A Batch Power Iteration Approach for the Iterative Quasi-Monte Carlo Method Using a Randomized-Halton Sequence

C.T. Kelley; Ilham Variansyah; Ryan G. McClarren; Samuel Pasmann

REVIEW 3 major objections 5 minor 18 references

A Batch Power Iteration Approach for the Iterative Quasi-Monte Carlo Method Using a Randomized-Halton Sequence

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read By replacing fixed-seed sampling with randomized Halton batches, the iQMC method removes ray-effect bias and preserves the O(N^{-1}) quasi-Monte Carlo convergence rate while cutting the required particles per batch by nearly two orders of…

desk verdict A plausible and clearly described batching variant of iQMC, but the O(N^{-1}) rate claim is under-evidenced: one benchmark, no fitted slope, hand-set batch counts, and a mesh plateau that truncates the range. read the letter →

arxiv 2501.06154 v1 pith:UGWBBZZ6 submitted 2025-01-10 physics.comp-ph

Samuel Pasmann , Ilham Variansyah , C.T. Kelley , Ryan G. McClarren This is my paper · ORCID

classification physics.comp-ph

keywords Quasi-MonteCarloNeutronTransportk-EigenvalueIterativeMethodHybridRandomizedHaltonsequenceBatchpoweriterationOwen'srandomization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that switching the Iterative Quasi-Monte Carlo (iQMC) neutron transport method from a fixed-seed to a batch approach with randomized Halton sequences preserves the theoretical QMC convergence rate of O($N^{{-1}}$) for k-effective and mean scalar flux error, while reducing the number of particle histories per batch by nearly two orders of magnitude. The authors argue that the fixed-seed approach, which resets particles to the same positions and directions each sweep, undersamples phase space and introduces ray-effect-like bias. They implement a batched power iteration that randomizes the Halton sequence with Owen's randomization each sweep, runs inactive batches to converge the source, then active batches to record statistics. In the Takeda-1 benchmark, the batch approach produces stable, accurate solutions with far fewer particles per sweep than fixed-seed, and averaged over 20 simulations the error converges at the QMC rate.

What carries the argument

The central object is the randomized Halton sequence produced by Owen's randomization, which constructs each sample point as individually uniform on [0,1]^D while collectively preserving low discrepancy, and can be called any number of times to generate unique sample sets. It is used inside a batch power iteration: each transport sweep draws a new RQMC sample set to initialize particle positions and directions, inactive batches converge the source strength, and active batches accumulate tally statistics. This mechanism removes the fixed-seed bias while retaining the O($N^{{-1}}$) convergence of QMC sampling.

What would settle it

Run the batched RQMC iQMC method on the Takeda-1 problem with a substantially finer mesh (or adaptive refinement near flux gradients) and check whether the k-effective and mean scalar flux errors continue to follow O($N^{{-1}}$) across a wider range of particle counts than shown in Figure 5. If the error curve flattens at the same particle count regardless of mesh, the claimed rate is an artifact of the mesh; if the curve continues, the claim is supported.

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Extended reading notes

Core claim

The central claim is that the batch RQMC iQMC method converges both the k-effective and the mean scalar flux error at the theoretical QMC convergence rate of O($N^{{-1}}$) despite the stochastic noise introduced by randomizing the low-discrepancy sequence each transport sweep. Using Owen's randomization of the Halton sequence to generate a fresh set of samples per batch, the method removes the under-sampling bias of the fixed-seed approach while the averaging over active batches reduces the introduced noise. On the Takeda-1 problem, the batch method produces stable and accurate solutions with nearly two orders of magnitude fewer particle histories per batch than the fixed-seed method, and the ensemble-averaged convergence follows the O($N^{{-1}}$) rate until the spatial mesh discretization error becomes the limiting factor.

Load-bearing premise

The O($N^{{-1}}$) convergence rate is measured on a single benchmark under the assumption that the scalar flux error is dominated by the RQMC sampling error, not by the coarse spatial mesh that visibly flattens the error at high particle counts.

