Pith. sign in

REVIEW 2 major objections 5 minor 35 references

Nonnegative Low-Rank Matrix Correction under an Orthogonality Constraint in Conservative Vlasov Simulations

T0 review · 2 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read Nonnegative low-rank corrections to Vlasov solutions reduce to an implicit singular-value thresholding rule, enabling five convex algorithms and a tangent-space method that preserve mass, momentum, and energy.

desk verdict A genuine new proximal characterization and a fast tangent-space projection algorithm, undercut by a false feasibility condition and under-specified dual gradient steps. read the letter →

arxiv 2607.26272 v1 pith:7SHKNGQK submitted 2026-07-28 math.NA cs.NAmath.OC

Yue Wu , Stephen Becker , Jingmei Qiu , Xiangxiong Zhang This is my paper · ORCID
classification math.NAcs.NAmath.OC MSC 65F5515A8390C2590C2665K1035Q83
keywords nonnegativelow-rankmatrixapproximationnuclearnormminimizationsingularvaluethresholdingorthogonalityconstraintVlasov–Poissonsystempositivitylimitertangent-spacealternatingprojectionsmomentconservation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tackles a defect in low-rank methods for the Vlasov equation: SVD truncation can make the computed distribution function negative, which is unphysical for a probability density. The authors recast the fix as a minimal correction matrix that must satisfy A+X≥0 and XB=0, where XB=0 encodes pointwise preservation of the three macroscopic moments. For the convex formulation based on squared nuclear norm minimization, they prove the proximal operator under the orthogonality constraint is an implicit singular-value soft-thresholding whose threshold can be found by bisection. That theorem drives five convex algorithms; a separate rank-constrained formulation yields a tangent-space accelerated alternating projection that needs only a 2r×2r SVD per iteration. Numerical tests show comparable correction quality, with the tangent-space method the cheapest at scale, and the correction works as a positivity limiter inside a conservative low-rank Vlasov solver without disturbing conserved mass, momentum, and energy.

What carries the argument

The central object is the proximal operator of g(Y)=||Y||_*^2+(a2/2)||Y||_F^2+I_{YB=0}; Theorem 1 characterizes it as implicit singular value soft-thresholding with threshold τ=2||Y||_*/(a2+η), computed from the root of 0=(a2+η)τ−2Σ_k max{σ_k−τ,0}. This single identity carries the convex algorithms (Douglas–Rachford splitting, restarted dual FISTA, accelerated gradient, PR+ conjugate gradient, L-BFGS). For the non-convex formulation, the carrying mechanism is the tangent-space projection P_{T_{M_{r,B}}(X)}(Y)=P_{T_{M_r}(X)}(YΠ^⊥_B), which reduces each alternating-projection step to a 2r×2r SVD while keeping the iterate in rank-r factored form and satisfying XB=0 exactly.

What would settle it

Compute the proximal operator for a small random instance (e.g., 2×3 with a chosen orthonormal B) by brute-force convex optimization and compare with the implicit threshold formula (25); a mismatch anywhere in the threshold bisection would refute Theorem 1. Separately, take moments with ρ>0 and eρ<m² and confirm the algorithms cannot return a nonnegative A+X, which would confirm the infeasibility boundary stated in Remark 2.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 1: for the correction problem min_X ||X||_*^2 + (a/2)||X||_F^2 subject to X≥−A and XB=0, the proximal operator of the orthogonality-constrained squared nuclear norm has the closed-form characterization Y = D_{2||Y||_*/(a2+η)}(η/(a2+η) M Π^⊥_B), an implicit equation because the threshold depends on the nuclear norm of the unknown minimizer. The paper shows the threshold is the unique root of a convex piecewise-linear scalar equation, computable by bisection after one SVD of MΠ^⊥_B. This makes the convex problem globally solvable by any of five proposed algorithms and makes the non-convex rank-constrained formulation solvable by tangent-space accelerated altern

Load-bearing premise

The admissible set must be nonempty: there has to exist a nonnegative matrix with the same first three velocity moments as the input; the paper gives the condition ρ>0 and eρ≥m² (Remark 2), and if it fails, no correction exists and every algorithm and the central claim collapse.

