field. Each volume treatment is thus + // paired with the update form that is exact or optimal for it. + if (volume == source_regions_.volume_naive(sr)) { + source_regions_.scalar_flux_new(sr, g) += + source_regions_.source_gradients(sr, g).dot( + source_regions_.centroid_offset(sr)); + } } // If a source region is small, then the moments are likely noisy, so we zero // them. This is reasonable, given that small regions can get by with a flat diff --git a/src/random_ray/source_region.cpp b/src/random_ray/source_region.cpp index 265840308f8..870faf41e82 100644 --- a/src/random_ray/source_region.cpp +++ b/src/random_ray/source_region.cpp @@ -245,8 +245,8 @@ void SourceRegionContainer::adjoint_reset() std::fill(centroid_iteration_.begin(), centroid_iteration_.end(), Position {0.0, 0.0, 0.0}); std::fill(centroid_t_.begin(), centroid_t_.end(), Position {0.0, 0.0, 0.0}); - std::fill(centroid_offset_.begin(), centroid_offset_.end(), - Position {0.0, 0.0, 0.0}); + std::fill( + centroid_offset_.begin(), centroid_offset_.end(), Position {0.0, 0.0, 0.0}); std::fill(mom_matrix_.begin(), mom_matrix_.end(), MomentMatrix {0.0, 0.0, 0.0, 0.0, 0.0, 0.0}); std::fill(mom_matrix_t_.begin(), mom_matrix_t_.end(), diff --git a/tests/regression_tests/random_ray_diagonal_stabilization/results_true.dat b/tests/regression_tests/random_ray_diagonal_stabilization/results_true.dat index 503d8cd02ce..034d7f7c64d 100644 --- a/tests/regression_tests/random_ray_diagonal_stabilization/results_true.dat +++ b/tests/regression_tests/random_ray_diagonal_stabilization/results_true.dat @@ -1,2 +1,2 @@ k-combined: -7.074914E-01 1.463204E-02 +7.134473E-01 1.422763E-02 diff --git a/tests/regression_tests/random_ray_fixed_source_linear/linear/results_true.dat b/tests/regression_tests/random_ray_fixed_source_linear/linear/results_true.dat index f461cb08bd6..a3f137499a2 100644 --- a/tests/regression_tests/random_ray_fixed_source_linear/linear/results_true.dat +++ b/tests/regression_tests/random_ray_fixed_source_linear/linear/results_true.dat @@ -1,9 +1,9 @@ tally 1: -2.346966E+00 -2.759162E-01 +2.347067E+00 +2.759398E-01 tally 2: -1.093918E-01 -6.109599E-04 +1.089827E-01 +6.069322E-04 tally 3: -7.306078E-03 -2.715682E-06 +7.300833E-03 +2.715941E-06 diff --git a/tests/regression_tests/random_ray_fixed_source_linear/linear_xy/results_true.dat b/tests/regression_tests/random_ray_fixed_source_linear/linear_xy/results_true.dat index 8f3f5eca7e4..58d5b110a25 100644 --- a/tests/regression_tests/random_ray_fixed_source_linear/linear_xy/results_true.dat +++ b/tests/regression_tests/random_ray_fixed_source_linear/linear_xy/results_true.dat @@ -1,9 +1,9 @@ tally 1: -2.345165E+00 -2.754957E-01 +2.345238E+00 +2.755122E-01 tally 2: -1.084489E-01 -6.008484E-04 +1.081885E-01 +5.983320E-04 tally 3: -7.293108E-03 -2.707065E-06 +7.295394E-03 +2.711862E-06 diff --git a/tests/regression_tests/random_ray_fixed_source_mesh/linear/results_true.dat b/tests/regression_tests/random_ray_fixed_source_mesh/linear/results_true.dat index 48618bd866a..66ca0826e6d 100644 --- a/tests/regression_tests/random_ray_fixed_source_mesh/linear/results_true.dat +++ b/tests/regression_tests/random_ray_fixed_source_mesh/linear/results_true.dat @@ -1,9 +1,9 @@ tally 1: -1.768878E+00 -2.095422E-01 +1.768881E+00 +2.095454E-01 tally 2: -8.105951E-02 -4.452456E-04 +8.228219E-02 +4.596629E-04 tally 3: -5.241401E-03 -1.858513E-06 +5.207806E-03 +1.834565E-06 diff --git a/tests/regression_tests/random_ray_fixed_source_subcritical/linear_xy/results_true.dat b/tests/regression_tests/random_ray_fixed_source_subcritical/linear_xy/results_true.dat index 3d6df4554d2..1bf4083061e 100644 --- a/tests/regression_tests/random_ray_fixed_source_subcritical/linear_xy/results_true.dat +++ b/tests/regression_tests/random_ray_fixed_source_subcritical/linear_xy/results_true.dat @@ -1,168 +1,168 @@ tally 1: -1.652767E+02 -1.097494E+03 +1.584451E+02 +1.008274E+03 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -6.126912E+01 -1.506521E+02 +5.894185E+01 +1.394021E+02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -2.097207E+01 -1.762968E+01 +2.028422E+01 +1.649342E+01 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -2.539966E+01 -2.586305E+01 +2.453913E+01 +2.413937E+01 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -5.426858E+01 -1.181176E+02 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a/tests/regression_tests/random_ray_linear/linear/results_true.dat b/tests/regression_tests/random_ray_linear/linear/results_true.dat index e2b8f641a41..77d41f3732c 100644 --- a/tests/regression_tests/random_ray_linear/linear/results_true.dat +++ b/tests/regression_tests/random_ray_linear/linear/results_true.dat @@ -1,171 +1,171 @@ k-combined: -9.796812E-01 2.615985E-02 +1.095967E+00 1.543581E-02 tally 1: -4.522874E+00 -1.044824E+00 -1.665801E+00 -1.416948E-01 -4.054226E+00 -8.393126E-01 -3.406263E+00 -5.883877E-01 -5.015978E-01 -1.275500E-02 -1.220789E+00 -7.555273E-02 -2.651378E+00 -3.561261E-01 -8.498131E-02 -3.656872E-04 -2.068275E-01 -2.166105E-03 -3.405589E+00 -5.880386E-01 -1.122982E-01 -6.391206E-04 -2.733114E-01 -3.785756E-03 -8.864924E+00 -3.986162E+00 -1.029817E-01 -5.376520E-04 -2.506399E-01 -3.184796E-03 -1.949219E+01 -1.921658E+01 -2.936048E-02 -4.356978E-05 -7.265045E-02 -2.667692E-04 -1.066457E+01 -5.710150E+00 -1.421683E-01 -1.014766E-03 -3.954339E-01 -7.850700E-03 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b/tests/regression_tests/random_ray_volume_estimator_linear/hybrid/results_true.dat index f461cb08bd6..a3f137499a2 100644 --- a/tests/regression_tests/random_ray_volume_estimator_linear/hybrid/results_true.dat +++ b/tests/regression_tests/random_ray_volume_estimator_linear/hybrid/results_true.dat @@ -1,9 +1,9 @@ tally 1: -2.346966E+00 -2.759162E-01 +2.347067E+00 +2.759398E-01 tally 2: -1.093918E-01 -6.109599E-04 +1.089827E-01 +6.069322E-04 tally 3: -7.306078E-03 -2.715682E-06 +7.300833E-03 +2.715941E-06 diff --git a/tests/regression_tests/random_ray_volume_estimator_linear/simulation_averaged/results_true.dat b/tests/regression_tests/random_ray_volume_estimator_linear/simulation_averaged/results_true.dat index 9a75655c304..1e8aa9fb75f 100644 --- a/tests/regression_tests/random_ray_volume_estimator_linear/simulation_averaged/results_true.dat +++ b/tests/regression_tests/random_ray_volume_estimator_linear/simulation_averaged/results_true.dat @@ -1,9 +1,9 @@ tally 1: -2.670984E+02 -4.433011E+05 +2.670850E+02 +4.432939E+05 tally 2: -1.122814E-01 -6.552230E-04 +1.116994E-01 +6.491358E-04 tally 3: -7.578232E-03 -2.957122E-06 +7.564527E-03 +2.947794E-06 diff --git a/tests/regression_tests/weightwindows_fw_cadis_mesh/linear/results_true.dat b/tests/regression_tests/weightwindows_fw_cadis_mesh/linear/results_true.dat index c1066ae8285..99c37d4fda6 100644 --- a/tests/regression_tests/weightwindows_fw_cadis_mesh/linear/results_true.dat +++ b/tests/regression_tests/weightwindows_fw_cadis_mesh/linear/results_true.dat @@ -7,2916 +7,2916 @@ RegularMesh Upper Right = [np.float64(30.0), np.float64(30.0), np.float64(30.0)] Width = [2. 