Pith. sign in

REVIEW 1 major objections 7 minor 2 cited by

A review of low-rank methods for time-dependent kinetic simulations

T0 review · 1 major / 7 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read This review claims that solutions of high-dimensional kinetic equations often concentrate on a low-rank manifold, and that dynamical low-rank and step-and-truncate methods can evolve those solutions with memory reduced from $O(n^6)$ to…

desk verdict A thorough, honest review that does its job; the thin evidence in strongly nonlinear 3+3D regimes is already conceded by the authors. read the letter →

arxiv 2412.05912 v2 pith:JLGL4VEL submitted 2024-12-08 math.NA cs.NAphysics.comp-phphysics.plasm-ph

Lukas Einkemmer , Katharina Kormann , Jonas Kusch , Ryan G. McClarren , Jing-Mei Qiu This is my paper · ORCID
classification math.NAcs.NAphysics.comp-phphysics.plasm-ph MSC 65M9965F5515A6935Q8382C40
keywords Low-rankmethodsdynamicallowrankapproximationStep-and-TruncatetensordecompositionkineticmodelsparticletransportVlasovequation
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 is a review organized around a central claim: for many time-dependent kinetic models, the distribution function living in a six-dimensional phase space is not effectively full-rank, but can be captured by a small number of coupled space-dependent and velocity-dependent factors. On that premise, it surveys two families of methods that evolve the low-rank representation in time: dynamical low-rank (DLR), which projects the equation onto the tangent space of the low-rank manifold, and Step-and-Truncate (SAT), which takes a step of a conventional discretization and then compresses via a singular value decomposition. The payoff is that memory and computational cost drop from $O(n^6)$ to $O(r n^3)$ per step, with a 4D Vlasov–Poisson bump-on-tail example showing about three orders of magnitude of compression. The review also covers time integrators, tensor-network extensions, conservation and asymptotic-preserving properties, and open problems such as error bounds that stay valid in stiff regimes and positivity preservation.

What carries the argument

The central object is the rank-$r$ separation $f(t,x,v) = \sum_{i,j} U_i(x) S_{ij}(t) V_j(v)$, together with its tensor-network extensions (tensor train and hierarchical Tucker), which split the six-dimensional phase space into low-dimensional factors. DLR carries the argument by projecting the kinetic equation onto the tangent space of the low-rank manifold using the projector $P(f)g = P_U g + P_V g - P_U P_V g$; integrators such as projector splitting and basis-update-and-Galerkin (BUG) make this projection well-conditioned even when $S$ has small singular values. SAT carries the argument by taking one explicit or implicit time step of a full-rank discretization, which raises the rank to a small multiple of $r$, and then truncating by SVD. The efficiency rests on the fact that all inner products in the projected equations factor into separate integrals over $x$ and $v$, so no operation ever touches the full $O(n^6)$ array.

What would settle it

Compute the full-rank solution of a standard bump-on-tail Vlasov–Poisson problem on a fine grid and plot the singular values of the discretized distribution over time; if the rank needed for $10^{-2}$ relative error in the electric field grows without bound as the grid is refined or as the simulation enters the nonlinear phase, the central claim of practical low-rankness fails.

Watch

Extended reading notes

Core claim

The central claim is that kinetic solutions, despite living in up to six-dimensional phase space, frequently concentrate on a low-dimensional manifold, so that $f(t,x,v) \approx \sum_{i,j} U_i(t,x) S_{ij}(t) V_j(t,v)$ for small rank $r$. The paper argues this is not accidental: in diffusive and fluid limits the distribution is provably low-rank (rank 1 to 4 up to a given order), and for weakly nonlinear collisionless problems the linearized solution of a rank-$r$ initial value stays at rank at most $r+1$. Based on this, DLR and SAT methods evolve only the low-rank factors, reducing storage from $O(n^6)$ to $O(r n^3)$ and per-step cost to $O(r^2 n^d)$, while numerical examples—shear Alfvén waves at rank 2, bump-on-tail instabilities with three orders of magnitude compression—show that practical accuracy is retained.

Load-bearing premise

The load-bearing premise is that the phase-space domain splits as a Cartesian product $\Omega_x \times \Omega_v$ and that the solution is well approximated by a small rank in that splitting; if either fails, the reviewed methods lose their advantage.

Editorial extensions

If this is right

  • Kinetic simulations that would otherwise require a supercomputer can run on a single workstation; the review cites a six-dimensional Vlasov–Poisson computation done on one workstation.
  • Low-rank evolution can be made locally conservative and asymptotic preserving when combined with macro-micro decompositions, conservative truncations, or augmented bases, so the reduced method still respects mass, momentum, energy, and the diffusive or fluid limit.
  • Tensor formats extend the same reduction beyond a single space-velocity split, and the choice of dimension tree materially changes compression, with pairing $(x_i,v_i)$ being best in some regimes and grouping all $x$ against all $v$ better in others.
  • Implicit and IMEX low-rank integrators solve much smaller linear systems than the full problem, making stiff kinetic and Fokker–Planck models computationally feasible.
  • Rank-adaptive variants grow or shrink $r$ during a simulation, matching the changing complexity of solutions such as the rank rise and fall seen in radiation therapy dose calculations.

Reading between the lines

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

  • If the low-rank premise holds, the rank itself could serve as a dimensionless measure of a problem's kinetic complexity, letting practitioners compare regimes and geometries by how many singular values are needed for a target accuracy.
  • The tensor-product domain restriction suggests that the largest gains will be in Cartesian and slab geometries; extending these methods to complex fusion-device geometry would require local low-rank patches or coordinate-adaptive ranks, which the paper leaves open.
  • The observation that dynamically evolving velocity bases beat fixed Hermite or spherical-harmonic bases suggests low-rank compression could also be used as an a posteriori diagnostic for how much kinetic detail a reduced model must retain.
  • Because each rank defines a different-fidelity model, low-rank solvers could be plugged into multi-fidelity optimization and uncertainty-quantification loops without extra offline training, a direction the paper notes but does not develop for kinetic equations.
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

1 major / 7 minor

Summary. This paper reviews low-rank matrix and tensor methods for time-dependent kinetic equations, focusing on the dynamical low-rank (DLR) and Step-and-Truncate (SAT) families. It explains the origin of low-rank structure in kinetic solutions (fluid and diffusive limits, weakly nonlinear regimes), derives factor evolution equations, surveys time integrators and tensor formats, and discusses discretization, linear stability, structure preservation, parallelization, and open problems. Numerical illustrations include compression ratios for a Vlasov-Poisson bump-on-tail instability, shear Alfvén wave simulations, and a six-dimensional hierarchical Tucker test.