Editorial extensions

If this is right

  • A positivity limiter based on this correction can be embedded in a conservative low-rank Vlasov solver; it removes truncation-induced negative entries while conserving mass, momentum, and energy pointwise.
  • Because all five convex algorithms provably reach the same global minimizer, the correction quality does not depend on which solver is chosen, only on the penalty parameter a.
  • The tangent-space accelerated alternating projection keeps XB=0 exactly at every iterate, so when used as a limiter it preserves the three conserved moments to machine precision, whereas the Douglas–Rachford limiter preserves them only to its convergence tolerance.
  • Residual negativity after storage is set by the re-truncation tolerance of the low-rank format, not by the correction itself; tightening the tolerance drives negativity toward machine precision at a modest, sub-linear rank increase.
  • The per-iteration cost gap between the convex methods (full m×n SVD) and TAP (2r×2r SVD) grows with problem size, so TAP becomes increasingly cost-competitive as grids refine.

Reading between the lines

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

  • The implicit-threshold proximal operator is a generic primitive: any optimization with a squared nuclear norm objective and a linear constraint XB=0 (e.g., matrix denoising with a known column subspace) could reuse this bisection-based solver.
  • The feasibility condition ρ>0 and eρ≥m², if violated, means no nonnegative distribution can have those first three moments; then the correction problem is infeasible and the solver would need to adjust the moments themselves, not just the matrix entries.
  • The TAP idea of projecting onto the tangent space of the rank-and-orthogonality manifold before truncation likely extends to tensor-train or other low-rank formats, where a full SVD is prohibitively expensive.
  • Because the correction is operator-agnostic, the same limiter should apply to Fokker–Planck or other kinetic equations where nonnegativity is required, not just Vlasov–Poisson.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proposes optimization-based post-processing to restore nonnegativity in low-rank Vlasov simulations while preserving the macroscopic moments (density, momentum, and energy). The correction X must satisfy A+X ≥ 0 and XB = 0, where B encodes the discrete moment basis. The central theoretical result (Theorem 1) characterizes the proximal operator of the squared nuclear norm subject to the orthogonality constraint XB = 0 as an implicit singular-value soft-thresholding map, with the threshold determined by a scalar root-finding problem. On this basis the authors develop Douglas–Rachford splitting and four dual first-order methods for the convex formulation (2), and a tangent-space accelerated alternating projection (TAP) for the rank-constrained nonconvex formulation (17). Numerical experiments on Landau damping and a bump-on-tail LoMaC low-rank run compare the algorithms, measure per-iteration cost scaling, and demonstrate the correction as a positivity limiter inside a time-dependent conservative solver.

Significance. If the proximal characterization in Theorem 1 is correct, it is a useful addition to the toolkit for spectral-norm regularization with linear constraints; the bisection threshold makes the proximal operator implementable. The TAP algorithm is attractive because it avoids full SVDs, keeps the orthogonality constraint exact at every iterate, and the numerical scaling study is informative. The paper also includes worthwhile fairness checks: a naive baseline, a randomized-SVD comparison, and tuned step sizes in the cost-scaling study. The main caveats are the incorrect discrete feasibility condition in Remark 2 and under-specified gradient evaluations for the dual algorithms; both affect the validity or reproducibility of the claims as currently stated.

