Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

Synthetic Acceleration Preconditioners for Parametric Radiative Transfer Equations based on Trajectory-Aware Reduced Order Models

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Trajectory-aware ROM preconditioners solve parametric radiative transfer in just a few online iterations.

desk verdict Trajectory-aware ROM acceleration for RTE is a real, well-scoped improvement over the authors' earlier ROMSAD; the empirical case is plausible but needs code-release and a fix to the pin-cell training-data issue before I'd trust the '2-3 iterations' headline. read the letter →

arxiv 2509.05001 v1 pith:UWXSPFWU submitted 2025-09-05 math.NA cs.NA

Ning Tang , Zhichao Peng This is my paper · ORCID
classification math.NAcs.NA MSC 65F1065F0865N2285A25
keywords parametricradiativetransfersyntheticaccelerationreducedordermodeltrajectory-awareROMsourceiterationFGMRESlow-rankstructuretransportsweeps
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

Solving radiative transfer equations repeatedly for many parameter values is a bottleneck in design, imaging, and uncertainty studies. This paper argues that the usual acceleration strategy—diffusion synthetic acceleration plus a reduced-order model (the ROMSAD scheme)—loses efficiency after the first iteration because the reduced model is built from residual trajectories generated by a fixed preconditioner, while online residuals are generated by the reduced model itself. The proposed trajectory-aware framework removes this mismatch by constructing a sequence of ROMs offline, each one using corrections from the previous ROM-based iteration, so the offline and online residual paths agree. In numerical tests the resulting preconditioner converges in 2–3 online iterations for a parametric lattice problem and substantially beats both DSA and ROMSAD, with only marginal extra offline cost.

What carries the argument

The central object is the trajectory-aware ROM sequence {U_rl^{(l)}}_{l=1}^{N_w}: one reduced basis per iteration, each obtained by SVD of correction snapshots δf_μ^{(l)} = f_μ − f_μ^{(l)} that were produced by running the earlier ROM-based corrections in the offline loop. The associated ROM synthetic-acceleration preconditioner for iteration l is M^{-1}_{ROM,l} = I + C^{-1}_{ROM,l} Σ_s, with C^{-1}_{ROM,l} built from the projected reduced operator (U_rl^{(l)})^T A_μ U_rl^{(l)} and the angular sums U_iso and U_ρ. After N_w iterations the solver switches to DSA. This construction keeps the reduced spaces consistent with the actual residual path of the preconditioned iteration; inside FGMRES,

What would settle it

Choose a test parameter at the edge of (or slightly outside) the training range of the lattice problem, run TAR-IG, and record the residual after each ROM correction and after the switch to DSA; if the residual after the first ROM correction is not smaller than DSA's and convergence takes more than 2–3 sweeps, the paper's central speed claim fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that the efficiency degradation of ROM-enhanced synthetic acceleration is caused by a trajectory mismatch, not by insufficient basis accuracy. In ROMSAD, correction snapshots δf_μ^{(l)} = f_μ − f_μ^{(l)} are generated offline with a fixed DSA preconditioner, so the basis encodes residuals evolving under DSA; online, the residual path is set by the ROM preconditioner itself, and the two diverge after the first correction. The trajectory-aware framework builds a separate reduced basis for each of the first N_w iterations, computing each new basis from snapshots taken with the preceding ROM-based corrections. This sequential construction makes the offline and online

Load-bearing premise

The correction snapshots built from training parameters must stay representative for unseen test parameters; the paper gives no theoretical bound tying reduced-basis dimension to convergence rate.

Editorial extensions

If this is right

  • For multi-query RTE workloads such as optimization, inverse problems, and uncertainty quantification, the dominant per-query cost can drop from tens of transport sweeps to a few sweeps.
  • Because the ROMs are tailored to the preconditioner's own trajectory, the method remains effective under loose POD truncation tolerances, where ROMSAD loses robustness.
  • The same trajectory-aware construction works for both source iteration and FGMRES, so robust Krylov solvers inherit the acceleration without extra theoretical assumptions.
  • Offline cost stays dominated by the training solves; the extra N_w transport sweeps per training parameter are reported to be small relative to one full solve, so online savings begin almost immediately in multi-query settings.