Significance. The review is timely and comprehensive, filling a gap by unifying the DLR and SAT literatures for kinetic problems. Its strengths are the balanced treatment of the two methodological families, the careful separation of established results from open questions, and the reproducible numerical illustrations with accessible code. The potential impact is high: if the reviewed methods mature, they could reduce memory from O(n^6) to O(r n^3) in many kinetic regimes. The concern about rank growth in strongly nonlinear 3+3D regimes is real, but the manuscript is transparent about this limitation in Section 8, so it does not undermine the overall reliability of the review.

major comments (1)
  1. [§3.1.1, Eqs. (9)-(11)] The displayed S-step integrates forward from t_n to t_{n+1} with initial condition S_ij(t_n) = S^⋆_ij, but S^⋆ is introduced just above as the output of the QR factorization of K(t_{n+1}), i.e., it is a quantity defined at t_{n+1}. The prose immediately following the algorithm correctly states that the S step is solved backward in time. This inconsistency makes the algorithm as printed non-executable. Please correct Eq. (10) so that the S-step is integrated backward from t_{n+1} to t_n with S_ij(t_{n+1}) = S^⋆_ij, and adjust the L-step initial data accordingly.
minor comments (7)
  1. [§5, Table 2] The term 'compression' is used ambiguously: the tabulated values around 10^-6 appear to denote the ratio of low-rank storage to full-grid storage, whereas in Figure 3 the 'compression ratio' is defined as full-rank memory divided by low-rank memory so that larger is better. Please define the quantity consistently in both places (for instance, call one a 'storage ratio' and the other a 'compression factor').
  2. [§6.2, near Eq. (31)] The Maxwellian is written as exp((v-u)^2/2) with a missing minus sign; it should be exp(-(v-u)^2/2). The same sign typo recurs in the following paragraph discussing the multiplicative decomposition.
  3. [§4.3, Eq. (25)] In the last term of the low-rank interpolation formula, the argument of the max function should be v_l rather than v_k, since v_k refers to the spatial grid while v_l is the velocity grid point.
  4. [§3.1.2, Step 3] The notation 'Û, = QR(...)' and 'The denotes the R part' is missing the discarded factor; the intended statement is '[Û, ~] = QR(...)' and '~ denotes the R part of the QR factorization that is discarded.'
  5. [§4.4 and elsewhere] The notation for the low-rank factors alternates between U and X (for example, Section 4.4 refers to X, S, V while Section 3 uses U, S, V). Please unify the notation to avoid confusion.
  6. [References] Reference [124] contains a typo: 'Trensor-Train decomposition' should read 'Tensor-Train decomposition.'
  7. [§4.3] The sentence 'We note that the semi-Lagrangian method does not require such a stability constraint on the time step' directly follows the linear-interpolation condition |Δt v| ≤ Δx, which is confusing; it should be clarified that the restriction applies only to the first-order interpolation formula and can be relaxed with higher-order interpolation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the review's complexity-reduction claim follows by degree-of-freedom counting from the low-rank representation, and its low-rank-effectiveness claims rest on external citations and independent numerical experiments, with limitations disclosed in Section 8.

full rationale

This is a survey, not a derivation from first principles, so the circularity patterns do not apply. The O(n^6) to O(r n^3) memory reduction is obtained by counting degrees of freedom in the representation f(t,x,v) = sum_i,j U_i(t,x) S_ij(t) V_j(t,v), and is therefore true by construction; but it is not presented as an empirical prediction. The empirical content, namely that r can be small, is supported by external references (e.g. [61] for the rank r+1 bound in the linearized Vlasov regime, [51] for rank 4 in the diffusive linear transport limit) and by the paper's own numerical experiments (Figures 2 and 3) performed with a publicly available solver. These are independently checkable results, not conclusions assumed in advance. The paper does define a threshold for low-rank in terms of storage savings, but it then measures, rather than assumes, the rank needed to meet a prescribed error tolerance in Figure 3; this is a benchmark experiment, not a fitted-input-called-prediction. Self-citations are frequent, but each is used as a pointer to an externally published proof or computation, and none is invoked as a uniqueness theorem to rule out alternatives. The paper is also transparent about the actual weakness: Section 8 states that 'robust error bounds ... are missing' and that 'whether a given problem is low-rank needs to be investigated specifically for that problem and there are few general statements that can be made.' That honest limitation is the opposite of a circular argument. No load-bearing step reduces to its own input.

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

The review introduces no free parameters or invented entities. It relies on standard mathematical tools (SVD, tensor networks) and on domain assumptions about low-rank structure and tensor-product phase spaces, which are explicitly stated and are the main scope-limiting premises.

assumptions (4)
  • domain assumption Kinetic phase-space domain is a tensor product Ωx × Ωv.
    Section 4 states: 'this tensor product decomposition between variables in x and v is a requirement for applying low-rank methods.' This restricts the applicability of all reviewed methods.
  • domain assumption Many kinetic solutions have a small low-rank representation.
    Section 2 provides examples (diffusive limit, weakly nonlinear regime) and numerical evidence; this is the central premise on which the efficiency claims rest.
  • standard math The tangent-space projection used in DLR is a valid approximation.
    Section 3.1 cites [90] for the projector formulation and assumes standard manifold and tangent-space properties in the low-rank matrix setting.
  • standard math SVD truncation is L2 stable and order-preserving.
    Assumed in SAT and rank-adaptive integrators; the review cites [136,100] for stability inheritance but does not prove it.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A review of low-rank methods for time-dependent kinetic simulations." pith.science (2026). https://pith.science/paper/JLGL4VEL

@misc{pith2026241205912,
  author       = {Pith},
  title        = {Pith review of: A review of low-rank methods for time-dependent kinetic simulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JLGL4VEL}},
  note         = {Machine review of arXiv:2412.05912}
}
read the original abstract

Time-dependent kinetic models are ubiquitous in computational science and engineering. The underlying integro-differential equations in these models are high-dimensional, comprised of a six--dimensional phase space, making simulations of such phenomena extremely expensive. In this article we demonstrate that in many situations, the solution to kinetics problems lives on a low dimensional manifold that can be described by a low-rank matrix or tensor approximation. We then review the recent development of so-called low-rank methods that evolve the solution on this manifold. The two classes of methods we review are the dynamical low-rank (DLR) method, which derives differential equations for the low-rank factors, and a Step-and-Truncate (SAT) approach, which projects the solution onto the low-rank representation after each time step. Thorough discussions of time integrators, tensor decompositions, and method properties such as structure preservation and computational efficiency are included. We further show examples of low-rank methods as applied to particle transport and plasma dynamics.

Figures

Figures reproduced from arXiv: 2412.05912 by the authors.

Figure 1
Figure 1. The rank-1 solution (3) for the simple advection equation is shown. The low-rank factor [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Simulation of kinetic shear Alfv´en waves. On the left the eight largest singular values of the full rank [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. A bump-on-tail instability (a test example e.g. for beam heating in plasma physics) is simulated using [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Electron therapy test case computed with the augmented BUG integrator of Section 3.1.2. Left: Isolines of [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: Illustration of projection mechanism in the DLR (left panel) and SAT (right panel) approach. [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: Dimension tree for different low-rank tensor formats. [PITH_FULL_IMAGE:figures/full_fig_p026_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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

  1. Asymptotic-Preserving Dynamical Low-Rank Method for the Stiff Nonlinear Boltzmann Equation

    math.NA 2025-02 conditional novelty 7.0 of 10

    A dynamical low-rank integrator (XL/sXL) for the stiff Boltzmann equation evaluates the collision operator r^2 times per step and is asymptotic-preserving in the fluid limit.