major comments (2)
  1. [§3.2, Remark 2] The claimed necessary and sufficient condition for non-emptiness of the admissible set is false on a discrete grid. For v ∈ {−1,0,1}, take one row of A to be [1,−1,1]. Then (ρ,m,e) = (1,0,2), so ρ > 0 and eρ ≥ m², yet no nonnegative vector y on this grid can have these moments: e = Σ v_j² y_j ≤ Σ y_j = ρ, while here e = 2 > ρ = 1. The finite-grid moment cone is strictly smaller than the continuous Hamburger cone. Since an empty admissible set makes every proposed correction algorithm meaningless, the paper should either state the correct discrete moment-cone condition or verify feasibility for the test data and restrict the existence claim to the continuous setting.
  2. [§4.3–§4.4, Eq. (42)] The gradient ∇g*(Λ) is used by dual FISTA variant II, dual accelerated gradient descent, PR+ conjugate gradient, and L-BFGS, but Eq. (42) defines X only implicitly: X = D_{2||X||_*/a2}(a2^{-1}ΛΠ_B^⊥). The manuscript never states that evaluating this map requires solving the scalar threshold equation a2 τ = 2 Σ max(σ_i−τ,0) by bisection, nor does it derive (42). As written, these four algorithms are not fully specified. Please add the derivation and the root-finding step, or explicitly reduce to the a2=0/η=a2 case of Theorem 1.
minor comments (5)
  1. [§4.2, Remark 3] The function h(τ) = (a2+η)τ − 2Σ max(σ_k−τ,0) is strictly increasing and piecewise linear, but it is concave, not convex: its derivative (a2+η) + 2·#{σ_k > τ} decreases as τ increases. Monotonicity is enough for the bisection argument; please correct the wording.
  2. [§4.2, Theorem 1 proof] The proof says it constructs a candidate via (25), but (25) is implicit and the threshold root (31) is only introduced in Remark 3 after the theorem. Reorganizing the proof as: define τ as the unique root of (31), set Y = D_τ(η/(a2+η) M Π_B^⊥), and then verify the optimality conditions, would remove the appearance of circularity and shorten the argument.
  3. [§3.2, Eqs. (33)–(34)] The conjugate formulas (33)–(34) are stated with almost no derivation. In particular, (34) relies on a spectral conjugate calculation that is not evident from the text. A short derivation or a reference to the spectral-conjugate calculus would improve reproducibility.
  4. [§4.3 and Table 1] The text says all convex algorithms converge to the same global minimizer, but Table 1 reports that PR+ conjugate gradient does not meet the tolerance within 1000 iterations and has not reached rank 33. Please qualify the statement to refer to the converged runs.
  5. [Data availability] The paper states that data and code are available from the authors upon request. For a numerical paper with a cost-scaling comparison, a persistent repository link would substantially aid reproducibility.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: Theorem 1 is derived in-text from KKT/subgradient conditions, algorithm outputs are measured against an algebraic lower bound, and the paper's self-citations are provenance/setup rather than load-bearing.

full rationale

The derivation chain is self-contained at its load-bearing point. Theorem 1 (Sec. 4.2.2) characterizes the proximal operator of the orthogonality-constrained squared nuclear norm as an implicit singular value thresholding map; the proof starts from the KKT system (23)-(27), uses Lemma 1 (proved in-text) and the standard subdifferential of the nuclear norm (Prop. 1), and verifies that a candidate built from the self-consistency equation (31) satisfies optimality. The proximal problem (22) (input) and the implicit SVT characterization (25) (output) are not equivalent by construction: the subgradient verification is genuine content, and the bisection in Remark 3 is a root-finding procedure on that derived fixed-point condition, not a fitted parameter. Formulation (2) is justified in-text by consistency and homogeneity (Example 1), so it is not a renamed known result. All five convex algorithms and TAP are standard optimization methods applied to (2)/(17); the reported corrections (3.29%/3.31% versus the algebraic lower bound 3.08% = ||min{A,0}||_F/||A||_F), conserved-moment deviations (~1e-14), and per-iteration cost scalings (Tables 2-3, Fig. 6) are measured, not fitted, and hyperparameters (a=1000, eta=7a, alpha=0.88) are explicitly reported as tuned. Self-citations ([13],[14] by Guo/Qiu for the LoMaC setup and solver; [35] by Zheng/Vandereycken/Zhang for standard fixed-rank tangent-space facts) are not load-bearing: the moment-preservation equivalence XB=0 <-> X sqrt(S) R^T = 0 is derived algebraically in Secs. 2.2-2.3, and conservation is independently measured in Table 5. The reviewer-flagged feasibility condition of Remark 2 (rho>0 and e*rho>=m^2 as necessary and sufficient) is a potential correctness gap on discrete velocity grids, not a circularity, since no output is derived from it and test-data feasibility is evidenced by the computed min(A+X)=0. The paper also states its own limitations honestly (DR enforces XB=0 only in the limit; fixed rank caps attainable nonnegativity; limited-solution rank grows sub-linearly, Sec. 5.5 and Fig. 8), which corroborates rather than undermines the non-circularity verdict.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The paper adds one optimization variable X and standard convex-analysis machinery; no new physical or mathematical entities are postulated. Its load-bearing assumptions are the low-rank moment-splitting from prior conservative methods and the physical realizability of the moments. Several numerical hyperparameters are tuned on the test problems.