Editorial extensions

If this is right

The batch method produces stable k-effective estimates in the Takeda-1 benchmark with roughly two orders of magnitude fewer particles per transport sweep than the fixed-seed method needs to avoid diverging.
Averaging the tally scores over active batches cancels most of the stochastic noise introduced by randomization, so the ensemble-mean error still follows the QMC O(N^{-1}) rate.
The scalar flux error plateaus at high particle counts because the 25x25x25 mesh limits accuracy; refining the mesh should extend the observable O(N^{-1}) regime.
By randomizing the sequence each batch, the method loses the linearity that fixed-seed iQMC used for Krylov solvers, so the batch approach trades solver acceleration for unbiased sampling.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the O(N^{-1}) rate reproduces on other benchmarks and meshes, batch RQMC iQMC could make quasi-Monte Carlo transport practical for deep-penetration or void-dominated problems where ray effects are severe.
Because the Halton sequence is less sensitive to sample count than Sobol, the batch scheme could naturally support varying particle numbers per batch, enabling adaptive refinement in the active phase.
The method effectively turns iQMC into a stochastic iterative map; existing theory of stochastic power iteration might help set inactive/active batch counts automatically rather than by user input.
One testable extension is to apply the same batch RQMC approach to the piecewise-linear source tilting from earlier iQMC work, which may push the convergence plateau to higher particle counts.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read the paper. The batched RQMC variant of iQMC is a genuine, small step forward: it replaces the fixed-seed sweep with Owen-scrambled Halton sequences per batch, removes the ray-effect bias the authors were chasing, and appears to stabilize k-effective with roughly two orders of magnitude fewer particles per batch. The algorithmic description is clear, the implementation in MC/DC is concrete, and the authors are honest that N_inactive and N_active are user inputs selected after inspecting the same problem, and that the scalar flux error plateaus on the 25^3 mesh.

What is new: this is the first application of RQMC batching to iQMC, as far as I can tell from the cited literature. The observation that the QMC O(N^{-1}) rate survives the added stochastic noise is useful if true. What is not established: the rate claim itself. Figure 5 is one benchmark, no fitted slope, no confidence band around the slope, and the O(N^{-1}) is a visual assertion. The plateau at high N, which the paper attributes to the coarse mesh, cuts off exactly the range where you would want to confirm the asymptotic rate. The k-effective curve is cleaner, but still one problem. The batch counts are not a priori; they are tuned on the same case, so the experiment is not a prediction. No code or data are included, so independent verification means reimplementation.

None of this is disqualifying. A plausible method with honest caveats and a clear numerical demonstration on a standard benchmark deserves referee time. I would send it for peer review and ask for a second benchmark, a fitted slope or at least a comparison to the theoretical rate on a range unaffected by the mesh, and a sensitivity study for N_inactive/N_active. If the code were released, that would help a lot.

Bottom line: useful for the neutron transport / hybrid MC community, not a major advance. I would not cite it in my own work unless I were actively working on iQMC, but I would recommend it to a student working on QMC transport. Yes to peer review.

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a batched power-iteration variant of the iterative Quasi-Monte Carlo (iQMC) method for multigroup neutron transport. Instead of resetting particles to the same low-discrepancy sequence every transport sweep (the fixed-seed approach), the new method uses Owen's randomization of the Halton sequence to generate fresh RQMC samples per sweep, with user-specified inactive and active batches to manage the stochastic noise. The method is implemented in MC/DC and tested on the Takeda-1 k-eigenvalue benchmark. The reported results show that the batched RQMC approach gives more stable and more accurate k-effective and scalar-flux estimates than fixed-seed iQMC, and the authors claim that the method converges at the theoretical QMC rate of O(N^-1) while needing roughly two orders of magnitude fewer particles per batch. However, the O(N^-1) claim rests on a single benchmark and is inferred visually without a fitted slope, and the paper itself notes that the scalar-flux error plateaus at high particle counts due to the 25x25x25 mesh.