  2. A Fast, Accurate and Oscillation-free Spectral Collocation Solver for High-dimensional Transport Problems

    math.NA 2025-06 conditional novelty 5.0 of 10

    A dimension-wise superconsistent spectral collocation method in tensor-train format solves six-dimensional linear transport problems with spectral accuracy and extreme compression in minutes on standard hardware.

Reference graph

Works this paper leans on

162 extracted references · 49 canonical work pages · cited by 2 Pith papers

  1. [1]

    M. L. Adams and E. W. Larsen , Fast iterative methods for discrete-ordinates particle transport calculations, Progress in nuclear energy, 40 (2002), pp. 3–159

    2002

  2. [2]

    Allmann-Rahn, R

    F. Allmann-Rahn, R. Grauer, and K. Kormann , A parallel low-rank solver for the six-dimensional Vlasov–Maxwell equations , Journal of Computational Physics, 469 (2022), p. 111562

    2022

  3. [3]

    Appel ¨o and Y

    D. Appel ¨o and Y. Cheng , Robust implicit adaptive low rank time-stepping methods for matrix differential equations , Journal of Scientific Computing, 102 (2025), p. 81

    2025

  4. [4]

    Armbruster, D

    D. Armbruster, D. Marthaler, and C. Ringhofer, Kinetic and fluid model hierarchies for supply chains , Multiscale Modeling & Simulation, 2 (2003), pp. 43–61

    2003

  5. [5]

    Armbruster and C

    D. Armbruster and C. Ringhofer , Thermalized kinetic and fluid models for reentrant supply chains , Multiscale Modeling & Simulation, 3 (2005), pp. 782–800

    2005

  6. [6]

    Bachmayr, Low-rank tensor methods for partial differential equations , Acta Numerica, 32 (2023), pp

    M. Bachmayr, Low-rank tensor methods for partial differential equations , Acta Numerica, 32 (2023), pp. 1–121. 37

    2023

  7. [7]

    Bachmayr, R

    M. Bachmayr, R. Bardin, and M. Schlottbom , Low-rank tensor product Richardson iteration for radiative transfer in plane-parallel geometry , arXiv preprint, arXiv:2403.14229, (2024)

    arXiv 2024

  8. [8]

    Ballani, L

    J. Ballani, L. Grasedyck, and M. Kluge , Black box approximation of tensors in hier- archical tucker format , Linear algebra and its applications, 438 (2013), pp. 639–657

    2013

Show all 162 references
  1. [9]

    Bardos, F

    C. Bardos, F. Golse, and C. Levermore , Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Stat. Phys., 63 (1991), pp. 323–344, https://doi.org/10.1007/bf 01026608

    1991 doi

  2. [10]

    Bardos, F

    C. Bardos, F. Golse, and C. Levermore , Fluid dynamic limits of kinetic equations II convergence proofs for the Boltzmann equation , Communications on Pure and Applied Mathematics, 46 (1993), pp. 667–753, https://doi.org/10.1002/cpa.3160460503

    1993 doi

  3. [11]

    Baumann, L

    L. Baumann, L. Einkemmer, C. Klingenberg, and J. Kusch , A stable multiplicative dynamical low-rank discretization for the linear Boltzmann-BGK equation , arXiv:2411.06844, (2024)

    2024 arXiv

  4. [12]

    Baumann, L

    L. Baumann, L. Einkemmer, C. Klingenberg, and J. Kusch , Energy Stable and Con- servative Dynamical Low-Rank Approximation for the Su–Olson Problem , SIAM Journal on Scientific Computing, 46 (2024), pp. B137–B158

    2024

  5. [13]

    Baumann, L

    L. Baumann, L. Einkemmer, C. Klingenberg, and J. Kusch , An energy stable and conservative multiplicative dynamical low-rank discretization for the Su-Olson problem , arXiv:2502.03008, (2025)

    2025 arXiv

  6. [14]

    Benner, S

    P. Benner, S. Gugercin, and K. Willcox , A survey of projection-based model reduction methods for parametric dynamical systems , SIAM review, 57 (2015), pp. 483–531

    2015

  7. [15]

    Benner, M

    P. Benner, M. Ohlberger, A. Patera, G. Rozza, K. Urban, et al. , Model reduction of parametrized systems, Springer, 2017

    2017

  8. [16]

    Berkooz, P

    G. Berkooz, P. Holmes, and J. L. Lumley , The proper orthogonal decomposition in the analysis of turbulent flows , Annual review of fluid mechanics, 25 (1993), pp. 539–575

    1993

  9. [17]

    Bernard, A

    F. Bernard, A. Iollo, and S. Riffaud , Reduced-order model for the bgk equation based on pod and optimal transport , Journal of Computational Physics, 373 (2018), pp. 545–570

    2018

  10. [18]

    A. M. Boelens, D. Venturi, and D. M. Tartakovsky , Parallel tensor methods for high-dimensional linear PDEs , Journal of Computational Physics, 375 (2018), pp. 519–539

    2018

  11. [19]

    A. M. Boelens, D. Venturi, and D. M. Tartakovsky , Tensor methods for the Boltzmann-BGK equation, Journal of Computational Physics, 421 (2020), p. 109744

    2020

  12. [20]

    X. Cai, S. Boscarino, and J.-M. Qiu , High order semi-lagrangian discontinuous galerkin method coupled with runge-kutta exponential integrators for nonlinear vlasov dynamics , Jour- nal of Computational Physics, 427 (2021), p. 110036. 38

    2021

  13. [21]

    Camporeale, G

    E. Camporeale, G. Delzanno, B. Bergen, and J. Moulton , On the velocity space discretization for the Vlasov–Poisson system: Comparison between implicit Hermite spectral and Particle-in-Cell methods , Comput. Phys. Commun., 198 (2016), pp. 47–58

    2016

  14. [22]

    L. L. Carter and E. D. Cashwell , Particle-transport simulation with the Monte Carlo method, tech. report, Los Alamos National Lab.(LANL), Los Alamos, NM (United States), 1975

    1975

  15. [23]

    Case and P

    K. Case and P. Zweifel , Linear Transport Theory, Addison-Wesley series in nuclear en- gineering, Addison-Wesley Publishing Company, 1967

    1967

  16. [24]

    Cassini and L

    F. Cassini and L. Einkemmer , Efficient 6D Vlasov simulation using the dynamical low- rank framework Ensign , Comput. Phys. Commun., 280 (2022), p. 108489

    2022

  17. [25]

    Ceruti, N

    G. Ceruti, N. Crouseilles, and L. Einkemmer , A Galerkin Alternating Projection Method for Kinetic Equations in the Diffusive Limit , arXiv:2505.19929, (2025)

    2025 arXiv

  18. [26]