Reading between the lines

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

  • The same sequential-consistency idea could transfer to other parameterized kinetic equations, such as frequency-dependent or anisotropic-scattering transport, where the ideal correction equation is also an expensive kinetic problem.
  • For non-affine parameter dependence, standard row-sampling hyper-reduction is incompatible with matrix-free transport sweeps, so a practical extension would need an oblique or sampling-free approximation of the parametric operator.
  • The 2–3 iteration behavior is empirical rather than proven; a scan of test parameters near the boundary of the training region would reveal how quickly the benefit degrades when the correction snapshots leave the offline span.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes trajectory-aware reduced-order-model (ROM) enhanced synthetic acceleration (SA) preconditioners for parametric radiative transfer equations. Building on the authors' earlier ROMSAD method, it constructs a sequence of ROMs offline, each conditioned on the correction produced by the previous ROM, so that the residual trajectory seen online matches the offline trajectory. The framework is developed for source iteration and for FGMRES, using a reformulation of the ideal correction equation to avoid directly solving the kinetic correction problem. Numerical tests in 1D slab, 2D variable scattering, pin-cell, and lattice configurations report substantial reductions in transport sweeps and wall-clock time relative to SI-DSA and ROMSAD, with the lattice problem claimed to converge in 2--3 iterations online.

Significance. The trajectory-aware construction is a conceptually clean and plausible remedy for the offline/online residual-trajectory mismatch identified in ROMSAD. The algebraic derivations in Appendix A are correct, and the numerical results are consistently in the claimed direction across four test problems. The additional offline cost over ROMSAD is reported as small relative to the cost of generating training snapshots. If the method generalizes as claimed, it would be a valuable tool for multi-query RTE applications. However, the evidence base is narrow: all online claims rest on 10 (or 20) random test parameters, only averages are reported, no error bars or per-parameter distributions are given, and no code or data are provided. The most serious concern is the pin-cell case in §5.3, where the reference solutions used to form correction snapshots are themselves not converged, directly calling into question the premise of the ROM construction. These issues, together with an unsupported amortization claim, prevent the paper from being accepted in its current form.

major comments (4)
  1. [§5.3, Table 6] SI-DSA fails to converge for the pin-cell problem within the 50-iteration limit (residual 1.09e-09 versus tolerance 1e-11), yet the correction snapshots δf^(l)_µ = f_µ − f^(l)_µ in Eq. (34) require the converged solution f_µ. If f_µ is not converged, the snapshots are not the ideal corrections of Eq. (22), and the offline ROM is built on inaccurate data. The paper explicitly acknowledges 'snapshots with limited accuracy' but does not explain why TAR-IG nevertheless reaches 1e-12. This is a load-bearing issue: the key premise of the method is violated in a test that is used to claim robustness. Please either demonstrate that the inaccuracy is benign (e.g., by showing that the reference solutions are converged for all training parameters, or by quantifying the effect of the unconverged reference on the ROM) or use a properly converged reference to generate snapshots.
  2. [§5.2, §5.4, Tables 3 and 8] The central performance claims, including the abstract's '2-3 iterations online', are based on averages over only 10 randomly selected test parameters, with no error bars, standard deviations, minima/maxima, or seed information. For example, Table 8 reports ¯nsweep=2.3 and ¯niter=3.4, but the reader cannot tell whether this is a tight distribution or a few lucky cases. Given the small test sets and the strong claims, the paper should report per-parameter convergence counts or a measure of spread, and ideally a larger or multiple-seed test set. This is needed to support the robustness and generalization claims.
  3. [§5.2, 'Offline efficiency' paragraph] The statement that 'even taking the offline computational time into account, our method starts to achieve computational savings when predicting the solution for the first new parameter' is not supported by Tables 5, 7, or 9. Those tables report only the cost of ROM construction relative to one linear solve; the dominant offline cost is the 50 (or 121) training solves needed to generate solution snapshots. A break-even analysis comparing total offline cost plus k online solves against k full-order solves is required to substantiate the multi-query efficiency claim. As written, the claim appears to overlook the fixed training cost.
  4. [§4.1, §4.2, Sec. 6] No theoretical or a posteriori guarantee is provided that the trajectory-aware reduced basis at each level spans the correction manifold for unseen test parameters. The paper explicitly defers theory on the relation between basis dimension and convergence to future work. While this is understandable for a numerical methods paper, the strong online claim makes the lack of any robustness indicator a practical risk. At minimum, add leave-one-out or cross-validation experiments on the training set, or report performance on test parameters near the boundary of the parameter domain, to give some evidence beyond 10 random interior samples.
minor comments (5)
  1. [General] There are several typos: 'syntehtic' in Section 3, 'gvien' in Section 4.1, 'Anorldi' in Algorithm 5. The caption of Figure 8 says 'Sec. 5.3' but the lattice problem is in Sec. 5.4.
  2. [Notation] The notation for the reduced basis is inconsistent: Alg. 3 and Alg. 4 use U^(l)_rl, while the text sometimes writes U^(l)_r or U^(l)_rl. Please unify the subscripts and clarify that r_l denotes the dimension at level l.
  3. [§4.2.1, Eq. (46)] The derivation of η^(l) in Eq. (46) uses H with subscripts k,l−1 and H_{l,l−1}; the indices appear slightly inconsistent with the standard Arnoldi recurrence in Alg. 5. A short explanation of the notation would improve readability.
  4. [§5.4, Table 8] The table reports ¯nsweep=2.3 for TAR-IG but the text later says 'converges in just 3.4 iterations' for FGMRES-TAR-IG; please ensure the reader understands how iterations and sweeps differ (Remark 5.1 partially covers this, but a reminder in the table caption would help).
  5. [Reproducibility] The manuscript provides no statement on data/code availability and no description of the random number generation used for selecting test parameters. Including this information would make the numerical results reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the trajectory-aware ROMs are trained on high-fidelity snapshots and tested out-of-sample; self-citations to prior ROMSAD work are not load-bearing.