2. 2.] Lower Bounds -1.77e-01 -1.81e-01 -1.81e-01 +1.79e-01 1.81e-01 -1.78e-01 -1.74e-01 -1.70e-01 -1.76e-01 1.80e-01 -1.88e-01 +1.82e-01 +1.79e-01 +1.74e-01 +1.73e-01 1.75e-01 -1.93e-01 -2.76e-01 +1.79e-01 +1.87e-01 +1.77e-01 +1.91e-01 +2.75e-01 4.28e-01 1.58e-01 -1.74e-01 -1.82e-01 +1.75e-01 +1.84e-01 1.81e-01 1.63e-01 -1.78e-01 +1.76e-01 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+2.12e-01 +2.12e-01 +2.29e-01 +2.37e-01 +2.22e-01 2.11e-01 2.10e-01 1.87e-01 1.89e-01 -1.88e-01 -1.79e-01 +1.93e-01 +1.80e-01 1.33e-01 -3.03e-02 +3.04e-02 1.06e-02 -2.08e-01 -2.14e-01 -2.12e-01 -2.27e-01 -2.27e-01 -2.05e-01 -2.29e-01 -2.21e-01 -1.93e-01 -2.02e-01 +2.07e-01 +2.13e-01 +2.15e-01 +2.26e-01 +2.28e-01 +2.06e-01 +2.28e-01 +2.20e-01 +1.92e-01 +1.98e-01 1.81e-01 -1.73e-01 -1.67e-01 -4.08e-02 +1.74e-01 +1.68e-01 +4.10e-02 1.39e-02 2.00e-01 -2.19e-01 -2.28e-01 -2.14e-01 +2.18e-01 +2.32e-01 +2.16e-01 2.28e-01 -2.06e-01 -2.17e-01 -1.92e-01 -1.79e-01 +2.07e-01 +2.18e-01 +1.95e-01 +1.78e-01 1.74e-01 1.72e-01 1.71e-01 1.60e-01 -4.62e-02 +4.60e-02 1.43e-02 -1.82e-01 -1.92e-01 -2.16e-01 -2.14e-01 +1.85e-01 +1.93e-01 +2.17e-01 +2.15e-01 2.12e-01 -2.26e-01 -1.94e-01 -1.68e-01 -1.64e-01 +2.27e-01 +1.97e-01 +1.70e-01 +1.66e-01 1.50e-01 1.55e-01 -1.48e-01 -1.04e-01 +1.49e-01 +1.05e-01 4.02e-02 -1.44e-02 +1.45e-02 1.56e-01 -1.60e-01 -1.93e-01 -1.87e-01 -1.94e-01 -2.01e-01 -2.07e-01 +1.59e-01 +1.90e-01 +1.89e-01 +1.95e-01 +2.02e-01 +2.06e-01 1.85e-01 -1.76e-01 -1.49e-01 -1.39e-01 -1.38e-01 +1.77e-01 +1.50e-01 +1.37e-01 +1.36e-01 1.04e-01 -2.84e-02 -1.17e-02 +2.81e-02 +1.16e-02 1.33e-01 1.46e-01 1.65e-01 -1.58e-01 -1.74e-01 -1.70e-01 -1.37e-01 +1.57e-01 +1.76e-01 +1.71e-01 +1.38e-01 1.94e-01 -1.93e-01 -1.64e-01 +1.96e-01 +1.63e-01 1.20e-01 1.02e-01 -6.63e-02 +6.62e-02 1.85e-02 8.29e-03 -6.84e-02 -4.09e-02 -5.11e-02 +6.83e-02 +4.08e-02 +5.13e-02 6.13e-02 -4.88e-02 -4.65e-02 +4.87e-02 +4.66e-02 3.93e-02 -4.74e-02 -8.51e-02 +4.75e-02 +8.49e-02 5.59e-02 2.40e-02 -2.44e-02 +2.43e-02 1.30e-02 -4.68e-03 -2.65e-03 +4.69e-03 +2.68e-03 2.59e-02 1.30e-02 1.81e-02 @@ -2926,84 +2926,84 @@ Lower Bounds 1.50e-02 1.42e-02 2.71e-02 -2.16e-02 +2.17e-02 1.05e-02 -7.72e-03 +7.70e-03 4.11e-03 2.35e-03 -9.64e-04 -2.21e-01 +9.65e-04 +2.20e-01 2.15e-01 2.72e-01 -1.92e-01 -3.01e-01 +1.93e-01 +3.00e-01 2.43e-01 -2.13e-01 +2.12e-01 2.18e-01 -1.88e-01 +1.87e-01 1.66e-01 1.13e-01 1.21e-01 -1.08e-01 +1.09e-01 2.92e-02 -8.37e-03 +8.36e-03 2.75e-01 -1.89e-01 +1.90e-01 3.28e-01 -3.64e-01 -2.91e-01 +3.63e-01 +2.90e-01 2.34e-01 1.86e-01 1.90e-01 1.53e-01 -1.35e-01 -1.21e-01 +1.36e-01 +1.22e-01 1.01e-01 -8.79e-02 -3.91e-02 +8.78e-02 +3.88e-02 1.37e-02 -2.37e-01 +2.36e-01 2.28e-01 -2.92e-01 -2.21e-01 +2.93e-01 +2.22e-01 1.70e-01 -2.29e-01 +2.30e-01 1.86e-01 1.95e-01 1.65e-01 1.42e-01 -1.73e-01 +1.74e-01 1.38e-01 -7.39e-02 +7.40e-02 4.16e-02 2.91e-02 2.73e-01 2.33e-01 -2.78e-01 +2.77e-01 3.06e-01 -2.31e-01 -2.52e-01 -2.69e-01 -2.39e-01 +2.30e-01 +2.53e-01 +2.70e-01 +2.40e-01 2.13e-01 1.63e-01 -1.20e-01 -8.39e-02 -5.83e-02 +1.21e-01 +8.45e-02 +5.85e-02 2.86e-02 1.32e-02 -2.63e-01 -1.92e-01 -2.54e-01 -3.11e-01 +2.61e-01 +1.91e-01 +2.53e-01 +3.10e-01 2.65e-01 -2.75e-01 +2.77e-01 2.71e-01 -2.21e-01 +2.23e-01 1.81e-01 1.68e-01 1.49e-01 -7.05e-02 +7.06e-02 5.03e-02 2.54e-02 1.28e-02 @@ -3011,294 +3011,294 @@ Lower Bounds 1.61e-01 2.21e-01 2.28e-01 -1.55e-01 -1.78e-01 -2.16e-01 +1.56e-01 +1.77e-01 +2.15e-01 1.50e-01 -1.32e-01 -1.76e-01 +1.33e-01 +1.77e-01 1.57e-01 1.01e-01 -5.61e-02 -2.32e-02 +5.65e-02 +2.33e-02 1.53e-02 1.50e-01 1.81e-01 -2.03e-01 -2.04e-01 +2.02e-01 +2.02e-01 1.79e-01 -1.94e-01 +1.95e-01 1.81e-01 1.52e-01 1.26e-01 1.13e-01 -9.16e-02 -6.72e-02 +9.18e-02 +6.70e-02 3.94e-02 2.35e-02 1.45e-02 -1.06e-01 +1.07e-01 1.43e-01 1.61e-01 1.57e-01 1.80e-01 1.80e-01 1.23e-01 -1.22e-01 +1.21e-01 8.18e-02 -7.30e-02 -9.33e-02 -7.39e-02 -3.89e-02 +7.29e-02 +9.31e-02 +7.41e-02 +3.87e-02 2.07e-02 -8.17e-03 +8.20e-03 1.13e-01 1.37e-01 1.61e-01 -2.20e-01 +2.22e-01 1.71e-01 1.51e-01 1.16e-01 -1.45e-01 +1.44e-01 1.13e-01 1.05e-01 -9.41e-02 -7.14e-02 -5.35e-02 -1.91e-02 -7.31e-03 -8.55e-02 +9.39e-02 +7.19e-02 +5.33e-02 +1.92e-02 +7.30e-03 +8.53e-02 1.58e-01 -2.08e-01 -1.60e-01 +2.09e-01 +1.59e-01 1.37e-01 -8.33e-02 +8.32e-02 1.16e-01 -1.39e-01 +1.38e-01 1.15e-01 9.22e-02 -8.69e-02 +8.68e-02 6.24e-02 4.30e-02 2.48e-02 -7.57e-03 -9.22e-02 +7.56e-03 +9.18e-02 1.13e-01 -1.60e-01 -1.04e-01 +1.61e-01 +1.05e-01 1.40e-01 1.35e-01 1.08e-01 -1.02e-01 -8.56e-02 -5.91e-02 -7.46e-02 -6.28e-02 +1.03e-01 +8.54e-02 +5.90e-02 +7.44e-02 +6.27e-02 3.81e-02 2.04e-02 -9.21e-03 -7.85e-02 -6.83e-02 +9.20e-03 +7.80e-02 +6.85e-02 1.01e-01 9.42e-02 -9.47e-02 +9.44e-02 1.38e-01 1.13e-01 -6.40e-02 -6.03e-02 -5.21e-02 +6.36e-02 +5.99e-02 +5.20e-02 4.63e-02 -4.67e-02 -3.57e-02 -9.24e-03 -3.46e-03 +4.65e-02 +3.58e-02 +9.31e-03 +3.45e-03 3.82e-02 -4.99e-02 -6.21e-02 +5.00e-02 +6.25e-02 4.80e-02 -5.79e-02 -8.37e-02 -6.86e-02 -6.66e-02 -5.19e-02 5.77e-02 -4.00e-02 -3.46e-02 -2.69e-02 -9.16e-03 -3.73e-03 -1.66e-02 +8.39e-02 +6.86e-02 +6.69e-02 +5.20e-02 +5.76e-02 +3.99e-02 +3.45e-02 +2.70e-02 +9.21e-03 +3.72e-03 +1.67e-02 1.94e-02 -2.26e-02 +2.27e-02 1.79e-02 -2.45e-02 -3.08e-02 +2.44e-02 +3.09e-02 2.00e-02 -2.89e-02 -3.35e-02 -2.83e-02 +2.92e-02 +3.36e-02 +2.82e-02 1.58e-02 -1.10e-02 -7.32e-03 -3.83e-03 -1.85e-03 -1.35e-02 -7.32e-03 -8.41e-03 -8.10e-03 -9.00e-03 +1.09e-02 +7.35e-03 +3.81e-03 +1.84e-03 +1.36e-02 +7.31e-03 +8.39e-03 +8.11e-03 +8.99e-03 1.12e-02 -5.57e-03 -9.30e-03 +5.56e-03 +9.32e-03 1.48e-02 -1.39e-02 +1.38e-02 6.30e-03 3.42e-03 2.99e-03 -1.40e-03 -5.26e-04 -7.56e-02 -7.51e-02 +1.41e-03 +5.27e-04 +7.59e-02 +7.52e-02 1.02e-01 6.63e-02 1.09e-01 -8.00e-02 +8.02e-02 7.46e-02 -7.69e-02 -7.23e-02 -5.75e-02 -3.77e-02 +7.68e-02 +7.20e-02 +5.73e-02 +3.78e-02 3.71e-02 -3.59e-02 -2.14e-02 -7.54e-03 +3.60e-02 +2.13e-02 +7.53e-03 1.13e-01 5.14e-02 1.11e-01 1.48e-01 1.23e-01 -8.67e-02 +8.69e-02 6.58e-02 -5.93e-02 -5.79e-02 -5.00e-02 +5.95e-02 +5.82e-02 +4.97e-02 3.78e-02 2.92e-02 -2.64e-02 +2.62e-02 1.76e-02 8.34e-03 8.71e-02 -6.82e-02 +6.78e-02 1.03e-01 -9.93e-02 -4.68e-02 -8.00e-02 +9.92e-02 +4.69e-02 +8.01e-02 6.54e-02 -5.73e-02 -5.88e-02 -4.17e-02 -6.05e-02 -5.26e-02 -2.82e-02 +5.75e-02 +5.86e-02 +4.15e-02 +6.06e-02 +5.25e-02 +2.81e-02 1.18e-02 -1.25e-02 -8.93e-02 -6.93e-02 +1.26e-02 +8.92e-02 +6.92e-02 1.05e-01 1.22e-01 7.82e-02 -8.44e-02 -9.53e-02 -9.08e-02 -8.61e-02 -6.38e-02 -4.26e-02 -2.94e-02 +8.43e-02 +9.54e-02 +9.05e-02 +8.62e-02 +6.35e-02 +4.27e-02 +2.95e-02 1.95e-02 -1.16e-02 -5.85e-03 +1.17e-02 +5.86e-03 1.07e-01 -7.48e-02 +7.50e-02 9.61e-02 -1.16e-01 +1.15e-01 1.00e-01 -9.56e-02 +9.60e-02 1.01e-01 -7.22e-02 -7.15e-02 -4.92e-02 -5.10e-02 +7.20e-02 +7.16e-02 +4.91e-02 +5.11e-02 2.14e-02 1.39e-02 8.55e-03 -5.42e-03 -6.30e-02 -5.46e-02 +5.41e-03 +6.28e-02 +5.47e-02 6.94e-02 -7.56e-02 +7.54e-02 5.76e-02 -4.99e-02 -6.77e-02 +5.00e-02 +6.78e-02 4.79e-02 4.29e-02 5.31e-02 -5.73e-02 -4.32e-02 +5.74e-02 +4.31e-02 2.01e-02 8.20e-03 -4.18e-03 -6.21e-02 -5.98e-02 -7.34e-02 -7.91e-02 -4.60e-02 +4.17e-03 +6.22e-02 +5.99e-02 +7.32e-02 +7.90e-02 +4.61e-02 5.72e-02 -6.01e-02 -5.26e-02 -5.01e-02 +6.00e-02 +5.25e-02 +4.99e-02 3.98e-02 3.12e-02 -2.62e-02 +2.63e-02 1.44e-02 -6.44e-03 -6.95e-03 +6.45e-03 +6.96e-03 2.95e-02 -5.38e-02 +5.39e-02 4.94e-02 -4.81e-02 -5.65e-02 -6.07e-02 -3.70e-02 -3.88e-02 -2.59e-02 +4.80e-02 +5.64e-02 +6.06e-02 +3.69e-02 +3.87e-02 +2.58e-02 2.05e-02 -2.51e-02 -2.29e-02 +2.50e-02 +2.30e-02 1.19e-02 1.08e-02 -8.34e-03 +8.36e-03 3.62e-02 4.00e-02 5.20e-02 6.63e-02 -6.42e-02 -5.90e-02 +6.43e-02 +5.89e-02 3.69e-02 4.87e-02 -4.00e-02 -3.31e-02 +3.99e-02 +3.30e-02 2.85e-02 -2.46e-02 +2.47e-02 2.05e-02 -8.72e-03 -3.45e-03 +8.69e-03 +3.46e-03 2.17e-02 4.81e-02 6.28e-02 -6.24e-02 -4.43e-02 +6.25e-02 +4.42e-02 2.41e-02 -3.73e-02 +3.71e-02 4.41e-02 4.25e-02 2.80e-02 @@ -3309,2320 +3309,2320 @@ Lower Bounds 6.90e-03 2.96e-02 4.88e-02 -5.84e-02 -2.83e-02 +5.85e-02 +2.84e-02 4.58e-02 3.78e-02 2.43e-02 -3.47e-02 -3.08e-02 -2.18e-02 -2.32e-02 +3.48e-02 +3.09e-02 +2.19e-02 +2.31e-02 1.70e-02 1.26e-02 -9.95e-03 -8.03e-03 -2.02e-02 +9.93e-03 +8.04e-03 +2.03e-02 2.21e-02 -3.09e-02 -2.79e-02 +3.10e-02 +2.80e-02 2.92e-02 4.73e-02 -3.85e-02 +3.84e-02 2.69e-02 1.89e-02 1.70e-02 -1.44e-02 +1.45e-02 1.42e-02 1.51e-02 4.21e-03 1.55e-03 1.72e-02 -1.70e-02 +1.69e-02 2.18e-02 -2.50e-02 -1.84e-02 -3.18e-02 -2.53e-02 +2.49e-02 +1.83e-02 +3.19e-02 +2.54e-02 2.01e-02 1.37e-02 1.89e-02 1.02e-02 1.15e-02 1.16e-02 -3.85e-03 +3.83e-03 1.22e-03 -5.45e-03 -9.53e-03 -9.49e-03 -6.68e-03 +5.44e-03 +9.51e-03 +9.52e-03 +6.67e-03 1.08e-02 -1.59e-02 +1.58e-02 1.34e-02 -1.53e-02 +1.54e-02 1.86e-02 1.03e-02 1.07e-02 6.50e-03 3.98e-03 -2.38e-03 +2.36e-03 2.29e-03 -4.46e-03 -5.06e-03 -5.22e-03 -4.55e-03 -5.72e-03 -1.00e-02 +4.45e-03 +5.03e-03 +5.23e-03 +4.58e-03 +5.70e-03 +9.99e-03 6.07e-03 -5.95e-03 -1.20e-02 -8.03e-03 -5.95e-03 +5.97e-03 +1.18e-02 +8.02e-03 +5.96e-03 3.61e-03 1.35e-03 -8.82e-04 +8.84e-04 1.47e-03 Upper Bounds -8.83e-01 -9.04e-01 -9.04e-01 +8.95e-01 9.06e-01 -8.90e-01 -8.71e-01 -8.51e-01 -8.82e-01 -8.99e-01 -9.42e-01 -8.75e-01 -9.67e-01 +8.98e-01 +9.09e-01 +8.95e-01 +8.69e-01 +8.65e-01 +8.74e-01 +8.94e-01 +9.36e-01 +8.83e-01 +9.56e-01 1.38e+00 2.14e+00 -7.89e-01 -8.70e-01 -9.11e-01 -9.06e-01 -8.15e-01 -8.92e-01 -9.06e-01 -8.84e-01 -8.83e-01 -8.68e-01 -8.58e-01 -9.07e-01 -9.12e-01 -1.29e+00 -1.63e+00 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3.22e-02 3.48e-02 -1.47e-01 -2.69e-01 +1.48e-01 +2.70e-01 2.47e-01 2.40e-01 2.82e-01 @@ -6652,109 +6652,109 @@ Upper Bounds 1.15e-01 5.95e-02 5.42e-02 -4.17e-02 +4.18e-02 1.81e-01 2.00e-01 2.60e-01 3.31e-01 3.21e-01 -2.95e-01 -1.85e-01 +2.94e-01 +1.84e-01 2.44e-01 -2.00e-01 -1.66e-01 -1.43e-01 +1.99e-01 +1.65e-01 +1.42e-01 1.23e-01 1.02e-01 -4.36e-02 -1.72e-02 -1.08e-01 +4.35e-02 +1.73e-02 +1.09e-01 2.40e-01 3.14e-01 3.12e-01 2.21e-01 -1.20e-01 +1.21e-01 1.86e-01 2.21e-01 -2.13e-01 +2.12e-01 1.40e-01 1.31e-01 1.05e-01 -5.58e-02 +5.59e-02 6.38e-02 3.45e-02 1.48e-01 2.44e-01 -2.92e-01 +2.93e-01 1.42e-01 2.29e-01 1.89e-01 1.21e-01 -1.73e-01 -1.54e-01 -1.09e-01 +1.74e-01 +1.55e-01 +1.10e-01 1.16e-01 -8.49e-02 -6.30e-02 -4.98e-02 +8.50e-02 +6.32e-02 +4.96e-02 4.02e-02 1.01e-01 -1.10e-01 +1.11e-01 1.55e-01 1.40e-01 1.46e-01 -2.37e-01 -1.93e-01 -1.34e-01 -9.47e-02 -8.48e-02 -7.21e-02 +2.36e-01 +1.92e-01 +1.35e-01 +9.45e-02 +8.49e-02 +7.23e-02 7.09e-02 7.53e-02 -2.10e-02 -7.74e-03 -8.60e-02 -8.49e-02 +2.11e-02 +7.76e-03 +8.61e-02 +8.47e-02 1.09e-01 -1.25e-01 -9.19e-02 -1.59e-01 +1.24e-01 +9.17e-02 +1.60e-01 1.27e-01 1.01e-01 6.87e-02 -9.44e-02 +9.46e-02 5.12e-02 -5.73e-02 -5.81e-02 +5.74e-02 +5.78e-02 1.92e-02 -6.09e-03 +6.08e-03 2.72e-02 -4.77e-02 -4.74e-02 +4.76e-02 +4.76e-02 3.34e-02 -5.41e-02 -7.93e-02 +5.42e-02 +7.92e-02 6.69e-02 -7.67e-02 -9.31e-02 +7.70e-02 +9.29e-02 5.17e-02 5.34e-02 3.25e-02 1.99e-02 -1.19e-02 +1.18e-02 1.14e-02 2.23e-02 -2.53e-02 -2.61e-02 -2.28e-02 -2.86e-02 -5.00e-02 +2.52e-02 +2.62e-02 +2.29e-02 +2.85e-02 +4.99e-02 3.04e-02 2.98e-02 -5.98e-02 -4.02e-02 -2.97e-02 +5.91e-02 +4.01e-02 +2.98e-02 1.80e-02 6.73e-03 -4.41e-03 +4.42e-03 7.35e-03 \ No newline at end of file From 9c73076ff8d2bc3c2da8aaf4f02e3e1ab799b31f Mon Sep 17 00:00:00 2001 From: John Tramm Date: Thu, 3 Sep 2026 16:35:18 +0000 Subject: [PATCH 3/9] Condense the linear source consistency note Co-Authored-By: Claude Fable 5 Claude-Session: https://claude.ai/code/session_01P9rxXRzNh13xQBGzePHEeK --- docs/source/methods/random_ray.rst | 55 ++++++++---------------------- 1 file changed, 15 insertions(+), 40 deletions(-) diff --git a/docs/source/methods/random_ray.rst b/docs/source/methods/random_ray.rst index a97ed5c1f59..0efa52510e3 100644 --- a/docs/source/methods/random_ray.rst +++ b/docs/source/methods/random_ray.rst @@ -1003,50 +1003,25 @@ completes the set of equations for LS. Consistency of the Scalar Flux Estimate ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -One subtlety of the linear source scheme deserves explicit treatment. During -a batch, the transport sweep evaluates the linear source of region :math:`i` -against the accumulated centroid estimate :math:`\mathbf{C}_i`, so every -segment's attenuation calculation subtracts the local source value -:math:`\bar{Q}_i + \vec{Q}_i \cdot (\mathbf{m}_s - \mathbf{C}_i)` at that -segment's midpoint :math:`\mathbf{m}_s`. Because the chord integral of a -linear function is exactly its midpoint value times the chord length, -summing the segment balance equations over a batch's tracks shows that the -tracks integrate a mean emission of +One subtlety of the linear source scheme deserves note. The transport sweep +evaluates each region's linear source against the accumulated centroid +:math:`\mathbf{C}_i`, but a batch's tracks average that source at their own +track-length-weighted centroid :math:`\mathbf{c}_{i,b}`, so the mean +emission the batch actually integrates is .. math:: :label: batch_sampled_source - \bar{Q}_i + \vec{Q}_i \cdot (\mathbf{c}_{i,b} - \mathbf{C}_i)\;, - -where :math:`\mathbf{c}_{i,b}` is the batch's track-length-weighted centroid, -which