    Ceruti, L

    G. Ceruti, L. Einkemmer, J. Kusch, and C. Lubich , A robust second-order low-rank BUG integrator based on the midpoint rule , BIT Numer. Math., 64 (2024), https://doi.or g/10.1007/s10543-024-01032-x

    2024 doi

  19. [27]

    Ceruti, J

    G. Ceruti, J. Kusch, and C. Lubich , A rank-adaptive robust integrator for dynamical low-rank approximation, BIT Numerical Mathematics, 62 (2022), pp. 1149–1174

    2022

  20. [28]

    Ceruti, J

    G. Ceruti, J. Kusch, and C. Lubich , A parallel rank-adaptive integrator for dynamical low-rank approximation, SIAM Journal on Scientific Computing, 46 (2024), pp. B205–B228

    2024

  21. [29]

    Ceruti, J

    G. Ceruti, J. Kusch, C. Lubich, and D. Sulz , A parallel basis update and galerkin integrator for tree tensor networks , arXiv preprint arXiv:2412.00858, (2024)

    2024 arXiv

  22. [30]

    Ceruti and C

    G. Ceruti and C. Lubich , An unconventional robust integrator for dynamical low-rank approximation, BIT Numerical Mathematics, 62 (2022), pp. 23–44

    2022

  23. [31]

    Ceruti, C

    G. Ceruti, C. Lubich, and D. Sulz , Rank-adaptive time integration of tree tensor net- works, SIAM Journal on Numerical Analysis, 61 (2023), pp. 194–222

    2023

  24. [32]

    Ceruti, C

    G. Ceruti, C. Lubich, and H. W alach, Time integration of tree tensor networks , SIAM Journal on Numerical Analysis, 59 (2021), pp. 289–313

    2021

  25. [33]

    Chen and G

    S. Chen and G. Doolen , Lattice Boltzmann method for fluid flows , Annu. Rev. Fluid Mech., 30 (1998), pp. 329–364

    1998

  26. [34]

    Cheng, A

    Y. Cheng, A. J. Christlieb, and X. Zhong , Energy-conserving discontinuous galerkin methods for the vlasov–amp` ere system , Journal of Computational Physics, 256 (2014), pp. 630–655

    2014

  27. [35]

    Clark, B

    K. Clark, B. Vendt, K. Smith, J. Freymann, J. Kirby, P. Koppel, S. Moore, S. Phillips, D. Maffitt, M. Pringle, et al. , The cancer imaging archive (tcia): main- taining and operating a public information repository , Journal of digital imaging, 26 (2013), pp. 1045–1057. 39

    2013

  28. [36]

    Coughlin and J

    J. Coughlin and J. Hu, Efficient dynamical low-rank approximation for the Vlasov-Amp` ere- Fokker-Planck system, J. Comput. Phys., 470 (2022), p. 111590, https://doi.org/10.101 6/j.jcp.2022.111590

    2022

  29. [37]

    Coughlin, J

    J. Coughlin, J. Hu, and U. Shumlak, Robust and conservative dynamical low-rank meth- ods for the vlasov equation via a novel macro-micro decomposition , Journal of Computational Physics, 509 (2024), p. 113055

    2024

  30. [38]

    De Lathauwer, B

    L. De Lathauwer, B. De Moor, and J. V andewalle , A multilinear singular value decomposition, SIAM Journal on Matrix Analysis and Applications, 21 (2000), pp. 1253– 1278, https://doi.org/10.1137/S0895479896305696

    2000 doi

  31. [39]

    Dektor, Collocation methods for nonlinear differential equations on low-rank manifolds , Linear Algebra and its Applications, 705 (2025), pp

    A. Dektor, Collocation methods for nonlinear differential equations on low-rank manifolds , Linear Algebra and its Applications, 705 (2025), pp. 143–184

    2025

  32. [40]

    Dektor and L

    A. Dektor and L. Einkemmer , Interpolatory dynamical low-rank approximation for the 3+3d Boltzmann–BGK equation , arXiv:2411.15990, (2024)

    2024 arXiv

  33. [41]

    Dektor and D

    A. Dektor and D. Venturi, Dynamic tensor approximation of high-dimensional nonlinear pdes, Journal of Computational Physics, 437 (2021), p. 110295

    2021

  34. [42]

    Dektor and D

    A. Dektor and D. Venturi , Tensor rank reduction via coordinate flows, Journal of Com- putational Physics, 491 (2023), p. 112378

    2023

  35. [43]

    Dektor and D

    A. Dektor and D. Venturi, Coordinate-adaptive integration of PDEs on tensor manifolds, Communications on Applied Mathematics and Computation, (2024), pp. 1–18

    2024

  36. [44]

    Z. Ding, L. Einkemmer, and Q. Li, Dynamical low-rank integrator for the linear boltzmann equation: error analysis in the diffusion limit , SIAM Journal on Numerical Analysis, 59 (2021), pp. 2254–2285

    2021

  37. [45]

    Dolgov, A

    S. Dolgov, A. Smirnov, and E. Tyrtyshnikov , Low-rank approximation in the numer- ical modeling of the Farley–Buneman instability in ionospheric plasma , Journal of Computa- tional Physics, 263 (2014), pp. 268–282, https://doi.org/https://doi.org/10.1016/j. jcp.2014.01.029, ht...

    2014 doi

  38. [46]

    S. V. Dolgov, B. N. Khoromskij, and I. V. Oseledets , Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with Initial Application to the Fokker–Planck Equation , SIAM Journal on Scientific Computing, 34 (2012), pp. A3016– A3038, https://doi.o...

    2012 doi

  39. [47]

    Ehrlacher and D

    V. Ehrlacher and D. Lombardi , A dynamical adaptive tensor method for the Vlasov- Poisson system , Journal of Computational Physics, 339 (2017), pp. 285–306

    2017

  40. [48]

    Einkemmer, A low-rank algorithm for weakly compressible flow , SIAM J

    L. Einkemmer, A low-rank algorithm for weakly compressible flow , SIAM J. Sci. Comput., 41 (2019), pp. A2795–A2814

    2019

  41. [49]

    Einkemmer, Accelerating the simulation of kinetic shear Alfv´ en waves with a dynamical low-rank approximation, J

    L. Einkemmer, Accelerating the simulation of kinetic shear Alfv´ en waves with a dynamical low-rank approximation, J. Comput. Phys., 501 (2024), p. 112757. 40

    2024

  42. [50]

    Einkemmer, J

    L. Einkemmer, J. Hu, and J. Kusch , Asymptotic-preserving and energy stable dynamical low-rank approximation, SIAM Journal on Numerical Analysis, 62 (2024), pp. 73–92

    2024

  43. [51]

    Einkemmer, J

    L. Einkemmer, J. Hu, and Y. W ang , An asymptotic-preserving dynamical low-rank method for the multi-scale multi-dimensional linear transport equation , Journal of Compu- tational Physics, 439 (2021), p. 110353

    2021

  44. [52]

    Einkemmer, J

    L. Einkemmer, J. Hu, and L. 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 (2021), pp. B1057–B1080

    2021

  45. [53]