full rationale

The derivation chain is self-contained. The ROM bases for the correction equation are built from high-fidelity training solutions via SVD (Eqs. 15-17, 34), and the online evaluations are performed on test parameters explicitly excluded from the training set (Sec. 5). The trajectory-aware construction is a legitimate training strategy: it simulates the online preconditioner sequence offline so that the offline and online residual trajectories are consistent for training parameters; this is not a definitional equivalence between the prediction and its input. The reformulation of the ideal correction equation in FGMRES is derived in the paper (Appendix A), not merely imported from the authors' prior work. Self-citations to [16,17] provide baseline methods and motivation, but the current paper reproduces the motivating failure mode (Figs. 1-2) and derives its own algorithm. The strongest empirical claim (2-3 iterations for the lattice problem) is an average over 10 out-of-sample test parameters (Table 8) and is not forced by construction; the paper explicitly notes in Sec. 6 that a theoretical basis-dimension/convergence-rate bound is still lacking. The Sec. 5.3 remark that SI-DSA produces 'snapshots with limited accuracy' yet TAR-IG still converges is a robustness observation about imperfect training data, not a circular step; it may indicate a correctness risk, but it does not reduce the claimed prediction to its inputs. Overall, no load-bearing argument reduces to a fitted parameter, a self-citation chain, or an ansatz smuggled in by citation.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The method's central claims rest on standard ROM assumptions (low-rank parameter manifolds, affine decomposition) and on practical solver ingredients (matrix-free sweeps, DSA). The only hand-chosen hyperparameters are the POD tolerance, the awareness level, and the initial-guess ROM dimension, none of which are fitted to the test data. The method introduces no new physical or mathematical entities.