fluctuates about :math:`\mathbf{C}_i` from batch to batch as the rays -sample the region unevenly. If the scalar flux update adds back only -:math:`\bar{Q}_i`, the difference (the source gradient dotted with the -per-batch centroid fluctuation) is silently absorbed by the flux estimate -as a zero-mean noise term that has no counterpart in flat source mode, as a -constant source has the same average over any set of tracks. - -When a region is updated with its own batch (iteration) volume, this term -is the only noise the update itself contributes. Adding back the full -batch-sampled source of Equation :eq:`batch_sampled_source` therefore makes -the update an exact per-batch track identity, in which the scalar flux -estimate equals the track-length average of the angular flux and inherits -its sign. Omitting the term breaks that exactness, and in near-cancellation -regions (scattering ratios near unity, as in the air-filled regions of -shielding problems) the absorbed noise can exceed the flux itself at modest -per-batch hit counts and ignite self-sustaining negativity. OpenMC -therefore includes the term whenever a region updates with its batch -volume. - -When a region is instead updated with the simulation-averaged volume, the -update omits the term and keeps its original form. This cannot bias the -accumulated result, because the accumulated centroid is built from the same -batch centroids and the omitted deviations sum to exactly zero over the -simulation. The original form also has a variance advantage there: it -samples the residual between the angular flux and the linear source model, -whose fluctuation is smaller than that of the flux itself wherever the -model tracks the field. Each volume treatment is thus paired with the -update form that is exact or optimal for it. + \bar{Q}_i + \vec{Q}_i \cdot (\mathbf{c}_{i,b} - \mathbf{C}_i)\;. + +A flux update that adds back only :math:`\bar{Q}_i` absorbs the difference +as gradient-scale noise, which in optically thin scatter-fed regions can +ignite self-sustaining negative fluxes. OpenMC therefore adds back the full +batch-sampled source whenever a region updates with its own batch +(iteration) volume, making that update exact for the batch's tracks. +Regions updating with the simulation-averaged volume keep the original +form, which is unbiased (the omitted deviations sum to zero against the +accumulated centroid) and carries less variance there. .. _methods-shannon-entropy-random-ray: From 8c28b68425d6d6f911a901d90dbabade337054b0 Mon Sep 17 00:00:00 2001 From: John Tramm Date: Thu, 3 Sep 2026 16:43:06 +0000 Subject: [PATCH 4/9] Open the consistency note with the statistics-mismatch intuition Co-Authored-By: Claude Fable 5 Claude-Session: https://claude.ai/code/session_01P9rxXRzNh13xQBGzePHEeK --- docs/source/methods/random_ray.rst | 15 ++++++++++----- 1 file changed, 10 insertions(+), 5 deletions(-) diff --git a/docs/source/methods/random_ray.rst b/docs/source/methods/random_ray.rst index 0efa52510e3..291373d5701 100644 --- a/docs/source/methods/random_ray.rst +++ b/docs/source/methods/random_ray.rst @@ -1003,11 +1003,16 @@ completes the set of equations for LS. Consistency of the Scalar Flux Estimate ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -One subtlety of the linear source scheme deserves note. The transport sweep -evaluates each region's linear source against the accumulated centroid -:math:`\mathbf{C}_i`, but a batch's tracks average that source at their own -track-length-weighted centroid :math:`\mathbf{c}_{i,b}`, so the mean -emission the batch actually integrates is +One subtlety of the linear source scheme deserves note. Intuitively, the +issue is a mismatch of statistics: the naive volume treatment updates the +flux from a single batch's rays, while the linear source is anchored to the +simulation-averaged centroid, so the two halves of the update describe +different sets of tracks and the exactness the naive treatment promises is +quietly broken. Concretely, the transport sweep evaluates each region's +linear source against the accumulated centroid :math:`\mathbf{C}_i`, but a +batch's tracks average that source at their own track-length-weighted +centroid :math:`\mathbf{c}_{i,b}`, so the mean emission the batch actually +integrates is .. math:: :label: batch_sampled_source From 0ebb4b334eff5651072b26200d2c42abc8010973 Mon Sep 17 00:00:00 2001 From: John Tramm Date: Thu, 3 Sep 2026 18:11:24 +0000 Subject: [PATCH 5/9] Remove a stale gradient limiter reference from the stability test Co-Authored-By: Claude Fable 5 Claude-Session: https://claude.ai/code/session_01P9rxXRzNh13xQBGzePHEeK --- .../random_ray_linear_source_stability/test.py | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/tests/regression_tests/random_ray_linear_source_stability/test.py b/tests/regression_tests/random_ray_linear_source_stability/test.py index 7eac708f301..b6946cad6e0 100644 --- a/tests/regression_tests/random_ray_linear_source_stability/test.py +++ b/tests/regression_tests/random_ray_linear_source_stability/test.py @@ -15,10 +15,10 @@ def _cleanup(self): def test_random_ray_linear_source_stability(): - # A linear source run in the regime that stresses the gradient limiter - # and the batch-consistent flux update: the naive volume estimator, an - # overlay source-region mesh whose cells receive modest per-batch hit - # counts, and the example's optically thin scattering-dominated interior. + # A linear source run in the regime that stresses the batch-consistent + # flux update: the naive volume estimator, an overlay source-region mesh + # whose cells receive modest per-batch hit counts, and the example's + # optically thin scattering-dominated interior. openmc.reset_auto_ids() model = random_ray_three_region_cube() model.settings.random_ray['source_shape'] = 'linear' From d7cc3550b6756383194bec41e5978f15c2aec3ba Mon Sep 17 00:00:00 2001 From: John Tramm Date: Thu, 10 Sep 2026 16:58:39 +0000 Subject: [PATCH 6/9] Carry the batch-consistency term into the strict estimator's rescue The strict adaptive estimator rescues a negative iterate by rescaling the transport part of the update to the batch volume, reconstructing the naive-volume update from the additive term. With the batch-consistent linear source update that term depends on the volume: a region updated with its batch volume adds the gradient dotted with the batch centroid offset. The additive term now takes the volume and the linear source solver overrides it, so the flux update and the rescue draw the term from one place and the rescued candidate is exactly the naive-volume update in linear mode too. Only the strict adaptive linear reference results move. Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_01P9rxXRzNh13xQBGzePHEeK --- .../openmc/random_ray/flat_source_domain.h | 2 +- .../openmc/random_ray/linear_source_domain.h | 1 + src/random_ray/flat_source_domain.cpp | 25 ++++---- src/random_ray/linear_source_domain.cpp | 62 +++++++++++-------- .../strict_adaptive/results_true.dat | 12 ++-- 5 files changed, 57 insertions(+), 45 deletions(-) diff --git a/include/openmc/random_ray/flat_source_domain.h b/include/openmc/random_ray/flat_source_domain.h index c87ff48f301..f1be1dded9f 100644 --- a/include/openmc/random_ray/flat_source_domain.h +++ b/include/openmc/random_ray/flat_source_domain.h @@ -242,7 +242,7 @@ class FlatSourceDomain { virtual void set_flux_to_flux_plus_source(int64_t sr, double volume, int g); void set_flux_to_source(int64_t sr, int g); virtual void set_flux_to_old_flux(int64_t sr, int g); - double flux_additive_term(int64_t sr, int g) const; + virtual double flux_additive_term(int64_t sr, int g, double volume) const; double stabilized_flux(int64_t sr, int g, double phi_new) const; //! Adaptive-estimator "strong source" test. Returns true if, in any group, diff --git a/include/openmc/random_ray/linear_source_domain.h b/include/openmc/random_ray/linear_source_domain.h index 0098c782001..4655f761d14 100644 --- a/include/openmc/random_ray/linear_source_domain.h +++ b/include/openmc/random_ray/linear_source_domain.h @@ -34,6 +34,7 @@ class LinearSourceDomain : public FlatSourceDomain { // Methods void set_flux_to_flux_plus_source(int64_t sr, double volume, int g) override; void set_flux_to_old_flux(int64_t sr, int g) override; + double flux_additive_term(int64_t sr, int g, double volume) const override; }; // class LinearSourceDomain diff --git a/src/random_ray/flat_source_domain.cpp b/src/random_ray/flat_source_domain.cpp index f3567e6f9db..0b1f9a757fe 100644 --- a/src/random_ray/flat_source_domain.cpp +++ b/src/random_ray/flat_source_domain.cpp @@ -342,13 +342,16 @@ void FlatSourceDomain::normalize_scalar_flux_and_volumes( } } -// The additive term of the flux update for a source region and group. A -// material region adds its reduced source q/Sigma_t. A void region has no -// such term and instead adds a bounded contribution from its external -// source, which is nonzero only in fixed source mode. The same term is used -// by the strict estimator's rescue, which rescales only the transport part -// of an update, so the two cannot drift apart. -double FlatSourceDomain::flux_additive_term(int64_t sr, int g) const +// The additive term of the flux update for a source region and group, +// given the volume the update divides the transport term by. A material +// region adds its reduced source q/Sigma_t. A void region has no such term +// and instead adds a bounded contribution from its external source, which +// is nonzero only in fixed source mode. The same term is used by the strict +// estimator's rescue, which rescales only the transport part of an update, +// so the two cannot drift apart. The linear source solver's term depends on +// the volume (see its override); the flat source term does not. +double FlatSourceDomain::flux_additive_term( + int64_t sr, int g, double volume) const { if (source_regions_.material(sr) == MATERIAL_VOID) { if (settings::run_mode == RunMode::FIXED_SOURCE) { @@ -373,7 +376,7 @@ void FlatSourceDomain::set_flux_to_flux_plus_source( source_regions_.density_mult(sr); source_regions_.scalar_flux_new(sr, g) /= (sigma_t * volume); } - source_regions_.scalar_flux_new(sr, g) += flux_additive_term(sr, g); + source_regions_.scalar_flux_new(sr, g) += flux_additive_term(sr, g, volume); } // Applies the "diagonal stabilization" technique developed by Gunow et al. @@ -665,9 +668,9 @@ int64_t FlatSourceDomain::add_source_to_scalar_flux() // flat shapes. if (is_strict && phi < 0.0) { if (volume != volume_iteration) { - double additive = flux_additive_term(sr, g); - double rescued = - (raw - additive) * (volume / volume_iteration) + additive; + double rescued = (raw - flux_additive_term(sr, g, volume)) * + (volume / volume_iteration) + + flux_additive_term(sr, g, volume_iteration); phi = stabilized_flux(sr, g, rescued); region_rescued = true; } diff --git a/src/random_ray/linear_source_domain.cpp b/src/random_ray/linear_source_domain.cpp index 838b9139593..e344fc09b28 100644 --- a/src/random_ray/linear_source_domain.cpp +++ b/src/random_ray/linear_source_domain.cpp @@ -189,6 +189,40 @@ void LinearSourceDomain::normalize_scalar_flux_and_volumes( } } +// A material region updated with its own batch (naive) volume adds back, +// on top of its flat source, the gradient part of the source that the +// batch's rays actually integrated. The sweep evaluates the linear source +// against the accumulated centroid, but the batch's tracks average it at +// their own centroid, so the mean emission the tracks sampled is the flat +// source plus the gradient dotted with the batch centroid offset. Including +// that term makes the naive-volume update an exact per-batch track +// identity, in which the flux estimate is the track-length average of the +// angular flux and so inherits its sign. Omitting it leaves gradient-scale +// noise in the flux with no flat-source counterpart, which in +// near-cancellation regions (scattering ratio near one) can exceed the flux +// itself and ignite self-sustaining negativity. +// +// A region updated with the simulation-averaged volume keeps the original +// form with no added term. The omitted term cannot bias the accumulated +// result, since the accumulated centroid is built from the same batch +// centroids and the deviations sum to exactly zero. The original form +// samples the residual between the angular flux and the linear source +// model, whose noise is smaller than the direct form's wherever the model +// tracks the field. Each volume treatment is thus paired with the update +// form that is exact or optimal for it. Void regions have no gradient term +// and take the flat source treatment. +double LinearSourceDomain::flux_additive_term( + int64_t sr, int g, double volume) const +{ + double term = FlatSourceDomain::flux_additive_term(sr, g, volume); + if (source_regions_.material(sr) != MATERIAL_VOID && + volume == source_regions_.volume_naive(sr)) { + term += source_regions_.source_gradients(sr, g).dot( + source_regions_.centroid_offset(sr)); + } + return term; +} + void LinearSourceDomain::set_flux_to_flux_plus_source( int64_t sr, double volume, int g) { @@ -197,33 +231,7 @@ void LinearSourceDomain::set_flux_to_flux_plus_source( FlatSourceDomain::set_flux_to_flux_plus_source(sr, volume, g); } else { source_regions_.scalar_flux_new(sr, g) /= volume; - source_regions_.scalar_flux_new(sr, g) += source_regions_.source(sr, g); - // A