    Einkemmer, J

    L. Einkemmer, J. Hu, and S. Zhang , Asymptotic-preserving dynamical low-rank method for the stiff nonlinear boltzmann equation , arXiv:2502.08951, (2025)

    2025 arXiv

  46. [54]

    Einkemmer and I

    L. Einkemmer and I. Joseph , A mass, momentum, and energy conservative dynamical low-rank scheme for the Vlasov equation , J. Comput. Phys., 443 (2021), p. 110495

    2021

  47. [55]

    Einkemmer, J

    L. Einkemmer, J. Kusch, and S. Schotth ¨ofer, Conservation properties of the aug- mented basis update & Galerkin integrator for kinetic problems , arXiv:2311.06399, (2023), https://doi.org/10.48550/ARXIV.2311.06399

    2023 doi arXiv

  48. [56]

    Einkemmer and C

    L. Einkemmer and C. Lubich , A low-rank projector-splitting integrator for the Vlasov– Poisson equation, SIAM J. Sci. Comput., 40 (2018), pp. B1330–B1360

    2018

  49. [57]

    Einkemmer and C

    L. Einkemmer and C. Lubich , A quasi-conservative dynamical low-rank algorithm for the Vlasov equation, SIAM J. Sci. Comput., 41 (2019), pp. B1061–B1081

    2019

  50. [58]

    Einkemmer, J

    L. Einkemmer, J. Mangott, and M. Prugger , A low-rank complexity reduction algo- rithm for the high-dimensional kinetic chemical master equation , J. Comput. Phys., (2024), p. 112827, https://doi.org/10.1016/j.jcp.2024.112827

    2024

  51. [59]

    Einkemmer, J

    L. Einkemmer, J. Mangott, and M. Prugger , A hierarchical dynamical low-rank al- gorithm for the stochastic description of large reaction networks , arXiv:2407.11792, (2024), https://doi.org/10.48550/ARXIV.2407.11792

    2024 doi arXiv

  52. [60]

    Einkemmer, J

    L. Einkemmer, J. Mangott, and M. Prugger , Automatic partitioning for the low-rank integration of stochastic Boolean reaction networks , arXiv:2501.04157, (2025)

    2025 arXiv

  53. [61]

    Einkemmer, A

    L. Einkemmer, A. Ostermann, and C. Piazzola , A low-rank projector-splitting inte- grator for the Vlasov–Maxwell equations with divergence correction , J. Comput. Phys., 403 (2020), p. 109063

    2020

  54. [62]

    Einkemmer, A

    L. Einkemmer, A. Ostermann, and C. Scalone , A robust and conservative dynamical low-rank algorithm, J. Comput. Phys., 484 (2023), p. 112060

    2023

  55. [63]

    El Kahza, J.-M

    H. El Kahza, J.-M. Qiu, L. Chacon, and W. Taitano , Sylvester-preconditioned adaptive-rank implicit time integrators for advection-diffusion equations with inhomogeneous coefficients, arXiv preprint: 2410.19662, (2024). 41

    2024

  56. [64]

    El Kahza, W

    H. El Kahza, W. Taitano, J.-M. Qiu, and L. Chac ´on, Krylov-based adaptive-rank implicit time integrators for stiff problems with application to nonlinear fokker-planck kinetic models, Journal of Computational Physics, 518 (2024), p. 113332

    2024

  57. [65]

    Etter , Parallel ALS Algorithm for Solving Linear Systems in the Hierarchical Tucker Representation, SIAM Journal on Scientific Computing, 38 (2016), pp

    S. Etter , Parallel ALS Algorithm for Solving Linear Systems in the Hierarchical Tucker Representation, SIAM Journal on Scientific Computing, 38 (2016), pp. A2585–A2609, https: //doi.org/10.1137/15M1038852

    2016 doi

  58. [66]

    Gamba, J

    I. Gamba, J. Haack, C. Hauck, and J. Hu , A fast spectral method for the Boltzmann collision operator with general collision kernels , SIAM J. Sci. Comput., 39 (2017), pp. B658– B674

    2017

  59. [67]

    Gamba and S

    I. Gamba and S. Tharkabhushanam , Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states , J. Comput. Phys., 228 (2009), pp. 2012–2036

    2009

  60. [68]

    Ghahremani and H

    B. Ghahremani and H. Babaee , A DEIM Tucker tensor cross algorithm and its appli- cation to dynamical low-rank approximation , Computer Methods in Applied Mechanics and Engineering, 423 (2024), p. 116879

    2024

  61. [69]

    Grasedyck, Hierarchical singular value decomposition of tensors , SIAM Journal on Ma- trix Analysis and Applications, 31 (2010), pp

    L. Grasedyck, Hierarchical singular value decomposition of tensors , SIAM Journal on Ma- trix Analysis and Applications, 31 (2010), pp. 2029–2054, https://doi.org/10.1137/0907 64189

    2010 doi

  62. [70]

    Grasedyck, D

    L. Grasedyck, D. Kressner, and C. Tobler , A literature survey of low-rank tensor approximation techniques, GAMM-Mitteilungen, 36 (2013), pp. 53–78

    2013

  63. [71]

    Grasedyck and C

    L. Grasedyck and C. L ¨obbert, Distributed hierarchical SVD in the Hierarchical Tucker format, Numerical Linear Algebra with Applications, 25 (2018), p. e2174, https://doi.or g/https://doi.org/10.1002/nla.2174

    2018 doi

  64. [72]

    Greif and K

    C. Greif and K. Urban , Decay of the kolmogorov n-width for wave problems , Applied Mathematics Letters, 96 (2019), pp. 216–222

    2019

  65. [73]

    Grella and C

    K. Grella and C. Schwab, Sparse tensor spherical harmonics approximation in radiative transfer, Journal of Computational Physics, 230 (2011), pp. 8452–8473

    2011

  66. [74]

    W. Guo, J. F. Ema, and J.-M. Qiu, A Local Macroscopic Conservative (LoMaC) Low Rank Tensor Method with the Discontinuous Galerkin Method for the Vlasov Dynamics , Commu- nications on Applied Mathematics and Computation, (2023), https://doi.org/10.1007/ s42967-023-00277-7

    2023

  67. [75]

    Guo and J.-M

    W. Guo and J.-M. Qiu , A low rank tensor representation of linear transport and nonlin- ear Vlasov solutions and their associated flow maps , Journal of Computational Physics, 458 (2022), p. 111089, https://doi.org/111089

    2022

  68. [76]

    Guo and J.-M

    W. Guo and J.-M. Qiu , A conservative low rank tensor method for the Vlasov dynamics , SIAM Journal on Scientific Computing, 46 (2024), pp. A232–A263

    2024

  69. [77]

    Guo and J.-M

    W. Guo and J.-M. Qiu, A local macroscopic conservative (LoMaC) low rank tensor method for the vlasov dynamics , Journal of Scientific Computing, 101 (2024), p. 61. 42

    2024

  70. [78]