free parameters (4)
  • POD truncation tolerance ǫPOD = 10^-7 for the headline lattice result; also 10^-5, 10^-6, 10^-9 in other tests
    SVD threshold in Eq. (17) sets the reduced basis dimension; the 2-3 iteration claims are for the specific tolerance values tested, and performance degrades with looser tolerances.
  • Total aware level Nw = 1 for lattice; 1 or 2 elsewhere
    Number of trajectory-aware ROMs constructed offline; controls when the method switches to DSA. Larger Nw adds offline sweeps and mainly helps zero-initial-guess cases.
  • ROM-based initial guess dimension rIG = 16 for lattice; 4-28 in other tests
    Dimension of the ROM used to produce the online initial guess; the rapid 2-3 iteration convergence in Tables 2,3,6,8 is achieved with this initial guess strategy.
  • Convergence tolerance ǫSISA / ǫGMRES = 10^-12 (1D and lattice) or 10^-11 (2D problems)
    Stopping criteria for the iterative solvers; standard numerical inputs, but they define what 'converged in 2-3 iterations' means.
assumptions (6)
  • domain assumption Affine parameter dependence of A_µ and b_µ (Eq. 14)
    Required for efficient online assembly of reduced operators; Remark 2.1 notes non-affine cases need hyper-reduction, which Remark 4.2 states is incompatible with matrix-free sweeps.
  • domain assumption Low-rank structure of the parametric solution and correction snapshot matrices (Eqs. 16, 34)
    POD truncation at ǫPOD assumes snapshot columns span unseen parameters' relevant directions; this is the core empirical premise of all ROM methods and is not proven.
  • domain assumption Existence of sufficiently accurate converged solutions f_µ for training
    Correction snapshots δf = f_µ - f^{(l)}_µ require f_µ; the pin-cell test in Sec. 5.3 reports SI-DSA failing to converge in 50 iterations, so the accuracy of training solutions is unclear.
  • standard math Upwind DG discretization yields block lower-triangular (D_j + Σ_t) enabling matrix-free transport sweeps
    Standard property of upwind DG sweeps, cited from [46,6]; used throughout Algorithms 1-6.
  • standard math FGMRES Arnoldi relations used to derive η^{(l)} (Eqs. 45-46)
    Standard Krylov subspace theory; the reformulation in Appendix A is algebraic and requires the flexible preconditioner to be the one actually applied.
  • domain assumption DSA is a robust fallback after the switch from ROM corrections
    The online stage switches to DSA after Nw iterations and relies on DSA handling the remaining error; this robustness is demonstrated empirically but not analyzed theoretically.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Synthetic Acceleration Preconditioners for Parametric Radiative Transfer Equations based on Trajectory-Aware Reduced Order Models." pith.science (2026). https://pith.science/paper/UWXSPFWU

@misc{pith2026250905001,
  author       = {Pith},
  title        = {Pith review of: Synthetic Acceleration Preconditioners for Parametric Radiative Transfer Equations based on Trajectory-Aware Reduced Order Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UWXSPFWU}},
  note         = {Machine review of arXiv:2509.05001}
}
abstract

The parametric radiative transfer equation (RTE) arises in multi-query applications, such as design optimization, inverse problems, and uncertainty quantification, which require solving the RTE multiple times for various parameters. Classical synthetic acceleration (SA) preconditioners are designed based on low-order approximations of a kinetic correction equation, e.g., its diffusion limit in diffusion synthetic acceleration (DSA). Despite their widespread success, these methods rely on empirical physical assumptions and do not leverage low-rank structures across parameters of the parametric problem. To address these limitations, our previous work introduced a reduced-order model (ROM) enhanced preconditioner called ROMSAD, which exploits low-rank structures across parameters and the original kinetic description of the correction equation. While ROMSAD improves overall efficiency compared with DSA, its efficiency reduces after the first iteration, because the construction of the underlying ROM ignores the preconditioner-dependence of the residual trajectory, leading to a mismatch between the offline and online residual trajectories. To overcome this issue, we introduce a trajectory-aware framework that iteratively constructs ROMs to eliminate the mismatch between offline and online residual trajectories. Numerical tests demonstrate superior efficiency over DSA, and substantial gains in both efficiency and robustness over ROMSAD. For a parametric lattice problem, trajectory-aware ROM preconditioners achieve rapid convergence within only $2$-$3$ iterations online.

Figures

Figures reproduced from arXiv: 2509.05001 by the authors.

Figure 1
Figure 1. The residual history for the 1D setup in (32) and [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. The residual of equation (20) in the SI step, [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. The flowcharts of the offline stage for the ROMSAD meth [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Results for the 1D two-material problem in 5.1. Left [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Left: the setup of scattering cross section for the [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]
Figure 6
Figure 6. Figure 6: Left: the setup for the pin-cell problem in Sec. 5.3 [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: History of residual for the 2D pin-cell problem in 5 [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]
Figure 8
Figure 8. Figure 8: Left: set up of cross section for the lattice proble [PITH_FULL_IMAGE:figures/full_fig_p026_8.png]
Figure 9
Figure 9. Figure 9: History of residual for the 2D lattice problem in 5. [PITH_FULL_IMAGE:figures/full_fig_p027_9.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Highly Efficient Rank-Adaptive Sweep-based SI-DSA for the Radiative Transfer Equation via Mild Space Augmentation

    math.NA 2026-03 unverdicted novelty 7.0 of 10

    Rank-adaptive sweep-based SI-DSA with mild space augmentation matches full-rank accuracy and outer iterations while cutting memory and runtime for steady-state RTE even at 30–45% effective rank.

Reference graph

Works this paper leans on

53 extracted references · 49 canonical work pages · cited by 1 Pith paper

  1. [32]

    N. D. Santo, S. Deparis, A. Manzoni, A. Quarteroni, Mult i space reduced basis preconditioners for large- scale parametrized PDEs, SIAM Journal on Scientific Computi ng 40 (2) (2018) A954–A983

    work page 2018

  2. [1]

    G. S. Abdoulaev, A. H. Hielscher, Three-dimensional opt ical tomography with the equation of radiative transfer, Journal of Electronic Imaging 12 (4) (2003) 594–6 01

    work page 2003

  3. [2]

    G. B. Rybicki, A. P. Lightman, Radiative processes in ast rophysics, John Wiley & Sons, 2024

    work page 2024

  4. [3]

    Spurr, T

    R. Spurr, T. Kurosu, K. Chance, A linearized discrete ord inate radiative transfer model for atmospheric remote-sensing retrieval, Journal of Quantitative Spectr oscopy and Radiative Transfer 68 (6) (2001) 689– 735

    work page 2001

  5. [4]

    G. C. Pomraning, The equations of radiation hydrodynami cs, International Series of Monographs in Natural Philosophy, Oxford: Pergamon Press (1973)

    work page 1973

  6. [5]

    M. L. Adams, E. W. Larsen, Fast iterative methods for disc rete-ordinates particle transport calculations, Progress in Nuclear Energy 40 (2002) 3–159

    work page 2002

  7. [6]

    Y. Azmy, E. Sartori, E. W. Larsen, J. E. Morel, Advances in discrete-ordinates methodology, Nuclear computational science: A century in review (2010) 1–84

    work page 2010

  8. [7]

    Kopp, Synthetic method solution of the transport equa tion, Nuclear Science and Engineering 17 (1) (1963) 65–74

    H. Kopp, Synthetic method solution of the transport equa tion, Nuclear Science and Engineering 17 (1) (1963) 65–74

    work page 1963

Show all 53 references
  1. [8]

    W. H. Reed, The effectiveness of acceleration techniques for iterative methods in transport theory, Nuclear Science and Engineering 45 (3) (1971) 245–254

    1971

  2. [9]

    R. E. Alcouffe, Diffusion synthetic acceleration methods for the diamond-differenced discrete-ordinates equations, Nuclear Science and Engineering 64 (2) (1977) 34 4–355. 29

    1977

  3. [10]

    L. J. Lorence Jr, J. Morel, E. W. Larsen, An S2 synthetic acceleration scheme for the one-dimensional sn equations with linear discontinuous spatial differencing, Nuclear Science and Engineering 101 (4) (1989) 341–351

    1989

  4. [11]

    V. Y. Gol’din, A quasi-diffusion method of solving the ki netic equation, Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki 4 (6) (1964) 1078–1087

    1964

  5. [12]

    D. Y. Anistratov, V. Y. Gol’Din, Nonlinear methods for s olving particle transport problems, Transport Theory and Statistical Physics 22 (2-3) (1993) 125–163

    1993

  6. [13]

    Olivier, W

    S. Olivier, W. Pazner, T. S. Haut, B. C. Yee, A family of in dependent Variable Eddington Factor methods with efficient preconditioned iterative solvers, Journal of Computational Physics 473 (2023) 111747

    2023

  7. [14]

    J. S. Warsa, T. A. Wareing, J. E. Morel, Krylov iterative methods and the degraded effectiveness of diffu- sion synthetic acceleration for multidimensional SN calcu lations in problems with material discontinuities, Nuclear science and engineering 147 (3) (2004) 218–248

    2004

  8. [15]

    K. Ren, R. Zhang, Y. Zhong, A fast algorithm for radiativ e transport in isotropic media, Journal of Computational Physics 399 (2019) 108958

    2019

  9. [17]

    Z. Peng, A flexible gmres solver with reduced order model enhanced synthetic acceleration preconditioenr for parametric radiative transfer equation, Journal of Com putational Physics (2025) 114004

    2025

  10. [18]

    A. G. Buchan, A. Calloo, M. G. Goffin, S. Dargaville, F. Fan g, C. C. Pain, I. M. Navon, A POD reduced order model for resolving angular direction in neutron/pho ton transport problems, J. Comput. Phys. 296 (2015) 138–157

    2015

  11. [19]

    A. C. Hughes, A. G. Buchan, A discontinuous and adaptive reduced order model for the angular discretiza- tion of the Boltzmann transport equation, International Jo urnal for Numerical Methods in Engineering 121 (24) (2020) 5647–5666

    2020

  12. [20]

    Y. Choi, P. Brown, W. Arrighi, R. Anderson, K. Huynh, Spa ce–time reduced order model for large-scale linear dynamical systems with application to boltzmann tra nsport problems, Journal of Computational Physics 424 (2021) 109845

    2021

  13. [21]

    J. M. Coale, D. Y. Anistratov, Reduced-order models for thermal radiative transfer based on pod-galerkin method and low-order quasidiffusion equations, arXiv prepr int arXiv:2102.08592 (2021)

    2021

  14. [22]

    M. Tano, J. Ragusa, D. Caron, P. Behne, Affine reduced-ord er model for radiation transport problems in cylindrical coordinates, Ann. Nucl. Energy. 158 (2021) 108 214

    2021

  15. [23]

    A. G. Buchan, I. M. Navon, L. Yang, A reduced order model d iscretisation of the space-angle phase-space dimensions of the Boltzmann transport equation with applic ation to nuclear reactor problems, J. Comput. Phys. 517 (2024) 113268

    2024

  16. [24]

    Z. K. Hardy, J. E. Morel, Proper Orthogonal Decompositi on Mode Coefficient Interpolation: A Non- Intrusive Reduced-Order Model for Parametric Reactor Kine tics, Nucl. Sci. Eng. 198 (4) (2024) 832–852

    2024

  17. [25]

    Tencer, K

    J. Tencer, K. Carlberg, R. Hogan, M. Larsen, Reduced ord er modeling applied to the discrete ordinates method for radiation heat transfer in participating media, in: Heat Transfer Summer Conference, Vol. 50336, American Society of Mechanical Engineers, 2016, p. V 002T15A011

    2016

  18. [26]

    Z. Peng, Y. Chen, Y. Cheng, F. Li, A reduced basis method f or radiative transfer equation, Journal of Scientific Computing 91 (1) (2022) 1–27. 30

    2022

  19. [27]

    Z. Peng, Y. Chen, Y. Cheng, F. Li, A micro-macro decompos ed reduced basis method for the time- dependent radiative transfer equation, Multiscale Model. Simul. 22 (1) (2024) 639–666

    2024

  20. [28]

    Matsuda, Y

    K. Matsuda, Y. Chen, Y. Cheng, F. Li, Reduced basis metho ds for parametric steady-state radiative transfer equation, paper under preparation

  21. [29]

    R. G. McClarren, Calculating time eigenvalues of the ne utron transport equation with dynamic mode decomposition, Nuclear Science and Engineering 193 (8) (20 19) 854–867

  22. [30]

    Smith, I

    E. Smith, I. Variansyah, R. McClarren, Variable dynami c mode decomposition for estimating time eigen- values in nuclear systems, Nucl. Sci. Eng. 197 (8) (2023) 176 9–1778

    2023

  23. [31]

    K. A. Dominesey, W. Ji, Reduced-order modeling of neutr on transport separated in space and angle via proper generalized decomposition, Nuclear Science and Eng ineering (2022)

    2022

  24. [33]

    Behne, J

    P. Behne, J. Vermaak, J. C. Ragusa, Minimally-invasive parametric model-order reduction for sweep-based radiation transport, Journal of Computational Physics 469 (2022) 111525

    2022

  25. [34]

    R. G. McClarren, T. S. Haut, Data-driven acceleration o f thermal radiation transfer calculations with the dynamic mode decomposition and a sequential singular value decomposition, Journal of Computational Physics 448 (2022) 110756

    2022

  26. [35]

    Dahmen, F

    W. Dahmen, F. Gruber, O. Mula, An adaptive nested source term iteration for radiative transfer equations, Math. Comput. 89 (324) (2020) 1605–1646

    2020

  27. [36]

    K. Chen, Q. Li, J. Lu, S. J. Wright, Random sampling and effi cient algorithms for multiscale pdes, SIAM Journal on Scientific Computing 42 (5) (2020) A2974–A3005

    2020

  28. [37]

    J. Fu, M. Tang, A fast offline/online forward solver for st ationary transport equation with multiple inflow boundary conditions and varying coefficients, arXiv preprin t arXiv:2401.03147 (2024)

    2024 arXiv

  29. [38]