Significance. The work addresses a genuine and well-known limitation of the iQMC method: the fixed-seed approach leaves parts of phase space permanently undersampled, producing ray-effect-like bias. Replacing fixed seeds with randomized low-discrepancy samples in a batch power iteration is a natural and promising remedy, and the authors implement it in MC/DC in a way that avoids costly exits from compiled code. The qualitative improvement over fixed-seed iQMC is clearly demonstrated in Figures 3 and 4, and the paper is generally clearly written. If the O(N^-1) convergence claim is confirmed with rigorous quantitative evidence, the contribution would be significant for the iQMC research community. As it stands, the central rate claim is not yet supported to the standard required by a journal publication, because the evidence is limited to one benchmark, the rate is not measured, and the observable convergence range is partly masked by the mesh-induced plateau.

major comments (3)

[Section 3, Figure 5] The central claim that the RQMC batch method converges both k-effective and mean scalar flux error at the theoretical QMC rate O(N^-1) is not quantitatively established. The paper reports no fitted slope, no confidence interval, and no comparison with the expected rate; the rate is inferred visually from a limited range of N. That range is further restricted by the scalar-flux plateau at high particle counts, which the text attributes to the coarse 25x25x25 mesh. Because the analog Monte Carlo reference is tallied on the same mesh, the comparison cannot cancel iQMC's piecewise-linear source discretization error, so the measured error mixes RQMC sampling error with spatial discretization error. Please fit the log-log slope in the pre-plateau region, report standard errors, and ideally repeat on at least one additional benchmark or refine the mesh to confirm that the plateau shifts and the asymptotic rate is reached.
[Section 2, Algorithm 1 and Section 3] The inactive and active batch counts (N_inactive = 200, N_active = 100) are user inputs selected after inspecting convergence on the same Takeda-1 problem, as the text acknowledges when it says these values 'can be approximated from studying the convergence of the solution in each problem.' This makes the experiment not an a priori test of the method, and the batch counts could be tuned to this particular benchmark. To support the general claim of iterative stability, either implement an adaptive stopping criterion based on the relative change of the source strength per batch, or provide a sensitivity study showing that the reported accuracy and convergence rate are robust to reasonable variations in N_inactive and N_active.
[Section 3, Figures 3-5] The paper claims 'nearly two orders of magnitude reduction in the number of particle histories per transport sweep' for the batch method relative to fixed-seed iQMC, but the quantitative support is unclear. The direct comparison in the text is between a fixed-seed run with N = 1.5E6 particles per sweep (Figure 3) and a batch run with N = 1E5 particles per sweep (Figure 4), which is a factor of 15, not two orders of magnitude. If the claim instead refers to total particle histories including inactive and active batches, that comparison should be made explicit with the relevant totals; otherwise the two-orders-of-magnitude statement should be revised.

minor comments (5)

[Algorithm 1] Line 14 of Algorithm 1 contains a typo: 'N_inactve' should be 'N_inactive'.
[Section 3, Figure 5] The caption states that solutions are 'shown with one standard deviation of the ensemble,' but the definition of the plotted 'mean scalar flux error' is never given. Please specify the normalization and the set of cells over which the error is averaged (e.g., relative L2 error over all mesh cells).
[Section 2] The text spells the library name as 'Scipy' in Section 2; the reference list correctly uses 'SciPy'. Please use a consistent spelling.
[Section 3, Figure 4] The caption uses 'randomized-halton' while the abstract and elsewhere use 'Randomized-Halton' or 'randomized-Halton'; please standardize the capitalization.
[General] The paper would benefit from a brief discussion of the trade-off that the batched RQMC approach destroys the linearity assumption used for Krylov solvers in prior iQMC work; this is mentioned in Section 1.2 but never revisited in the conclusions, where the authors state the method 'negate[s] the use of linear Krylov solvers' without commenting on the practical cost of that loss.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the batched RQMC iQMC convergence claim is evaluated against an independent analog Monte Carlo benchmark rather than derived from the method's own inputs.