    Hackbusch and S

    W. Hackbusch and S. K ¨uhn, A new scheme for the tensor representation , Journal of Fourier Analysis and Applications, 15 (2009), p. 706–722, https://doi.org/10.1007/s000 41-009-9094-9 , http://dx.doi.org/10.1007/s00041-009-9094-9

    2009 doi

  71. [79]

    Hatch, D

    D. Hatch, D. del Castillo-Negrete, and P. Terry , Analysis and compression of six- dimensional gyrokinetic datasets using higher order singular value decomposition , Journal of Computational Physics, 231 (2012), pp. 4234–4256, https://doi.org/10.1016/j.jcp.20 12.02.007

    2012 doi

  72. [80]

    C. D. Hauck and S. Schnake , A predictor-corrector strategy for adaptivity in dynamical low-rank approximations , SIAM Journal on Matrix Analysis and Applications, 44 (2023), pp. 971–1005

    2023

  73. [81]

    Heenatigala and W

    S. Heenatigala and W. Guo, A Local Macroscopic Conservative (LoMaC) low rank tensor method for the Vlasov-Maxwell system , Journal of Scientific Computing, (in revision, 2024)

    2024

  74. [82]

    J. S. Hesthaven, G. Rozza, B. Stamm, et al. , Certified reduced basis methods for parametrized partial differential equations, vol. 590, Springer, 2016

    2016

  75. [83]

    Hochbruck, M

    M. Hochbruck, M. Neher, and S. Schrammer , Rank-adaptive dynamical low-rank in- tegrators for first-order and second-order matrix differential equations, BIT Numerical Math- ematics, 63 (2023), p. 9

    2023

  76. [84]

    Holtz, T

    S. Holtz, T. Rohwedder, and R. Schneider , The alternating linear scheme for tensor optimization in the tensor train format , SIAM Journal on Scientific Computing, 34 (2012), pp. A683–A713, https://doi.org/10.1137/100818893

    2012 doi

  77. [85]

    Hu and Y

    J. Hu and Y. W ang, An Adaptive Dynamical Low Rank Method for the Nonlinear Boltzmann Equation, Journal of Scientific Computing, 92 (2022), https://doi.org/10.1007/s10915 -022-01934-4

    2022 doi

  78. [86]

    Jahnke and W

    T. Jahnke and W. Huisinga , A dynamical low-rank approach to the chemical master equation, Bull. Math. Biol., 70 (2008), pp. 2283–2302, https://doi.org/10.1007/s11538 -008-9346-x

    2008 doi

  79. [87]

    Jin and D

    S. Jin and D. Levermore , The discrete-ordinate method in diffusive regimes , Transport theory and statistical physics, 20 (1991), pp. 413–439

    1991

  80. [88]

    Kieri, C

    E. Kieri, C. Lubich, and H. W alach , Discretized dynamical low-rank approximation in the presence of small singular values , SIAM J. Numer. Anal., 54 (2016), pp. 1020–1038

    2016

  81. [89]

    Kieri and B

    E. Kieri and B. V andereycken, Projection methods for dynamical low-rank approximation of high-dimensional problems , Computational Methods in Applied Mathematics, 19 (2019), pp. 73–92

    2019

  82. [90]

    Koch and C

    O. Koch and C. Lubich, Dynamical low-rank approximation, SIAM J. Matrix Anal. Appl., 29 (2007), pp. 434–454, https://doi.org/10.1137/050639703 , http://epubs.siam.org /doi/pdf/10.1137/050639703. 43

    2007 doi

  83. [91]

    Koch and C

    O. Koch and C. Lubich , Dynamical tensor approximation , SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2360–2375, https://doi.org/10.1137/09076578X , http://epubs.siam.o rg/doi/pdf/10.1137/09076578X

    2010 doi

  84. [92]

    Koellermeier, P

    J. Koellermeier, P. Krah, and J. Kusch, Macro-micro decomposition for consistent and conservative model order reduction of hyperbolic shallow water moment equations: A study using POD-Galerkin and dynamical low rank approximation , Advances in Computational Mathematics, 50 (20...

    2024

  85. [93]

    Kormann, A semi-Lagrangian Vlasov solver in tensor train format , SIAM J

    K. Kormann, A semi-Lagrangian Vlasov solver in tensor train format , SIAM J. Sci. Com- put., 37 (2015), pp. 613–632, https://doi.org/10.1137/140971270 , http://epubs.siam .org/doi/pdf/10.1137/140971270

    2015 doi

  86. [94]

    Kormann and E

    K. Kormann and E. Sonnendr ¨ucker, Sparse grids for the Vlasov–Poisson equation , Sparse Grids Appl., (2014), pp. 163–190, http://pubman.mpdl.mpg.de/pubman/item/e scidoc:2308864/component/escidoc:2448133/Kormann{\_}Sparse.pdf

    2014

  87. [95]

    Kowalski and M

    J. Kowalski and M. Torrilhon, Moment approximations and model cascades for shallow flow, Communications in Computational Physics, 25 (2018), pp. 669–702, https://doi.or g/10.4208/cicp.OA-2017-0263

    2018 doi

  88. [96]

    Kressner and C

    D. Kressner and C. Tobler , Krylov subspace methods for linear systems with tensor product structure, SIAM journal on matrix analysis and applications, 31 (2010), pp. 1688– 1714

    2010

  89. [97]

    Kunisch and S

    K. Kunisch and S. Volkwein , Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics , SIAM Journal on Numerical analysis, 40 (2002), pp. 492– 515

    2002

  90. [98]

    Kusch , Second-order robust parallel integrators for dynamical low-rank approximation , arXiv preprint arXiv:2403.02834, (2024)

    J. Kusch , Second-order robust parallel integrators for dynamical low-rank approximation , arXiv preprint arXiv:2403.02834, (2024)

    2024 arXiv

  91. [99]

    Kusch, G

    J. Kusch, G. Ceruti, L. Einkemmer, and M. Frank, Dynamical low-rank approximation for Burgers’ equation with uncertainty , International Journal for Uncertainty Quantification, 12 (2022)

    2022

  92. [100]

    Kusch, L

    J. Kusch, L. Einkemmer, and G. Ceruti , On the stability of robust dynamical low-rank approximations for hyperbolic problems , SIAM Journal on Scientific Computing, 45 (2023), pp. A1–A24

    2023

  93. [101]

    Kusch and M

    J. Kusch and M. Frank, Intrusive methods in uncertainty quantification and their connec- tion to kinetic theory , International Journal of Advances in Engineering Sciences and Applied Mathematics, 10 (2018), pp. 54–69

    2018

  94. [102]

    Kusch and P

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

    2023

  95. [103]

    Kusch, B

    J. Kusch, B. Whewell, R. McClarren, and M. Frank , A low-rank power iteration scheme for neutron transport criticality problems , Journal of Computational Physics, 470 (2022), p. 111587. 44

    2022

  96. [104]

    M. P. Laiu, C. D. Hauck, R. G. McClarren, D. P. O’Leary, and A. L. Tits, Positive filtered P n closures for linear kinetic equations , SIAM Journal on Numerical Analysis, 54 (2016), pp. 3214–3238

    2016

  97. [105]

    E. W. Larsen, J. E. Morel, and W. F. Miller Jr , Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes , Journal of Computational Physics, 69 (1987), pp. 283–324

    1987

  98. [106]

    Lubich , From quantum to classical molecular dynamics: reduced models and numerical analysis, European Mathematical Society, 2008

    C. Lubich , From quantum to classical molecular dynamics: reduced models and numerical analysis, European Mathematical Society, 2008

    2008

  99. [107]

    Lubich and I

    C. Lubich and I. Oseledets , A projector-splitting integrator for dynamical low-rank ap- proximation, BIT Numer. Math., 54 (2014), pp. 171–188

    2014

  100. [108]

    Lubich, B

    C. Lubich, B. V andereycken, and H. W alach , Time integration of rank-constrained Tucker tensors , SIAM J. Numer. Anal., 56 (2018), pp. 1273–1290

    2018

  101. [109]

    Donello and G

    M. Donello and G. Palkar and M. H. Naderi and D. C. Del Rey Fern ´andez and H. Babaee , Oblique projection for scalable rank-adaptive reduced-order modelling of non- linear stochastic partial differential equations with time-dependent bases , Proceedings of the Royal Society A...

    2023

  102. [110]

    Mandal and S

    J. Mandal and S. Deshpande, Kinetic flux vector splitting for Euler equations , Computers & fluids, 23 (1994), pp. 447–478

    1994

  103. [111]

    R. G. McClarren , Uncertainty quantification and predictive computational science , Springer, 2018

    2018

  104. [112]

    R. G. McClarren , Calculating time eigenvalues of the neutron transport equation with dynamic mode decomposition, Nuclear Science and Engineering, (2019)

    2019

  105. [113]

    R. G. McClarren, M. L. Adams, P. A. V aquer, and C. Strack, The asymptotic drift- diffusion limit of thermal neutrons , Journal of Computational and Theoretical Transport, 43 (2014), pp. 402–417

    2014

  106. [114]

    R. G. McClarren, J. P. Holloway, and T. A. Brunner , On solutions to the pn equa- tions for thermal radiative transfer , Journal of Computational Physics, 227 (2008), pp. 2864– 2885

    2008

  107. [115]

    R. G. McClarren and T. J. Urbatsch , A modified implicit Monte Carlo method for time-dependent radiative transfer with adaptive material coupling , Journal of Computational Physics, 228 (2009), pp. 5669–5686

    2009

  108. [116]

    S. Meng, D. Appelo, and Y. Cheng , Preconditioning Low Rank Generalized Minimal Residual Method (GMRES) for Implicit Discretizations of Matrix Differential Equations , arXiv preprint arXiv:2410.07465, (2024)

    2024 arXiv

  109. [117]

    Meyer, F

    H.-D. Meyer, F. Gatti, and G. A. Worth , Multidimensional quantum dynamics , John Wiley & Sons, 2009. 45

    2009

  110. [118]

    Meyer, U

    H.-D. Meyer, U. Manthe, and L. S. Cederbaum , The multi-configurational time- dependent Hartree approach, Chem. Phys. Letters, 165 (1990), pp. 73–78

    1990

  111. [119]

    Nakao, G

    J. Nakao, G. Ceruti, and L. Einkemmer , A Reduced Augmentation Implicit Low-rank (RAIL) integrator for solving three-dimensional diffusion and advection-diffusion equations in the Tucker tensor format , arXiv:2503.04932, (2025)

    2025 arXiv

  112. [120]

    Nakao, J.-M

    J. Nakao, J.-M. Qiu, and L. Einkemmer , Reduced Augmentation Implicit Low-rank (RAIL) integrators for advection-diffusion and Fokker-Planck models , SIAM Journal on Sci- entific Computing, 47 (2025), pp. A1145–A1169

    2025

  113. [121]

    Nessyahu and E

    H. Nessyahu and E. Tadmor , Non-oscillatory central differencing for hyperbolic conser- vation laws , J. Comput. Phys., 87 (1990), pp. 408–463, https://doi.org/10.1016/0021-9 991(90)90260-8

    1990 doi

  114. [122]

    Nonnenmacher and C

    A. Nonnenmacher and C. Lubich , Dynamical low-rank approximation: applications and numerical experiments, Mathematics and Computers in Simulation, 79 (2008), pp. 1346–1357

    2008

  115. [123]

    OHLBERGER and S

    M. OHLBERGER and S. RA VE, Reduced basis methods: Success, limitations and future challenges, in Proceedings of ALGORITMY, 2016, pp. 1–12

    2016

  116. [124]

    I. V. Oseledets, Trensor-Train decomposition, SIAM J. Sci. Comput., 33 (2011), pp. 2295– 2317

    2011

  117. [125]

    Pareschi and G

    L. Pareschi and G. Russo , Numerical solution of the Boltzmann equation I: Spectrally accurate approximation of the collision operator, SIAM J. Numer. Anal., 37 (2000), pp. 1217– 1245

    2000

  118. [126]

    Patwardhan, M

    C. Patwardhan, M. Frank, and J. Kusch, Asymptotic-preserving and energy stable dy- namical low-rank approximation for thermal radiative transfer equations, Multiscale Modeling & Simulation, 23 (2025), pp. 278–312

    2025

  119. [127]

    Patwardhan and J

    C. Patwardhan and J. Kusch, A parallel, energy-stable low-rank integrator for nonlinear multi-scale thermal radiative transfer , arXiv preprint arXiv:2502.20883, (2025)

    2025

  120. [128]

    Peherstorfer, K

    B. Peherstorfer, K. Willcox, and M. Gunzburger, Survey of Multifidelity Methods in Uncertainty Propagation, Inference, and Optimization, SIAM Review, 60 (2018), pp. 550–591, https://doi.org/10.1137/16M1082469

    2018 doi

  121. [129]

    Peng and R

    Z. Peng and R. G. McClarren, A high-order/low-order (HOLO) algorithm for preserving conservation in time-dependent low-rank transport calculations, J. Comput. Phys., 447 (2021)

    2021

  122. [130]

    Peng and R

    Z. Peng and R. G. McClarren , A sweep-based low-rank method for the discrete ordinate transport equation, Journal of Computational Physics, 473 (2023), p. 111748

    2023

  123. [131]

    Z. Peng, R. G. McClarren, and M. Frank , A low-rank method for two-dimensional time-dependent radiation transport calculations, J. Comput. Phys., 421 (2020)

    2020

  124. [132]

    J. L. Proctor, S. L. Brunton, and J. N. Kutz , Dynamic mode decomposition with control, SIAM Journal on Applied Dynamical Systems, 15 (2016), pp. 142–161. 46

    2016

  125. [133]

    Prugger, L

    M. Prugger, L. Einkemmer, and C. Lopez , A dynamical low-rank approach to solve the chemical master equation for biological reaction networks , J. Comput. Phys., (2023), p. 112250, https://doi.org/10.1016/j.jcp.2023.112250