region updated with its own batch (naive) volume also adds back the - // gradient part of the source that the batch's rays actually - // integrated. The sweep evaluates the linear source against the - // accumulated centroid, but the batch's tracks average it at their own - // centroid, so the mean emission the tracks sampled is the flat source - // plus the gradient dotted with the batch centroid offset. Including - // that term makes the naive-volume update an exact per-batch track - // identity, in which the flux estimate is the track-length average of - // the angular flux and so inherits its sign. Omitting it leaves - // gradient-scale noise in the flux with no flat-source counterpart, - // which in near-cancellation regions (scattering ratio near one) can - // exceed the flux itself and ignite self-sustaining negativity. - // - // A region updated with the simulation-averaged volume keeps the - // original form with no added term. The omitted term cannot bias the - // accumulated result, since the accumulated centroid is built from the - // same batch centroids and the deviations sum to exactly zero. The - // original form samples the residual between the angular flux and the - // linear source model, whose noise is smaller than the direct form's - // wherever the model tracks the field. Each volume treatment is thus - // paired with the update form that is exact or optimal for it. - if (volume == source_regions_.volume_naive(sr)) { - source_regions_.scalar_flux_new(sr, g) += - source_regions_.source_gradients(sr, g).dot( - source_regions_.centroid_offset(sr)); - } + source_regions_.scalar_flux_new(sr, g) += flux_additive_term(sr, g, volume); } // If a source region is small, then the moments are likely noisy, so we zero // them. This is reasonable, given that small regions can get by with a flat diff --git a/tests/regression_tests/random_ray_volume_estimator_linear/strict_adaptive/results_true.dat b/tests/regression_tests/random_ray_volume_estimator_linear/strict_adaptive/results_true.dat index f9bb6717569..13730adf337 100644 --- a/tests/regression_tests/random_ray_volume_estimator_linear/strict_adaptive/results_true.dat +++ b/tests/regression_tests/random_ray_volume_estimator_linear/strict_adaptive/results_true.dat @@ -1,9 +1,9 @@ tally 1: -2.286190E+00 -2.736953E-01 +2.282401E+00 +2.727850E-01 tally 2: -1.350210E-01 -1.108740E-03 +1.297855E-01 +1.024290E-03 tally 3: -8.853520E-03 -5.120776E-06 +8.691375E-03 +5.002539E-06 From 2140a7169b46c028cbe33f1f41885bdf3b0be410 Mon Sep 17 00:00:00 2001 From: John Tramm Date: Thu, 10 Sep 2026 17:25:20 +0000 Subject: [PATCH 7/9] Pass the flux update's volume choice explicitly The linear source solver decided whether an update used the batch volume by comparing the volume it was handed with the region's batch volume, a floating-point equality that also holds by coincidence on the first batch for every estimator. The decision is already made at the call site, so it is now passed as a flag through the flux update and the additive term instead of being reconstructed. Results are unchanged. The comment on the simulation-averaged path no longer claims the omitted term sums to exactly zero: each batch's term is taken against the running centroid with that batch's gradient, so it has no persistent sign and its contribution shrinks with the batch count, which is what the pairing relies on. A stale cross-reference is fixed. Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_01P9rxXRzNh13xQBGzePHEeK --- .../openmc/random_ray/flat_source_domain.h | 5 ++- .../openmc/random_ray/linear_source_domain.h | 6 ++- src/random_ray/flat_source_domain.cpp | 18 +++++---- src/random_ray/linear_source_domain.cpp | 37 ++++++++++--------- 4 files changed, 36 insertions(+), 30 deletions(-) diff --git a/include/openmc/random_ray/flat_source_domain.h b/include/openmc/random_ray/flat_source_domain.h index f1be1dded9f..c1f2fdf39a0 100644 --- a/include/openmc/random_ray/flat_source_domain.h +++ b/include/openmc/random_ray/flat_source_domain.h @@ -239,10 +239,11 @@ class FlatSourceDomain { int target_material_id, const vector& instances); void apply_external_source_to_cell_and_children( int32_t i_cell, int src_idx, int32_t target_material_id); - virtual void set_flux_to_flux_plus_source(int64_t sr, double volume, int g); + virtual void set_flux_to_flux_plus_source( + int64_t sr, double volume, bool batch_volume, int g); void set_flux_to_source(int64_t sr, int g); virtual void set_flux_to_old_flux(int64_t sr, int g); - virtual double flux_additive_term(int64_t sr, int g, double volume) const; + virtual double flux_additive_term(int64_t sr, int g, bool batch_volume) const; double stabilized_flux(int64_t sr, int g, double phi_new) const; //! Adaptive-estimator "strong source" test. Returns true if, in any group, diff --git a/include/openmc/random_ray/linear_source_domain.h b/include/openmc/random_ray/linear_source_domain.h index 4655f761d14..bdcb006d4a2 100644 --- a/include/openmc/random_ray/linear_source_domain.h +++ b/include/openmc/random_ray/linear_source_domain.h @@ -32,9 +32,11 @@ class LinearSourceDomain : public FlatSourceDomain { protected: //---------------------------------------------------------------------------- // Methods - void set_flux_to_flux_plus_source(int64_t sr, double volume, int g) override; + void set_flux_to_flux_plus_source( + int64_t sr, double volume, bool batch_volume, int g) override; void set_flux_to_old_flux(int64_t sr, int g) override; - double flux_additive_term(int64_t sr, int g, double volume) const override; + double flux_additive_term( + int64_t sr, int g, bool batch_volume) const override; }; // class LinearSourceDomain diff --git a/src/random_ray/flat_source_domain.cpp b/src/random_ray/flat_source_domain.cpp index 0b1f9a757fe..d2f13f6b7d9 100644 --- a/src/random_ray/flat_source_domain.cpp +++ b/src/random_ray/flat_source_domain.cpp @@ -343,15 +343,16 @@ void FlatSourceDomain::normalize_scalar_flux_and_volumes( } // The additive term of the flux update for a source region and group, -// given the volume the update divides the transport term by. A material +// given whether the update divides the transport term by the region's own +// batch volume (as opposed to its simulation-averaged volume). A material // region adds its reduced source q/Sigma_t. A void region has no such term // and instead adds a bounded contribution from its external source, which // is nonzero only in fixed source mode. The same term is used by the strict // estimator's rescue, which rescales only the transport part of an update, // so the two cannot drift apart. The linear source solver's term depends on -// the volume (see its override); the flat