free parameters (5)
  • Penalty parameter a = 1000
    Weight in objective (2); chosen after experiments in Sec 5.1 as a compromise between correction size and rank. Controls the trade-off and affects all numerical comparisons.
  • DR splitting parameters a1, a2 = a/2 each
    Split of a in the Douglas-Rachford decomposition (19). The paper notes the choice has no visible effect, so this is a mild parameter.
  • DR step size η and relaxation α = η=7a, α=0.88
    Tuned on the 64×128 Landau data for fastest convergence (Sec 5.2); in the scaling study η/a is tuned per grid from 2.0 to 11.2. Quantitative efficiency claims depend on these choices.
  • TAP rank r = 20 (fixed test), N+3 (limiter)
    User-specified rank constraint in the non-convex problem. If fixed too low it caps attainable nonnegativity; the limiter application uses r=N+3 to track the solution.
  • Re-truncation tolerance εtr = 1e-4 or 1e-10
    LoMaC truncation tolerance in the limiter experiments; sets the residual negativity floor after the corrected field is re-expressed in low-rank form.
assumptions (6)
  • domain assumption F = F1 + F2 with F1 rank-3 carrying the macroscopic moments and F2 having zero velocity moments (F2 R^T = 0)
    Borrowed from conservative low-rank methods [13,14]; if this splitting is not exact, XB=0 does not preserve the actual physical moments. Sections 2.1-2.2.
  • domain assumption Rescaling by the positive diagonal matrix S maps the weighted velocity-space inner product to the Frobenius inner product
    Section 2.3; positivity of S makes A≥0 equivalent to F≥0 and converts the conservative projection into A1=ABB^T.
  • domain assumption Feasibility of the admissible set: there exists a nonnegative matrix with the same first three velocity moments; iff ρ>0 and e·ρ ≥ m²
    Remark 2; needed for the optimization problem to have a solution. In physical Vlasov settings this is typically satisfied.
  • standard math Subdifferential of the nuclear norm (Proposition 1) and Moreau decomposition
    Used throughout Theorem 1 and the dual derivations.
  • standard math Tangent-space characterization and least-squares projection formulas for fixed-rank manifolds from [29,35]
    Lemmas 3-4 are cited from prior work and assumed in the TAP algorithm.
  • domain assumption The low-rank solution remains genuinely low rank after correction and re-truncation
    Empirical; Section 5.5 shows the rank grows but sub-linearly under mesh refinement, so the low-rank storage argument holds.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Nonnegative Low-Rank Matrix Correction under an Orthogonality Constraint in Conservative Vlasov Simulations." pith.science (2026). https://pith.science/paper/7SHKNGQK

@misc{pith2026260726272,
  author       = {Pith},
  title        = {Pith review of: Nonnegative Low-Rank Matrix Correction under an Orthogonality Constraint in Conservative Vlasov Simulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7SHKNGQK}},
  note         = {Machine review of arXiv:2607.26272}
}
abstract

In low-rank numerical methods for Vlasov dynamics, the SVD-type truncation procedure may introduce negative entries into the numerical solution. Such negative values are unphysical because the solution is a probability distribution function. We design optimization-based post-processing algorithms to recover nonnegativity while preserving the macroscopic quantities (density, momentum, and energy) pointwise. The preservation of the macroscopic quantities is written as an orthogonality constraint on the correction term. For a convex formulation based on squared nuclear norm minimization, we show that the proximal operator with the orthogonality constraint is characterized by an implicit singular value thresholding equation, and the threshold can be computed efficiently by bisection. Based on this result, we develop five algorithms for the convex formulation: Douglas--Rachford splitting, restarted dual FISTA, restarted dual accelerated gradient descent, dual PR+ conjugate gradient, and dual L-BFGS. We also consider a non-convex formulation with an explicit rank constraint and develop a tangent-space accelerated alternating projection algorithm that only requires a \(2r \times 2r\) SVD per iteration. Numerical results for a Landau damping test case show that the proposed algorithms give comparable correction quality. Among them, the tangent-space accelerated alternating projection is the most cost-efficient, increasingly so as the problem size grows. We further demonstrate the correction as a positivity limiter inside a time-dependent conservative low-rank Vlasov solver, where it removes the negativity introduced by the SVD-type truncation while preserving the conserved mass, momentum, and energy.

Figures

Figures reproduced from arXiv: 2607.26272 by the authors.