    Bachmayr, R

    M. Bachmayr, R. Bardin, M. Schlottbom, Low-rank tensor product Richardson iteration for radiative transfer in plane-parallel geometry, arXiv:2403.14229 (2 024)

    arXiv

  30. [39]

    W. Guo, Z. Peng, An Inexact Low-Rank Source Iteration fo r Steady-State Radiative Transfer Equation with Diffusion Synthetic Acceleration, arXiv:2509.00805 ( 2025)

    2025 arXiv

  31. [40]

    Einkemmer, J

    L. Einkemmer, J. Hu, Y. Wang, An asymptotic-preserving dynamical low-rank method for the multi-scale multi-dimensional linear transport equation, Journal of C omputational Physics 439 (2021) 110353

    2021

  32. [41]

    Z. Peng, R. G. McClarren, M. Frank, A low-rank method for two-dimensional time-dependent radiation transport calculations, Journal of Computational Physics 421 (2020) 109735

    2020

  33. [42]

    P. Yin, E. Endeve, C. Hauck, S. Schnake, Towards dynamic al low-rank approximation for neutrino kinetic equations. Part I: Analysis of an idealized relaxation mode l, Math. Comput. (2024)

    2024

  34. [43]

    Einkemmer, J

    L. Einkemmer, J. Hu, J. Kusch, Asymptotic-Preserving a nd Energy Stable Dynamical Low-Rank Approx- imation, SIAM Journal on Numerical Analysis 62 (1) (2024) 73 –92

    2024

  35. [44]

    W. A. Sands, W. Guo, J.-M. Qiu, T. Xiong, High-order Adap tive Rank Integrators for Multi-scale Linear Kinetic Transport Equations in the Hierarchical Tucker For mat, arXiv:2406.19479 (2024)

    2024 arXiv

  36. [45]

    Bardos, R

    C. Bardos, R. Santos, R. Sentis, Diffusion approximatio n and computation of the critical size, Transactions of the american mathematical society 284 (2) (1984) 617–649 . 31

    1984

  37. [46]

    M. L. Adams, Discontinuous finite element transport sol utions in thick diffusive problems, Nuclear science and engineering 137 (3) (2001) 298–333

    2001

  38. [47]

    Guermond, G

    J.-L. Guermond, G. Kanschat, Asymptotic analysis of up wind discontinuous Galerkin approximation of the radiative transport equation in the diffusive limit, SIA M Journal on Numerical Analysis 48 (1) (2010) 53–78

    2010

  39. [48]

    Barrault, Y

    M. Barrault, Y. Maday, N. C. Nguyen, A. T. Patera, An ‘emp irical interpolation’method: application to efficient reduced-basis discretization of partial differe ntial equations, Comptes Rendus Mathematique 339 (9) (2004) 667–672

    2004

  40. [49]

    Chaturantabut, D

    S. Chaturantabut, D. C. Sorensen, Nonlinear model redu ction via discrete empirical interpolation, SIAM Journal on Scientific Computing 32 (5) (2010) 2737–2764

    2010

  41. [50]

    E. W. Larsen, Unconditionally stable diffusion-synthe tic acceleration methods for the slab geometry dis- crete ordinates equations. Part I: Theory, Nuclear Science and Engineering 82 (1) (1982) 47–63

    1982

  42. [51]

    G. L. Ramone, M. L. Adams, P. F. Nowak, A transport synthe tic acceleration method for transport iterations, Nuclear science and engineering 125 (3) (1997) 257–283

    1997

  43. [52]

    Peng, Reduced order model enhanced source iteration with synthetic acceleration for parametric ra- diative transfer equation, Journal of Computational Physi cs 517 (2024) 113303

    Z. Peng, Reduced order model enhanced source iteration with synthetic acceleration for parametric ra- diative transfer equation, Journal of Computational Physi cs 517 (2024) 113303

    2024

  44. [53]

    Saad, A flexible inner-outer preconditioned GMRES al gorithm, SIAM Journal on Scientific Computing 14 (2) (1993) 461–469

    Y. Saad, A flexible inner-outer preconditioned GMRES al gorithm, SIAM Journal on Scientific Computing 14 (2) (1993) 461–469

    1993

  45. [54]

    Chen, An integrated finite element method package in m atlab, California: University of California at Irvine 10 (2009)

    L. Chen, An integrated finite element method package in m atlab, California: University of California at Irvine 10 (2009). 32

    2009

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.