source term does not. +// the volume choice (see its override); the flat source term does not. double FlatSourceDomain::flux_additive_term( - int64_t sr, int g, double volume) const + int64_t sr, int g, bool batch_volume) const { if (source_regions_.material(sr) == MATERIAL_VOID) { if (settings::run_mode == RunMode::FIXED_SOURCE) { @@ -364,7 +365,7 @@ double FlatSourceDomain::flux_additive_term( } void FlatSourceDomain::set_flux_to_flux_plus_source( - int64_t sr, double volume, int g) + int64_t sr, double volume, bool batch_volume, int g) { int material = source_regions_.material(sr); int temp = source_regions_.temperature_idx(sr); @@ -376,7 +377,8 @@ void FlatSourceDomain::set_flux_to_flux_plus_source( source_regions_.density_mult(sr); source_regions_.scalar_flux_new(sr, g) /= (sigma_t * volume); } - source_regions_.scalar_flux_new(sr, g) += flux_additive_term(sr, g, volume); + source_regions_.scalar_flux_new(sr, g) += + flux_additive_term(sr, g, batch_volume); } // Applies the "diagonal stabilization" technique developed by Gunow et al. @@ -648,7 +650,7 @@ int64_t FlatSourceDomain::add_source_to_scalar_flux() // Hit this iteration: the flat source from the previous iteration plus // this iteration's transport contribution, normalized by the chosen // volume, then stabilized. - set_flux_to_flux_plus_source(sr, volume, g); + set_flux_to_flux_plus_source(sr, volume, use_naive_volume, g); double raw = source_regions_.scalar_flux_new(sr, g); double phi = stabilized_flux(sr, g, raw); // The strict adaptive estimator applies a per-batch fixup to @@ -668,9 +670,9 @@ int64_t FlatSourceDomain::add_source_to_scalar_flux() // flat shapes. if (is_strict && phi < 0.0) { if (volume != volume_iteration) { - double rescued = (raw - flux_additive_term(sr, g, volume)) * + double rescued = (raw - flux_additive_term(sr, g, false)) * (volume / volume_iteration) + - flux_additive_term(sr, g, volume_iteration); + flux_additive_term(sr, g, true); phi = stabilized_flux(sr, g, rescued); region_rescued = true; } diff --git a/src/random_ray/linear_source_domain.cpp b/src/random_ray/linear_source_domain.cpp index e344fc09b28..3d045dca85c 100644 --- a/src/random_ray/linear_source_domain.cpp +++ b/src/random_ray/linear_source_domain.cpp @@ -157,9 +157,9 @@ void LinearSourceDomain::normalize_scalar_flux_and_volumes( // the transport sweep evaluated the linear source against (the value // held here before this batch is folded in). The flux update uses it to // add back the source as actually sampled by this batch's tracks (see - // set_flux_to_flux_plus_source). Regions with no prior accumulated - // volume used per-segment midpoints in place of a centroid, for which - // the batch offset is identically zero. + // flux_additive_term). Regions with no prior accumulated volume used + // per-segment midpoints in place of a centroid, for which the batch + // offset is identically zero. if (source_regions_.volume_t(sr) > 0.0 && source_regions_.volume(sr) > 0.0) { source_regions_.centroid_offset(sr) = @@ -203,20 +203,20 @@ void LinearSourceDomain::normalize_scalar_flux_and_volumes( // itself and ignite self-sustaining negativity. // // A region updated with the simulation-averaged volume keeps the original -// form with no added term. The omitted term cannot bias the accumulated -// result, since the accumulated centroid is built from the same batch -// centroids and the deviations sum to exactly zero. The original form -// samples the residual between the angular flux and the linear source -// model, whose noise is smaller than the direct form's wherever the model -// tracks the field. Each volume treatment is thus paired with the update -// form that is exact or optimal for it. Void regions have no gradient term -// and take the flat source treatment. +// form with no added term. Each batch's centroid scatters about the +// accumulated centroid it feeds, so the omitted term has no persistent +// sign and its contribution to the accumulated flux shrinks with the +// number of batches, while the original form samples the residual between +// the angular flux and the linear source model, whose noise is smaller +// than the direct form's wherever the model tracks the field. Each volume +// treatment is thus paired with the update form that is exact or better +// for it. Void regions have no gradient term and take the flat source +// treatment. double LinearSourceDomain::flux_additive_term( - int64_t sr, int g, double volume) const + int64_t sr, int g, bool batch_volume) const { - double term = FlatSourceDomain::flux_additive_term(sr, g, volume); - if (source_regions_.material(sr) != MATERIAL_VOID && - volume == source_regions_.volume_naive(sr)) { + double term = FlatSourceDomain::flux_additive_term(sr, g, batch_volume); + if (batch_volume && source_regions_.material(sr) != MATERIAL_VOID) { term += source_regions_.source_gradients(sr, g).dot( source_regions_.centroid_offset(sr)); } @@ -224,14 +224,15 @@ double LinearSourceDomain::flux_additive_term( } void LinearSourceDomain::set_flux_to_flux_plus_source( - int64_t sr, double volume, int g) + int64_t sr, double volume, bool batch_volume, int g) { int material = source_regions_.material(sr); if (material == MATERIAL_VOID) { - FlatSourceDomain::set_flux_to_flux_plus_source(sr, volume, g); + FlatSourceDomain::set_flux_to_flux_plus_source(sr, volume, batch_volume, g); } else { source_regions_.scalar_flux_new(sr, g) /= volume; - source_regions_.scalar_flux_new(sr, g) += flux_additive_term(sr, g, volume); + source_regions_.scalar_flux_new(sr, g) += + flux_additive_term(sr, g, batch_volume); } // If a source region is small, then the moments are likely noisy, so we zero // them. This is reasonable, given that small regions can get by with a flat From df51897897c9c7814ca0fcfc1e3a94f9834deb95 Mon Sep 17 00:00:00 2001 From: John Tramm Date: Thu, 10 Sep 2026 17:25:20 +0000 Subject: [PATCH 8/9] Starve the consistency test and drop its inert overlay mesh The test's 12x12x12 overlay mesh coincided with the example's lattice and subdivided nothing, and with 90 rays no region was ever missed, so the "modest per-batch hit counts" its comment described did not exist. It now runs 20 rays over the example's 1,728 lattice regions, missing about 4% of them per batch, which is the regime the batch-consistent update targets and where its effect is ten times larger. The reference results are regenerated; the test runs in under 0.2 s at 2 threads, holds at 1, 2 and 4 threads, and fails against develop's binary. Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_01P9rxXRzNh13xQBGzePHEeK --- .../inputs_true.dat | 12 +----------- .../results_true.dat | 12 ++++++------ .../random_ray_linear_source_stability/test.py | 15 ++++++--------- 3 files changed, 13 insertions(+), 26 deletions(-) diff --git a/tests/regression_tests/random_ray_linear_source_stability/inputs_true.dat b/tests/regression_tests/random_ray_linear_source_stability/inputs_true.dat index 73379ec3240..26e066cea1f 100644 --- a/tests/regression_tests/random_ray_linear_source_stability/inputs_true.dat +++ b/tests/regression_tests/random_ray_linear_source_stability/inputs_true.dat @@ -191,7 +191,7 @@ fixed source - 90 + 20 60 30 @@ -217,17 +217,7 @@ true linear naive - - - - - - - 12 12 12 - 0.0 0.0 0.0 - 30.0 30.0 30.0 - diff --git a/tests/regression_tests/random_ray_linear_source_stability/results_true.dat b/tests/regression_tests/random_ray_linear_source_stability/results_true.dat index 1401ddad731..7bbb63f87e2 100644 --- a/tests/regression_tests/random_ray_linear_source_stability/results_true.dat +++ b/tests/regression_tests/random_ray_linear_source_stability/results_true.dat @@ -1,9 +1,9 @@ tally 1: -3.518081E+00 -4.139523E-01 +3.332488E+00 +3.785882E-01 tally 2: -1.729928E-01 -1.032644E-03 +1.864784E-01 +1.281925E-03 tally 3: -1.136650E-02 -4.448920E-06 +1.313058E-02 +6.808347E-06 diff --git a/tests/regression_tests/random_ray_linear_source_stability/test.py b/tests/regression_tests/random_ray_linear_source_stability/test.py index b6946cad6e0..bd4a1e5e44a 100644 --- a/tests/regression_tests/random_ray_linear_source_stability/test.py +++ b/tests/regression_tests/random_ray_linear_source_stability/test.py @@ -16,19 +16,16 @@ def _cleanup(self): def test_random_ray_linear_source_stability(): # A linear source run in the regime that stresses the batch-consistent - # flux update: the naive volume estimator, an overlay source-region mesh - # whose cells receive modest per-batch hit counts, and the example's - # optically thin scattering-dominated interior. + # flux update: the naive volume estimator updates every hit region with + # its own batch volume, so the consistency term applies throughout, and + # a ray count low enough to starve the example's 1728 regions (about 4% + # of them missed per batch) makes the batch centroids scatter widely + # about the accumulated ones. openmc.reset_auto_ids() model = random_ray_three_region_cube() model.settings.random_ray['source_shape'] = 'linear' model.settings.random_ray['volume_estimator'] = 'naive' - mesh = openmc.RegularMesh() - mesh.lower_left = (0.0, 0.0, 0.0) - mesh.upper_right = (30.0, 30.0, 30.0) - mesh.dimension = (12, 12, 12) - model.settings.random_ray['source_region_meshes'] = [ - (mesh, [model.geometry.root_universe])] + model.settings.particles = 20 model.settings.inactive = 30 model.settings.batches = 60 harness = MGXSTestHarness('statepoint.60.h5', model) From 1425ff86ba9aff0a919b273ff8578eb0130e0ae4 Mon Sep 17 00:00:00 2001 From: John Tramm Date: Thu, 10 Sep 2026 17:25:20 +0000 Subject: [PATCH 9/9] State the consistency note in the section's notation The note used a barred flat source, a new centroid symbol and no group index where the surrounding linear source section writes the region source as Q_{i,g}, the centroid as r_c and the flux update's additive term as Q/Sigma_t. It now uses the section's symbols and equations, and describes the omitted term on the simulation-averaged path as having no persistent sign rather than summing to exactly zero. Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_01P9rxXRzNh13xQBGzePHEeK --- docs/source/methods/random_ray.rst | 40 ++++++++++++++++-------------- 1 file changed, 22 insertions(+), 18 deletions(-) diff --git a/docs/source/methods/random_ray.rst b/docs/source/methods/random_ray.rst index 80951ca61f9..c139f05d29a 100644 --- a/docs/source/methods/random_ray.rst +++ b/docs/source/methods/random_ray.rst @@ -1039,29 +1039,33 @@ Consistency of the Scalar Flux Estimate ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ One subtlety of the linear source scheme deserves note. Intuitively, the -issue is a mismatch of statistics: the naive volume treatment updates the -flux from a single batch's rays, while the linear source is anchored to the -simulation-averaged centroid, so the two halves of the update describe -different sets of tracks and the exactness the naive treatment promises is -quietly broken. Concretely, the transport sweep evaluates each region's -linear source against the accumulated centroid :math:`\mathbf{C}_i`, but a +issue is a mismatch of statistics: the naive volume treatment of Equation +:eq:`phi_naive` updates the flux from a single batch's rays, while the +linear source is anchored to the simulation-averaged centroid, so the two +halves of the update describe different sets of tracks and the exactness +the naive treatment promises is quietly broken. Concretely, the transport +sweep evaluates each region's linear source of Equation :eq:`region_source` +against the accumulated centroid :math:`\mathbf{r}_{\mathrm{c}}`, but a batch's tracks average that source at their own track-length-weighted -centroid :math:`\mathbf{c}_{i,b}`, so the mean emission the batch actually -integrates is +centroid :math:`\mathbf{r}_{\mathrm{c},b}`, so the mean source the batch +actually integrates is .. math:: :label: batch_sampled_source - \bar{Q}_i + \vec{Q}_i \cdot (\mathbf{c}_{i,b} - \mathbf{C}_i)\;. - -A flux update that adds back only :math:`\bar{Q}_i` absorbs the difference -as gradient-scale noise, which in optically thin scatter-fed regions can -ignite self-sustaining negative fluxes. OpenMC therefore adds back the full -batch-sampled source whenever a region updates with its own batch -(iteration) volume, making that update exact for the batch's tracks. -Regions updating with the simulation-averaged volume keep the original -form, which is unbiased (the omitted deviations sum to zero against the -accumulated centroid) and carries less variance there. + Q_{i,g} + \boldsymbol{\vec{Q}}_{i,g} \cdot \left(\mathbf{r}_{\mathrm{c},b} + - \mathbf{r}_{\mathrm{c}}\right)\;. + +A flux update that adds back only :math:`Q_{i,g} / \Sigma_{t,i,g}` absorbs +the difference as gradient-scale noise, which in optically thin scatter-fed +regions can ignite self-sustaining negative fluxes. OpenMC therefore adds +back the full batch-sampled source (divided by :math:`\Sigma_{t,i,g}`) +whenever a region updates with its own batch volume, making that update +exact for the batch's tracks. Regions updating with the simulation-averaged +volume keep the original form: each batch's centroid scatters about the +accumulated centroid it feeds, so the omitted term has no persistent sign +and its contribution to the accumulated flux shrinks with the number of +batches, while the original form carries less variance there. .. _methods-shannon-entropy-random-ray: