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REVIEW 2 major objections 4 minor 53 references

An Asymptotic-Preserving Dynamical Low-Rank Semi-Lagrangian Method for Multiscale Linear Kinetic Transport Equations

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A low-rank semi-Lagrangian scheme keeps the diffusion limit while cutting cost

desk verdict A solid, genuinely useful combination of AP semi-Lagrangian and low-rank BUG with QDEIM angular sampling; the main gap is that the implemented NNLS weight fallback is not covered by the sampling-error analysis, though the diffusive-limit claim probably survives. read the letter →

arxiv 2607.27736 v1 pith:NJHAQRXI submitted 2026-07-30 math.NA cs.NA

Shun Li , Yan Jiang , Mengping Zhang , Tao Xiong This is my paper · ORCID
classification math.NAcs.NA MSC 65M2235Q2065M12
keywords kinetictransportequationasymptotic-preservingsemi-Lagrangiandynamicallow-rankQDEIMangularsamplingdiffusionlimitmultiscaleapproximation
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

The paper reports a numerical method that solves multiscale linear kinetic transport equations at lower cost by representing the particle distribution on a low-rank manifold while retaining the large-time-step and asymptotic-preserving (AP) behavior of a semi-Lagrangian discretization. The authors claim that the central obstacle—the nonseparable semi-Lagrangian flux derivative—can be handled by QDEIM-selected angular directions with effective weights, and they quantify the sampling error. They prove unconditional stability of the full-quadrature low-rank scheme in the constant-coefficient case, and show that both low-rank schemes inherit the diffusion limit as the Knudsen number goes to zero. If correct, the method would let kinetic solvers take time steps set by the macroscopic scale, not the mean free path, at a per-step cost reduced from O(N^{d_x+d_Omega}) to O(N^{max{d_x,d_Omega}}).

What carries the argument

The central object is the weighted low-rank representation Y = F M = X S V^T of the kinetic distribution, with M diagonal in square-root quadrature weights, advanced by a BUG integrator. The second key mechanism is the QDEIM-sampled angular quadrature for the backtracked semi-Lagrangian flux derivative: an angular target space Z is built from the low-rank velocity basis such as [1, Q1, Q M^{-1} V], column-pivoted QR selects sampled directions, and effective weights enforce exactness on Z. The sampling error is then controlled by the distance of the flux vector to Z (Lemma 3), and in the diffusive limit the prefactor alpha_1/epsilon suppresses it (Lemma 4 and Proposition 6). The unconditional

What would settle it

In the non-equilibrium transition-regime test (Example 5.2), monitor the l1 norm of the QDEIM effective weights and the NNLS residual on successive mesh refinements with fixed rank and Delta t = Delta x. If the weight norm grows without bound or the density error fails to track SL-DLR(full) as the mesh refines, the claimed sampling-error bound and AP inheritance do not hold.

Watch

Extended reading notes

Core claim

The central claim is that the proposed SL-DLR scheme couples an approximate macroscopic density update from a semi-Lagrangian reformulation of the transport equation with a basis-update-Galerkin dynamical low-rank integrator for the microscopic distribution, and evaluates the semi-Lagrangian flux derivative using QDEIM-sampled angular directions with effective weights that enforce exactness on a target angular space. The paper establishes that the full-quadrature low-rank scheme is unconditionally stable in the constant-coefficient case, that the angular-sampling error is bounded by the target-space approximation error, and that both low-rank schemes are asymptotic-preserving: as the Knudsen

Load-bearing premise

The load-bearing premise is that the QDEIM effective weights have a uniformly bounded l1 norm and satisfy the exactness condition with negligible residual, because the sampling-error and AP analyses (Lemma 3 and Propositions 6-7) rely on those bounds; when negative weights trigger the nonnegative least-squares fallback, the resulting residual is not included in the error estimates.

Editorial extensions

If this is right

  • If correct, kinetic solvers can take large time steps set by the macroscopic dynamics rather than by the small mean free path, because the diffusive limit is preserved.
  • Per-step computational cost drops from O(N^{d_x+d_Omega}) to O(N^{max{d_x,d_Omega}}) for fixed rank, making higher-dimensional computations such as 3D2V feasible.
  • The sampled angular quadrature with effective weights gives a systematic way to avoid the full angular sweep in semi-Lagrangian low-rank solvers, with an error controlled by the chosen target space.
  • The unconditional stability result in the constant-coefficient case suggests the scheme remains robust under mesh refinement and large time steps.
  • The AP inheritance means a single implementation can handle kinetic, transition, and diffusive regimes without resolving epsilon explicitly.

Reading between the lines

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

  • The analysis suggests a practical recipe for the transition regime: enlarging the target space to include v^2-weighted basis terms reduces sampling error, so an adaptive target-space selection driven by the estimated backtracking correction could be worthwhile.
  • Because the sampling error enters through alpha_1/epsilon, near-diffusive regimes may tolerate much coarser angular sampling for the same accuracy, allowing the number of sampled directions to be adapted as epsilon changes.
  • When the nonnegative least-squares fallback is used, the exactness condition is violated; monitoring the NNLS residual could provide an a posteriori indicator for when to enlarge the target space or refine the sampling.
  • The complexity gain assumes bounded GMRES iteration counts; in strongly anisotropic or high-contrast scattering settings, preconditioning may become the real bottleneck rather than the flux assembly.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper proposes SL-DLR, a low-rank semi-Lagrangian method for multiscale linear kinetic transport equations. It couples a macroscopic density update obtained from the AP finite-difference semi-Lagrangian reformulation of Zhang–Zhu–Xiong with a BUG low-rank integrator for a weighted microscopic distribution. To avoid a full angular sweep in the non-separable backtracked flux derivative, the flux is evaluated on QDEIM-selected angular directions with effective quadrature weights satisfying an exactness condition on a target angular space. The authors establish unconditional energy stability of the full-quadrature low-rank scheme in the constant-coefficient setting, derive a sampling-error estimate in terms of the target-space approximation error, and state an AP inheritance result for both full-quadrature and sampled schemes. Numerical experiments in 1D, 2D, and 3D demonstrate agreement with the full-rank SL reference and diffusion limits, and report a per-step cost reduction from O(N^{d_x+d_Ω}) to O(N^{max{d_x,d_Ω}}).

Significance. If the main results are correct, this is a valuable contribution to the low-rank kinetic solver literature. The paper addresses a real bottleneck: the semi-Lagrangian flux derivative is direction-dependent and therefore not directly separable in the low-rank factors. The energy-stability proof in Appendix A is nontrivial and appears internally coherent; the sampling-error framework, which reduces the error to the distance of the angular flux vector to a target space, is conceptually appealing; and the numerical section is extensive, including a comparison with an existing IMEX-S-BUG scheme and high-dimensional tests. The complexity tables and timings support the claimed cost reduction. However, the sampled scheme as implemented is not fully covered by the theory: the nonnegative least-squares fallback for the effective weights breaks the exactness condition and the uniform l1-bound assumption on which Lemmas 3–4, Proposition 6, and the last step of Proposition 7 rely. The AP claim for the sampled scheme is therefore not established as it stands. The significance of the paper is conditional on closing this gap.

major comments (2)
  1. [Section 3.3, Eq. (9); Lemmas 3–4; Prop. 6] The sampling-error analysis assumes that the effective weights satisfy the exactness condition (9) and the uniform l1 bound ||ŵ^n||_1 ≤ C_w (see Eq. (10), Eq. (11), Lemma 4, and Prop. 6). However, the algorithm replaces the exact solve by the NNLS solve 'min_{ŵ≥0} ||(P^T Z)^T ŵ − Z^T w||_2' whenever negative entries appear. This NNLS solution does not, in general, enforce (9); the residual is not included in the error estimates. Moreover, no bound on ||ŵ^n||_1 is provided, and nothing prevents the NNLS weights from growing as ε→0 or Δx→0, which would invalidate the suppression argument in (15). Since the final step of Prop. 7 ('SL-DLR has the same limiting diffusion equation as SL-DLR(full)') invokes Prop. 6, the AP claim for the implemented sampled scheme does not follow. To fix this, the authors should either modify the quadrature construction so that nonnegativity and exactness are gu
  2. [Section 4.3, Prop. 7] The AP proof is a formal asymptotic argument. The statements 'the leading ε^{-2} K-step balance and L-step balance give ...' are asserted without deriving the o(1) terms or showing that they are uniform in the fixed-mesh, fixed-Δt limit ε→0. In particular, the initial transient (K^n and L^n are arbitrary non-equilibrium low-rank factors), the lower-order upwind terms of size 1/ε, and the QR normalization after the K/L steps are not controlled. Since AP is a central claim of the manuscript, this gap is load-bearing. The proof should either quantify the error in ρ^{n+1,*} and Y^{n+1} with explicit ε-dependence (constants allowed to depend on the data but not on ε), or be explicitly labeled as a formal asymptotic derivation, with the rigorous AP property inherited only from the full-rank scheme [51] and verified numerically for the low-rank schemes.
minor comments (4)
  1. [Section 5, Figures 2–3] Captions should define the notation Z^(1) and Z^(2) locally; the text defines them only in Section 3.3. This would make the comparison in Example 5.2 easier to read.
  2. [Section 4.2.2, Lemma 4] The assumption max_j ||f^n_j − ρ^n||∞ ≤ C_g is an assumption on the numerical solution, not on the scheme. For diffusive-limit propagation over many steps, the uniformity in n should be clarified or justified; as written, the bound is a one-step estimate.
  3. [Section 3.5, Table 1] The complexity table reports the 'Sampling construction' cost as O(N_Ω r^2). Since QDEIM is applied to the target matrix Z^n and the effective weights are recomputed each step, please state whether the pivoting is done on the (N_Ω × O(r)) matrix directly; the cost O(N_Ω r^2) is then plausible for fixed r but should be stated explicitly for the O(r) target dimension.
  4. [Remark 1] For velocity-dependent sources, the macroscopic equation is modified with an additional flux term containing ⟨vΦ⟩. In the manufactured-solution tests, the paper says the source is 'obtained by substituting (18) into the model equation'; please clarify precisely how the discrete source term is inserted into both the macroscopic and microscopic updates so that the comparison is unambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the low-rank and sampling derivations are self-contained; the AP reformulation is imported from an independent published source and the sampling analysis is a direct projection/approximation argument.

full rationale

The derivation chain is not circular. The macroscopic AP reformulation is explicitly imported from the independently published FD-SL scheme [51] (“We recall the semi-Lagrangian AP reformulation [51, 4]”), and the new paper proves its low-rank variant’s constant-coefficient energy stability in Appendix A rather than assuming the low-rank result. The sampling analysis (Lemma 3, Eq. (10)–(11), Proposition 6) is a standard projection argument: the sampled quadrature weights are constructed to satisfy the exactness condition (9) on a target space Z^n, and the error is bounded by inf_{ξ∈Z^n} ||h^n_i − ξ||_∞. The target space is chosen to contain the leading angular structure of the backtracked flux derivative via span(Q1, QM^{-1}V^n), which is a model-reduction construction rather than a circular definition. No fitted parameter is renamed as a prediction, and Proposition 7 proves AP inheritance through explicit limiting balances rather than by invoking a uniqueness theorem or ansatz from the authors’ prior work. The only substantive caveat is the NNLS fallback in Section 3.3: when negative weights occur, the exactness condition (9) and the uniform l1 bound assumed in Lemmas 3–4 and Proposition 6 are not guaranteed, so the AP/sampling-error estimates are conditional on those hypotheses. That is a correctness/completeness gap in the implemented algorithm, not a circularity. Numerical comparisons against the diffusion-limit reference and the external analytical benchmark provide independent support. Hence the central claims do not reduce to their inputs by construction.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities. The effective quadrature weights and sampled angular directions are algorithmic constructs, not new physics. The central derivation relies on the imported AP-SL reformulation, the low-rank ansatz, and the formal asymptotic AP balance, plus the exactness and bounded-weight assumptions for the sampled quadrature.

free parameters (3)
  • low-rank r = fixed per run (e.g., r=50,8,3 in 1D; r=8,4,32,64 in 2D/3D)
    User-prescribed truncation rank. It controls the approximation error and the per-step cost, and is assumed fixed throughout the theory. Not fitted to data.
  • sampled angular set size m_n = r+2 for Z^(1); 2r+3 for Z^(2)
    Chosen to match the dimension of the target angular space in Section 3.3. The complexity claim depends on m_n = O(r).
  • target angular space Z^n = Z^(1) = [1, Q1, Q M^{-1} V^n] or Z^(2) = Z^(1) + [Q^2 1, Q^2 M^{-1} V^n]
    A modeling choice in Section 3.3. It determines the sampling-error estimate (11) and is not justified by an a priori error bound independent of the solution.
assumptions (6)
  • domain assumption The semi-Lagrangian AP reformulation (3) from [51] is accepted: slowly varying factors inside the integrals are frozen at (x*, t*).
    Section 2.2. The macroscopic update, and hence the AP property, inherit this approximation. Reference [51] shares an author with the present paper but is an independent published result.
  • domain assumption The solution is well approximated on the rank-r manifold and the BUG tangent-space projection error is acceptable.
    Sections 2.3 and 3.2. The theory provides no rank-adaptive error control; numerical tests fix r by hand.
  • ad hoc to paper The effective quadrature weights satisfy the exactness condition (9) and the uniform l1 bound ||\tilde{w}^n||_1 <= C_{\tilde{w}} independent of epsilon, Delta x, Delta t.
    Lemmas 3 and 4. The algorithm may switch to nonnegative least squares in Section 3.3, which only approximately satisfies (9); the residual is not analyzed.
  • domain assumption In the diffusive limit, the nonequilibrium part satisfies max_j ||f^n_j - rho^n||_infinity <= C_g independent of epsilon.
    Lemma 4. This is a well-prepared-data assumption; it is not proven for all time.
  • ad hoc to paper Formal asymptotic balance: in the epsilon^{-2} K/L/S steps, the o(1) terms are uniform enough to imply rho^{n+1,*} in span(X^{n+1}) + o(1) and M1 in span(V^{n+1}) + o(1).
    Proposition 7, Section 4.3. The proof states 'the leading epsilon^{-2} balance gives' without a quantified asymptotic error estimate.
  • domain assumption Constant-coefficient, periodic, source-free setting for the energy stability analysis.
    Section 4.1 and Appendix A. Variable coefficients, boundaries, and sources are outside the stability theory.

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Cite this review

Pith. "Pith review of An Asymptotic-Preserving Dynamical Low-Rank Semi-Lagrangian Method for Multiscale Linear Kinetic Transport Equations." pith.science (2026). https://pith.science/paper/NJHAQRXI

@misc{pith2026260727736,
  author       = {Pith},
  title        = {Pith review of: An Asymptotic-Preserving Dynamical Low-Rank Semi-Lagrangian Method for Multiscale Linear Kinetic Transport Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NJHAQRXI}},
  note         = {Machine review of arXiv:2607.27736}
}
read the original abstract

In this paper, we develop an asymptotic-preserving (AP) dynamical low-rank semi-Lagrangian method for multiscale linear kinetic transport equations. The method combines the large-time-step capability of semi-Lagrangian discretizations with the storage and cost reduction provided by low-rank representations. The proposed scheme couples an approximate macroscopic density update with the basis update Galerkin integrator for the kinetic distribution. To retain the reduced complexity in the semi-Lagrangian flux evaluation, the flux derivative is computed through a sampled angular quadrature strategy. We establish an unconditional stability analysis of the full-quadrature low-rank scheme in the constant-coefficient case. The error induced by angular sampling in the flux derivative is quantified. The resulting scheme is shown to be AP in the diffusive limit. Numerical experiments, including high-dimensional test cases, demonstrate that the proposed method is AP, stable under large time steps, and computationally efficient across kinetic and diffusive regimes.

Figures

Figures reproduced from arXiv: 2607.27736 by the authors.

Figure 1
Figure 1. Example 5.1. Density profiles in the kinetic ( [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Example 5.2. Density comparison for SL, SL-DLR(full), SL-DLR with default target [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Example 5.3. Top row: L 1 density error. Bottom row: average wall time per step. From left to right, ε = 1, 10−2 , and 10−6 [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Example 5.4. SL-DLR density (left), density profile along [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Example 5.5. Varying scattering coefficient. Plot of [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: Example 5.5. SL-DLR density contours (top row) and density profiles along [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: Example 5.5. Density profiles along y = 0 for SL-DLR and IMEX-S-BUG at T = 0.006 [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: Example 5.6. SL-DLR density slices at z = 0 (top row) and center-line profiles along x = y = 0 (bottom row). From left to right: T = 0.002, T = 0.006, T = 0.010 [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]

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Reference graph

Works this paper leans on

53 extracted references · 4 linked inside Pith

  1. [51]

    Asymptotic Preserving and Uniformly Un- conditionally Stable Finite Difference Schemes for Kinetic Transport Equations.SIAM Journal on Scientific Computing, 45(5):B697–B730, 2023

    Guoliang Zhang, Hongqiang Zhu, and Tao Xiong. Asymptotic Preserving and Uniformly Un- conditionally Stable Finite Difference Schemes for Kinetic Transport Equations.SIAM Journal on Scientific Computing, 45(5):B697–B730, 2023

    2023

  2. [1]

    Optimizing cubature for efficient integration of subspace deformations.ACM transactions on graphics (TOG), 27(5):1–10, 2008

    Steven S An, Theodore Kim, and Doug L James. Optimizing cubature for efficient integration of subspace deformations.ACM transactions on graphics (TOG), 27(5):1–10, 2008

    2008

  3. [2]

    Low-rank tensor methods for partial differential equations.Acta Numerica, 32:1–121, 2023

    Markus Bachmayr. Low-rank tensor methods for partial differential equations.Acta Numerica, 32:1–121, 2023. 26

    2023

  4. [3]

    Implicit-explicit runge–kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit.SIAM Journal on Scientific Computing, 35(1):A22–A51, 2013

    Sebastiano Boscarino, Lorenzo Pareschi, and Giovanni Russo. Implicit-explicit runge–kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit.SIAM Journal on Scientific Computing, 35(1):A22–A51, 2013

    2013

  5. [4]

    Asymptotic preserving semi- Lagrangian discontinuous Galerkin methods for multiscale kinetic transport equations.Journal of Computational Physics, 513:113190, 2024

    Yi Cai, Guoliang Zhang, Hongqiang Zhu, and Tao Xiong. Asymptotic preserving semi- Lagrangian discontinuous Galerkin methods for multiscale kinetic transport equations.Journal of Computational Physics, 513:113190, 2024

    2024

  6. [5]

    K. M. Case and P. F. Zweifel.Linear Transport Theory. Addison-Wesley, 1967

    1967

  7. [6]

    Springer, 1988

    Carlo Cercignani.The Boltzmann Equation and Its Applications, volume 67 ofApplied Math- ematical Sciences. Springer, 1988

    1988

  8. [7]

    A galerkin alternating projection method for kinetic equations in the diffusive limit.arXiv preprint arXiv:2505.19929, 2025

    Gianluca Ceruti, Nicolas Crouseilles, and Lukas Einkemmer. A galerkin alternating projection method for kinetic equations in the diffusive limit.arXiv preprint arXiv:2505.19929, 2025

    arXiv 2025

Show all 53 references
  1. [8]

    A rank-adaptive robust integrator for dynamical low-rank approximation.BIT Numerical Mathematics, 62(4):1149–1174, 2022

    Gianluca Ceruti, Jonas Kusch, and Christian Lubich. A rank-adaptive robust integrator for dynamical low-rank approximation.BIT Numerical Mathematics, 62(4):1149–1174, 2022

    2022

  2. [9]

    An unconventional robust integrator for dynamical low-rank approximation.BIT Numerical Mathematics, 62:23–44, 2022

    Gianluca Ceruti and Christian Lubich. An unconventional robust integrator for dynamical low-rank approximation.BIT Numerical Mathematics, 62:23–44, 2022

    2022

  3. [10]

    Chandrasekhar.Radiative Transfer

    S. Chandrasekhar.Radiative Transfer. Dover Publications, 1960

    1960

  4. [11]

    Nonlinear model reduction via discrete empirical interpolation.SIAM Journal on Scientific Computing, 32(5):2737–2764, 2010

    Saifon Chaturantabut and Danny C Sorensen. Nonlinear model reduction via discrete empirical interpolation.SIAM Journal on Scientific Computing, 32(5):2737–2764, 2010

    2010

  5. [12]

    Interpolatory dynamical low-rank approximation for the 3+3d Boltzmann-BGK equation.arXiv preprint arXiv:2411.15990, 2024

    Alec Dektor and Lukas Einkemmer. Interpolatory dynamical low-rank approximation for the 3+3d Boltzmann-BGK equation.arXiv preprint arXiv:2411.15990, 2024

    2024 arXiv

  6. [13]

    Numerical methods for kinetic equations.Acta Numerica, 23:369–520, 2014

    Giacomo Dimarco and Lorenzo Pareschi. Numerical methods for kinetic equations.Acta Numerica, 23:369–520, 2014

    2014

  7. [14]

    Accuracy and stability analysis of the semi- lagrangian method for stiff hyperbolic relaxation systems and kinetic BGK model.Multiscale Modeling & Simulation, 21(1):143–167, 2023

    Mingchang Ding, Jing-Mei Qiu, and Ruiwen Shu. Accuracy and stability analysis of the semi- lagrangian method for stiff hyperbolic relaxation systems and kinetic BGK model.Multiscale Modeling & Simulation, 21(1):143–167, 2023

    2023

  8. [15]

    Dynamical low-rank integrator for the linear boltzmann equation: Error analysis in the diffusion limit.SIAM Journal on Numerical Anal- ysis, 59(4):2254–2285, 2021

    Zhiyan Ding, Lukas Einkemmer, and Qin Li. Dynamical low-rank integrator for the linear boltzmann equation: Error analysis in the diffusion limit.SIAM Journal on Numerical Anal- ysis, 59(4):2254–2285, 2021

    2021

  9. [16]

    A new selection operator for the discrete empirical inter- polation method—improved a priori error bound and extensions.SIAM Journal on Scientific Computing, 38(2):A631–A648, 2016

    Zlatko Drmac and Serkan Gugercin. A new selection operator for the discrete empirical inter- polation method—improved a priori error bound and extensions.SIAM Journal on Scientific Computing, 38(2):A631–A648, 2016

    2016

  10. [17]

    Asymptotic-preserving and energy stable dynamical low-rank approximation.SIAM Journal on Numerical Analysis, 62(1):73–92, 2024

    Lukas Einkemmer, Jingwei Hu, and Jonas Kusch. Asymptotic-preserving and energy stable dynamical low-rank approximation.SIAM Journal on Numerical Analysis, 62(1):73–92, 2024

    2024

  11. [18]

    An asymptotic-preserving dynamical low-rank method for the multi-scale multi-dimensional linear transport equation.Journal of Computa- tional Physics, 439:110353, 2021

    Lukas Einkemmer, Jingwei Hu, and Yubo Wang. An asymptotic-preserving dynamical low-rank method for the multi-scale multi-dimensional linear transport equation.Journal of Computa- tional Physics, 439:110353, 2021. 27

    2021

  12. [19]

    An efficient dynamical low-rank algorithm for the Boltzmann-BGK equation close to the compressible viscous flow regime.SIAM Journal on Scientific Computing, 43(5):B1057–B1080, 2021

    Lukas Einkemmer, Jingwei Hu, and Lexing Ying. An efficient dynamical low-rank algorithm for the Boltzmann-BGK equation close to the compressible viscous flow regime.SIAM Journal on Scientific Computing, 43(5):B1057–B1080, 2021

    2021

  13. [20]

    McClarren, and Jing-Mei Qiu

    Lukas Einkemmer, Katharina Kormann, Jonas Kusch, Ryan G. McClarren, and Jing-Mei Qiu. A review of low-rank methods for time-dependent kinetic simulations.Journal of Computa- tional Physics, 538:114191, 2025

    2025

  14. [21]

    A low-rank projector-splitting integrator for the Vlasov–Poisson equation.SIAM Journal on Scientific Computing, 40(5):B1330–B1360, 2018

    Lukas Einkemmer and Christian Lubich. A low-rank projector-splitting integrator for the Vlasov–Poisson equation.SIAM Journal on Scientific Computing, 40(5):B1330–B1360, 2018

    2018

  15. [22]

    Asymptotic-preserving and energy stable dynamical low-rank approximation for thermal radiative transfer equations.Multiscale Modeling & Simulation, 23(1):278–312, 2025

    Martin Frank, Jonas Kusch, and Chinmay Patwardhan. Asymptotic-preserving and energy stable dynamical low-rank approximation for thermal radiative transfer equations.Multiscale Modeling & Simulation, 23(1):278–312, 2025

    2025

  16. [23]

    Ganapol.Analytical Benchmarks for Nuclear Engineering Applications: Case Studies in Neutron Transport Theory

    Barry D. Ganapol.Analytical Benchmarks for Nuclear Engineering Applications: Case Studies in Neutron Transport Theory. Number NEA/DB/DOC(2008)1 in OECD/NEA Data Bank. Nuclear Energy Agency, Organisation for Economic Co-operation and Development, Issy-les- Moulineaux, France, 2...

    2008

  17. [24]

    A DEIM Tucker tensor cross algorithm and its application to dynamical low-rank approximation.Computer Methods in Applied Mechanics and Engineering, 423:116879, 2024

    Behzad Ghahremani and Hessam Babaee. A DEIM Tucker tensor cross algorithm and its application to dynamical low-rank approximation.Computer Methods in Applied Mechanics and Engineering, 423:116879, 2024

    2024

  18. [25]

    A local macroscopic conservative (LoMaC) low rank tensor method with the discontinuous galerkin method for the Vlasov dynamics

    Wei Guo, Jannatul Ferdous Ema, and Jing-Mei Qiu. A local macroscopic conservative (LoMaC) low rank tensor method with the discontinuous galerkin method for the Vlasov dynamics. Communications on Applied Mathematics and Computation, 6(1):550–575, 2024

    2024

  19. [26]

    An inexact low-rank source iteration for steady-state radiative transfer equation with diffusion synthetic acceleration.arXiv preprint arXiv:2509.00805, 2025

    Wei Guo and Zhichao Peng. An inexact low-rank source iteration for steady-state radiative transfer equation with diffusion synthetic acceleration.arXiv preprint arXiv:2509.00805, 2025

    2025 arXiv

  20. [27]

    Highly efficient rank-adaptive sweep-based SI-DSA for the radia- tive transfer equation via mild space augmentation.arXiv preprint arXiv:2603.25233, 2026

    Wei Guo and Zhichao Peng. Highly efficient rank-adaptive sweep-based SI-DSA for the radia- tive transfer equation via mild space augmentation.arXiv preprint arXiv:2603.25233, 2026

    2026 arXiv

  21. [28]

    A conservative low rank tensor method for the Vlasov dynamics

    Wei Guo and Jing-Mei Qiu. A conservative low rank tensor method for the Vlasov dynamics. SIAM Journal on Scientific Computing, 46(1):A232–A263, 2024

    2024

  22. [29]

    Efficient SN-like and PN-like dynamic low rank methods for thermal radiative transfer

    Terry Haut, John Loffeld, Lukas Einkemmer, Pierson Guthrey, Stefan Brunner, and William Schill. Efficient SN-like and PN-like dynamic low rank methods for thermal radiative transfer. arXiv preprint arXiv:2601.18705, 2026

    2026

  23. [30]

    Dimensional hyper- reduction of nonlinear finite element models via empirical cubature.Computer methods in applied mechanics and engineering, 313:687–722, 2017

    Joaquin Alberto Hernandez, Manuel Alejandro Caicedo, and Alex Ferrer. Dimensional hyper- reduction of nonlinear finite element models via empirical cubature.Computer methods in applied mechanics and engineering, 313:687–722, 2017

    2017

  24. [31]

    J. Jang, F. Li, J.-M. Qiu, and T. Xiong. High order asymptotic preserving DG-IMEX schemes for discrete-velocity kinetic equations in a diffusive scaling.Journal of Computational Physics, 281:199–224, 2015

    2015

  25. [32]

    Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations

    Shi Jin. Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM Journal on Scientific Computing, 21:441–454, 1999. 28

    1999

  26. [33]

    Asymptotic-preserving schemes for multiscale physical problems.Acta Numerica, 31:1–82, 2022

    Shi Jin. Asymptotic-preserving schemes for multiscale physical problems.Acta Numerica, 31:1–82, 2022

    2022

  27. [34]

    Discretized dynamical low-rank approxima- tion in the presence of small singular values.SIAM Journal on Numerical Analysis, 54:1020– 1038, 2016

    Emil Kieri, Christian Lubich, and Hanna Walach. Discretized dynamical low-rank approxima- tion in the presence of small singular values.SIAM Journal on Numerical Analysis, 54:1020– 1038, 2016

    2016

  28. [35]

    Dynamical low-rank approximation.SIAM Journal on Matrix Analysis and Applications, 29(2):434–454, 2007

    Othmar Koch and Christian Lubich. Dynamical low-rank approximation.SIAM Journal on Matrix Analysis and Applications, 29(2):434–454, 2007

    2007

  29. [36]

    A semi-lagrangian Vlasov solver in tensor train format.SIAM Journal on Scientific Computing, 37(4):B613–B632, 2015

    Katharina Kormann. A semi-lagrangian Vlasov solver in tensor train format.SIAM Journal on Scientific Computing, 37(4):B613–B632, 2015

    2015

  30. [37]

    A robust collision source method for rank adaptive dynamical low-rank approximation in radiation therapy.ESAIM: Mathematical Modelling and Numerical Analysis, 57(2):865–891, 2023

    Jonas Kusch and Pia Stammer. A robust collision source method for rank adaptive dynamical low-rank approximation in radiation therapy.ESAIM: Mathematical Modelling and Numerical Analysis, 57(2):865–891, 2023

    2023

  31. [38]

    McClarren, and Martin Frank

    Jonas Kusch, Benjamin Whewell, Ryan G. McClarren, and Martin Frank. A low-rank power iteration scheme for neutron transport criticality problems.Journal of Computational Physics, 470:111587, 2022

    2022

  32. [39]

    Larsen and Jim E

    Edward W. Larsen and Jim E. Morel. Advances in discrete-ordinates methodology.Nuclear Computational Science: A Century in Review, pages 1–84, 2009

    2009

  33. [40]

    A new asymptotic preserving scheme based on micro- macro formulation for linear kinetic equations in the diffusion limit.SIAM Journal on Scientific Computing, 31:334–368, 2008

    Mohammed Lemou and Luc Mieussens. A new asymptotic preserving scheme based on micro- macro formulation for linear kinetic equations in the diffusion limit.SIAM Journal on Scientific Computing, 31:334–368, 2008

    2008

  34. [41]

    John Wiley and Sons, Inc., New York, NY, 1983

    Elmer Eugene Lewis and Warren F Miller.Computational Methods of Neutron Transport. John Wiley and Sons, Inc., New York, NY, 1983

    1983

  35. [42]

    Temporal-stability-enhanced and energy-stable dynamical low-rank approximation for multiscale linear kinetic transport equa- tions.arXiv preprint arXiv:2602.12337, 2026

    Shun Li, Yan Jiang, Mengping Zhang, and Tao Xiong. Temporal-stability-enhanced and energy-stable dynamical low-rank approximation for multiscale linear kinetic transport equa- tions.arXiv preprint arXiv:2602.12337, 2026

    2026

  36. [43]

    Analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit.SIAM Journal on Numerical Analysis, 48:1474–1491, 2010

    Jian-Guo Liu and Luc Mieussens. Analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit.SIAM Journal on Numerical Analysis, 48:1474–1491, 2010

    2010

  37. [44]

    Oseledets

    Christian Lubich and Ivan V. Oseledets. A projector-splitting integrator for dynamical low- rank approximation.BIT Numerical Mathematics, 54:171–188, 2014

    2014

  38. [45]

    Asymptotic preserving IMEX-DG-S schemes for linear kinetic transport equations based on schur complement.SIAM Journal on Scientific Computing, 43(2):A1194–A1220, 2021

    Zhichao Peng and Fengyan Li. Asymptotic preserving IMEX-DG-S schemes for linear kinetic transport equations based on schur complement.SIAM Journal on Scientific Computing, 43(2):A1194–A1220, 2021

    2021

  39. [46]

    McClarren

    Zhuogang Peng and Ryan G. McClarren. A sweep-based low-rank method for the discrete ordinate transport equation.Journal of Computational Physics, 473:111748, 2023

    2023

  40. [47]

    McClarren, and Martin Frank

    Zhuogang Peng, Ryan G. McClarren, and Martin Frank. A low-rank method for two- dimensional time-dependent radiation transport calculations.Journal of Computational Physics, 421:109735, 2020. 29

    2020

  41. [48]

    William A Sands, Wei Guo, Jing-Mei Qiu, and Tao Xiong. High-order adaptive rank integra- tors for multiscale linear kinetic transport equations in the hierarchical tucker format.SIAM Journal on Scientific Computing, 47(6):A3383–A3412, 2025

    2025

  42. [49]

    Sands, Jing-Mei Qiu, Daniel Hayes, and Nanyi Zheng

    William A. Sands, Jing-Mei Qiu, Daniel Hayes, and Nanyi Zheng. An adaptive-rank ap- proach with greedy sampling for multi-scale BGK equations.Journal of Computational Physics, 547:114523, 2026

    2026

  43. [50]

    Tyrtyshnikov, Sergei A

    Eugene E. Tyrtyshnikov, Sergei A. Goreinov, and Nikolai L. Zamarashkin. Pseudo-skeleton approximations.Doklady Akademii Nauk, 343(2):151–152, 1995

    1995

  44. [52]

    A semi-lagrangian adaptive- rank (SLAR) method for linear advection and nonlinear Vlasov-Poisson system.Journal of Computational Physics, 532:113970, 2025

    Nanyi Zheng, Daniel Hayes, Andrew Christlieb, and Jing-Mei Qiu. A semi-lagrangian adaptive- rank (SLAR) method for linear advection and nonlinear Vlasov-Poisson system.Journal of Computational Physics, 532:113970, 2025

    2025

  45. [53]

    Sands, Daniel Hayes, Andrew J

    Nanyi Zheng, William A. Sands, Daniel Hayes, Andrew J. Christlieb, and Jing-Mei Qiu. A semi-lagrangian adaptive rank (SLAR) method for high-dimensional Vlasov dynamics.arXiv preprint arXiv:2510.24861, 2025. 30

    2025

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