full rationale

The central claim is that batched randomized-Halton iQMC converges k-effective and mean scalar flux error at O(N^{-1}). The supporting evidence is the Takeda-1 benchmark comparison in Section 3, Figure 5, where iQMC results are compared with an independent analog multigroup Monte Carlo reference generated in MC/DC. The O(N^{-1}) rate is asserted from the observed convergence of the ensemble-averaged error; it is not obtained by fitting a parameter to the data and then relabeling the fit as a prediction. The fixed-seed iQMC framework and mesh-error observations are supported by self-citations ([1], [2], [6], [7]), but these citations do not carry the RQMC-batching claim: the improvement is demonstrated by direct numerical comparison against fixed-seed and pseudo-random batching on the same problem. The inactive/active batch counts (200/100) are user-selected algorithmic inputs chosen after inspecting convergence, not parameters fitted to force the rate; they affect statistical precision but do not by construction produce the claimed O(N^{-1}) trend. The paper itself notes that the scalar-flux error plateaus at high N due to the coarse 25x25x25 mesh (Section 3, Figure 5), which limits the quantitative support for the rate, but this is a correctness/evidence concern, not circularity. No equation or definition in the paper reduces the claimed result to its own inputs.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard QMC theory and numerical stability assumptions. The only hand-set constants are the inactive and active batch counts, chosen by inspecting the same benchmark problem. No new physical entities are introduced.

free parameters (2)

N_inactive = 200
User input; set to 200 for the final experiment by observing that k-effective stabilized around 200 sweeps in Figure 3 of the same Takeda-1 problem.
N_active = 100
User input; set to 100 active batches in the final experiment. No convergence criterion or sensitivity study is provided.

assumptions (4)

standard math The theoretical QMC convergence rate is O(N^{-1}) for sufficiently smooth integrands.
Relied on throughout Sections 1 and 3 as the benchmark for the empirical convergence claim.
domain assumption Owen's randomization of the Halton sequence produces point sets that are individually uniform and collectively low-discrepancy, so each batch is an unbiased sample.
Needed so that averaging active batches converges to the true solution; cited from reference 12 but not proved or numerically checked in this paper.
domain assumption The iQMC power iteration remains convergent when per-batch RQMC noise is added, provided enough inactive batches are run.
The paper selects 200 inactive batches empirically and assumes the source strength stabilizes before active averaging begins (Sections 2 and 3).
domain assumption The analog MC/DC reference solution is accurate enough to define scalar flux error.
The reference uses 50 million histories per batch with 10 inactive and 20 active batches (Section 3), and no statistical uncertainty on the reference is reported.

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Pith. "Pith review of A Batch Power Iteration Approach for the Iterative Quasi-Monte Carlo Method Using a Randomized-Halton Sequence." pith.science (2026). https://pith.science/paper/UGWBBZZ6

@misc{pith2026250106154,
  author       = {Pith},
  title        = {Pith review of: A Batch Power Iteration Approach for the Iterative Quasi-Monte Carlo Method Using a Randomized-Halton Sequence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UGWBBZZ6}},
  note         = {Machine review of arXiv:2501.06154}
}