    2023

  126. [134]

    Rodgers, Tucker-MPI, 2019

    A. Rodgers, Tucker-MPI, 2019. https://github.com/akrodger/htucker-mpi

    2019

  127. [135]

    Rodgers, A

    A. Rodgers, A. Dektor, and D. Venturi , Adaptive integration of nonlinear evolution equations on tensor manifolds , Journal of Scientific Computing, 92 (2022), p. 39

    2022

  128. [136]

    Rodgers and D

    A. Rodgers and D. Venturi , Stability analysis of hierarchical tensor methods for time- dependent PDEs, Journal of Computational Physics, 409 (2020), p. 109341

    2020

  129. [137]

    Rodgers and D

    A. Rodgers and D. Venturi , Implicit integration of nonlinear evolution equations on tensor manifolds , J. Sci. Comput., (2023)

    2023

  130. [138]

    Sands, W

    W. Sands, W. Guo, J.-M. Qiu, and T. Xiong , High-order Adaptive Rank Integrators for Multi-Scale Linear Kinetic Transport Equations in the Hierarchical Tucker Format, arXiv preprint arXiv:2406.19479, (2024)

    2024 arXiv

  131. [139]

    sands, J.-M

    W. sands, J.-M. Q. Qiu, D. Hayes, and N. Zheng , An Adaptive-Rank Approach with Greedy Sampling for Multi-Scale BGK Equations , arXiv, (2025), p. 2505.17191

    2025 arXiv

  132. [140]

    Scalone, L

    C. Scalone, L. Einkemmer, J. Kusch, and R. McClarren , A multi-fidelity adaptive dynamical low-rank based optimization algorithm for fission criticality problems , Journal of Scientific Computing, 104 (2025), pp. 1–18

    2025

  133. [141]

    P. J. Schmid, Dynamic mode decomposition of numerical and experimental data , Journal of fluid mechanics, 656 (2010), pp. 5–28

    2010

  134. [142]

    P. J. Schmid , Dynamic mode decomposition and its variants , Annual Review of Fluid Me- chanics, 54 (2022), pp. 225–254

    2022

  135. [143]

    J. W. Schumer and J. P. Holloway , Vlasov simulations using velocity-scaled Hermite representations, Journal of Computational Physics, 144 (1998), pp. 626–661

    1998

  136. [144]

    T. Shi, D. Hayes, and J.-M. Qiu , Distributed memory parallel adaptive tensor-train cross approximation, arXiv preprint arXiv:2407.11290, (2024)

    2024 arXiv

  137. [145]

    Shu, High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM review, 51 (2009), pp

    C.-W. Shu, High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM review, 51 (2009), pp. 82–126

    2009

  138. [146]

    Simoncini, Computational methods for linear matrix equations , SIAM Review, 58 (2016), pp

    V. Simoncini, Computational methods for linear matrix equations , SIAM Review, 58 (2016), pp. 377–441

    2016

  139. [147]

    R. C. Smith , Uncertainty quantification: theory, implementation, and applications , vol. 12, Siam, 2013

    2013

  140. [148]

    P. Stammer, Uncertainty quantification and numerical methods in charged particle radiation therapy, PhD thesis, Dissertation, Karlsruhe, Karlsruher Institut f¨ ur Technologie (KIT), 2023, 2023. 47

    2023

  141. [149]

    W. T. Taitano and S. J. Araki , A hybrid AMR low-rank tensor approach for solving the Boltzmann equation, in SIAM Conference on Computational Science and Engineering, 2023, https://www.siam.org/Portals/0/Conferences/CSE23/CSE23_ABSTRACTS.pdf

    2023

  142. [150]

    Tang and L

    X. Tang and L. Ying , Solving high-dimensional Fokker-Planck equation with functional hierarchical tensor, Journal of Computational Physics, 511 (2024), p. 113110

    2024

  143. [151]

    D. P. Truong, M. I. Ortega, I. Boureima, G. Manzini, K. O. Rasmussen, and B. S. Alexandrov , Tensor networks for solving the time-independent Boltzmann neutron transport equation, Journal of Computational Physics, 507 (2024), p. 112943, https://doi. org/10.1016/j.jcp.2024.112943

    2024

  144. [152]

    Uschmajew and A

    A. Uschmajew and A. Zeiser , Dynamical low-rank approximation of the Vlasov-Poisson equation with piecewise linear spatial boundary, BIT Numerical Mathematics, 64 (2024), p. 19

    2024

  145. [153]

    Uschmajew and A

    A. Uschmajew and A. Zeiser , Discontinuous galerkin discretization of conservative dy- namical low-rank approximation schemes for the vlasov-poisson equation , arXiv preprint arXiv:2503.10562, (2025)

    2025 arXiv

  146. [154]

    J. P. Verboncoeur , Particle simulation of plasmas: review and advances , Plasma Phys. Control. Fusion, 47 (2005), p. A231

    2005

  147. [155]

    W ang and M

    H. W ang and M. Thoss, Multilayer formulation of the multiconfiguration time-dependent Hartree theory, The Journal of Chemical Physics, 119 (2003), pp. 1289–1299, https://doi. org/10.1063/1.1580111

    2003 doi

  148. [156]

    J. S. W arsa, T. A. W areing, and J. E. Morel , Krylov iterative methods and the de- graded effectiveness of diffusion synthetic acceleration for multidimensional sn calculations in problems with material discontinuities , Nuclear science and engineering, 147 (2004), pp. 218– 248

    2004

  149. [157]

    S. R. White , Density matrix formulation for quantum renormalization groups , Phys. Rev. Lett., 69 (1992), pp. 2863–2866, https://doi.org/10.1103/PhysRevLett.69.2863 , https://link.aps.org/doi/10.1103/PhysRevLett.69.2863

    1992 doi

  150. [158]

    K. Xu, L. Martinelli, and A. Jameson , Gas-kinetic finite volume methods, flux-vector splitting, and artificial diffusion , Journal of computational physics, 120 (1995), pp. 48–65

    1995

  151. [159]

    Ye and N

    E. Ye and N. Loureiro , Quantized tensor networks for solving the Vlasov-Maxwell equa- tions, 2024

    2024

  152. [160]

    Ye and N

    E. Ye and N. F. Loureiro, Quantum-inspired method for solving the Vlasov-Poisson equa- tions, Physical Review E, 106 (2022), p. 035208

    2022

  153. [161]

    P. Yin, E. Endeve, C. Hauck, and S. Schnake, Towards dynamical low-rank approxima- tion for neutrino kinetic equations. Part I: Analysis of an idealized relaxation model , Mathe- matics of Computation, (2024), https://doi.org/https://doi.org/10.1090/mcom/3997

    2024 doi

  154. [162]

    Zheng, D

    N. Zheng, D. Hayes, A. Christlieb, and J.-M. Qiu , A semi-lagrangian adaptive-rank (slar) method for linear advection and nonlinear vlasov-poisson system , Journal of Computa- tional Physics, 532 (2025), p. 113970. 48

    2025

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.