abstract

The Iterative Quasi-Monte Carlo (iQMC) method is a recently developed hybrid method for neutron transport simulations. iQMC replaces standard quadrature techniques used in deterministic linear solvers with Quasi-Monte Carlo simulation for accurate and efficient solutions to the neutron transport equation. Previous iQMC studies utilized a fixed-seed approach wherein particles were reset to the same initial position and direction of travel at the start of every transport sweep. While the QMC samples offered greatly improved uniformity compared to pseudo-random samples, the fixed-seed approach meant that some regions of the problem were under-sampled and resulted in errors similar to ray effects observed in discrete ordinates methods. This work explores using randomized-Quasi Monte Carlo techniques (RQMC) to generate unique sets of QMC samples for each transport sweep and gain a much-improved sampling of the phase space. The use of RQMC introduces some stochastic noise to iQMC's iterative process, which was previously absent. To compensate, we adopt a ``batch'' approach similar to typical Monte Carlo k-eigenvalue problems, where the iQMC source is converged over $N_\text{inactive}$ batches, then results from $N_\text{active}$ batches are recorded and used to calculate the average and standard deviation of the solution. The RQMC batch method was implemented in the Monte Carlo Dynamic Code (MC/DC) and is shown to be a large improvement over the fixed-seed method. The batch method was able to provide iteratively stable and more accurate solutions with nearly two orders of magnitude reduction in the number of particle histories per batch. Notably, despite introducing some stochastic noise to the solution, the RQMC batch approach converges both the k-effective and mean scalar flux error at the theoretical QMC convergence rate of $O(N^{-1})$.

Figures

Figures reproduced from arXiv: 2501.06154 by the authors.

**Figure 2.** The Takeda-1 k-Eigenvalue benchmark problem [17]. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Updated k-Effective approximation after each transport sweep from the Takeda-1 problem [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Updated k-Effective approximation after each transport sweep from the Takeda-1 problem [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: Batch iQMC solution convergence averaged across an ensemble of 20 simulations and shown [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

A Quasi–Monte Carlo Method With Krylov Linear Solvers for Multigroup Neutron Transport Simulations

S. Pasmann, I. Variansyah, C. Kelley, and R. McClarren. “A Quasi–Monte Carlo Method With Krylov Linear Solvers for Multigroup Neutron Transport Simulations.” Nuclear Science and Engineering , volume 197(6), pp. 1159–1173 (2023)

work page 2023

[2] [2]

Pasmann, I

S. Pasmann, I. Variansyah, C. Kelley, and R. G. McClarren. “Mitigating Spatial Error in the Iterative Quasi–Monte Carlo (iQMC) Method for Neutron Transport Simulations with Linear Discontinuous Source Tilting and Effective Scattering and Fission Rate Tallies.” Nuclear Science and Engineering, pp. 1–16 (2024)

work page 2024

[3] [3]

On the efficiency of certain quasi-random sequences of points in evaluating multi- dimensional integrals

J. H. Halton. “On the efficiency of certain quasi-random sequences of points in evaluating multi- dimensional integrals.” Numerische Mathematik, volume 2, pp. 84–90 (1960)

work page 1960

[4] [4]

On the distribution of points in a cube and the approximate evaluation of integrals

I. M. Sobol’. “On the distribution of points in a cube and the approximate evaluation of integrals.” Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, volume 7(4), pp. 784–802 (1967)

work page 1967

[5] [5]

A comparison of three methods for selecting values of input variables in the analysis of output from a computer code

M. D. McKay, R. J. Beckman, and W. J. Conover. “A comparison of three methods for selecting values of input variables in the analysis of output from a computer code.” Technometrics, volume 42(1), pp. 55–61 (2000)

work page 2000

[6] [6]

iQMC: Iterative Quasi-Monte Carlo for k-Eigenvalue Neutron Transport Simulations

S. Pasmann, I. Variansyah, C. Kelley, and R. G. McClarren. “iQMC: Iterative Quasi-Monte Carlo for k-Eigenvalue Neutron Transport Simulations.” In International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering. Niagara Falls, Ontario, Canada (2023)

work page 2023

[7] [7]

Convergent transport source iteration calculations with Quasi-Monte Carlo

S. Pasmann, I. Variansyah, and R. G. McClarren. “Convergent transport source iteration calculations with Quasi-Monte Carlo.” Transactions of the American Nuclear Society, volume 124, pp. 192–195 (2021)

work page 2021

[8] [8]

McClarren

R. McClarren. Computational Nuclear Engineering and Radiological Science Using Python. Academic Press (2018)

work page 2018