Figure 1
Figure 1. Input matrix F, scaled matrix A, and their negative parts. For all algorithms applied to the convex formulation (2), after experimentation (see [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Convergence of two convex algorithms for different values of a. (a) Douglas–Rachford splitting. (b) Dual FISTA with fixed restart, overlaying Variant I (lines, gradient of f ∗ and proximal operator of g ∗ ) and Variant II (circles, gradient of g ∗ and proximal operator of f ∗ ), with curves of the same color corresponding to the same value of a. Each row shows the orthogonality violation ∥XB∥F , the rank of X, and t… view at source ↗
Figure 3
Figure 3. Convergence of all nine convex algorithms at the single penalty a = 1000 (a1 = a2 = a/2): the orthogonality violation ∥XB∥F , the rank of X, and the relative change between iterates versus iteration. Color encodes the method family (Douglas–Rachford, accelerated gradient, FISTA, PR+ conjugate gradient, L-BFGS) and line style encodes the restart strategy (fixed, function-value, gradient), with dual FISTA shown for Va… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p025_4.png]
Figure 5
Figure 5. Figure 5: Low-rank Landau damping solution F (top) and its negative part − min{F, 0} (bottom) on the 64 × 128 (left) and 256 × 512 (right) grids, in the style of the left column of [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]
Figure 6
Figure 6. Figure 6: Cost scaling versus matrix size mn = NxNv (log–log). (a) Per-iteration cost of the Douglas–Rachford splitting (convex) and the TAP algorithm (non-convex), plotted as the median over 20 runs with error bars of ±1 standard deviation. The DR cost follows the predicted (mn…
Figure 7
Figure 7. Figure 7: Positivity limiter in the LoMaC low-rank scheme for the bump-on-tail run (128 × 256, T = 30), applied every 1000 steps, comparing the convex DR limiter (no rank cap) with the rank￾constrained TAP limiter at rank r = N + 3. (a) Negativity ∥min{F, 0}∥F / ∥F∥F of the stor…
Figure 8
Figure 8. Figure 8: Rank of the limited solution versus mesh resolution, addressing whether it approaches full rank under refinement. Top: the coarse-grid (32 × 64) counterpart of [PITH_FULL_IMAGE:figures/full_fig_p034_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

35 extracted references · 1 linked inside Pith

  1. [1]

    A. Beck. First-Order Methods in Optimization , volume 25 of MOS-SIAM Series on Optimiza- tion. SIAM, Philadelphia, PA, 2017

    2017

  2. [2]

    Beck and M

    A. Beck and M. Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences , 2(1):183–202, 2009

    2009

  3. [3]

    J.-F. Cai, E. J. Candès, and Z. Shen. A singular value thresholding algorithm for matrix completion. SIAM Journal on optimization , 20(4):1956–1982, 2010

    1956

  4. [4]

    Ceruti, J

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

    2022

  5. [5]

    Ceruti and C

    G. Ceruti and C. Lubich. An unconventional robust integrator for dynamical low-rank approx- imation. BIT Numerical Mathematics , 62(1):23–44, 2022

    2022

  6. [6]

    Y. Chen, D. Xiu, and X. Zhang. On enforcing non-negativity in polynomial approximations in high dimensions. SIAM Journal on Scientific Computing , 47(2):A866–A888, 2025

    2025

  7. [7]

    Dektor and L

    A. Dektor and L. Einkemmer. Interpolatory dynamical low-rank approximation for the 3+ 3d Boltzmann-BGK equation. Journal of Computational Physics , 547:114515, 2026

    2026

  8. [8]

    Einkemmer and I

    L. Einkemmer and I. Joseph. A mass, momentum, and energy conservative dynamical low-rank scheme for the Vlasov equation. Journal of Computational Physics , 443:110495, 2021. 36

    2021

Show all 35 references
  1. [9]

    Einkemmer, K

    L. Einkemmer, K. Kormann, J. Kusch, R. G. McClarren, and J.-M. Qiu. A review of low- rank methods for time-dependent kinetic simulations. Journal of Computational Physics , 538:114191, 2025

    2025

  2. [10]

    S. A. Goreinov, I. V. Oseledets, D. V. Savostyanov, E. E. Tyrtyshnikov, and N. L. Zamarashkin. How to find a good submatrix. In V. Olshevsky and E. Tyrtyshnikov, editors, Matrix Methods: Theory, Algorithms and Applications , pages 247–256. World Scientific, 2010

    2010

  3. [11]

    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. Communications on Applied Mathematics and Computation , 6(1):550–575, 2024

    2024

  4. [12]

    Guo and J.-M

    W. Guo and J.-M. Qiu. A low rank tensor representation of linear transport and nonlinear Vlasov solutions and their associated flow maps. Journal of Computational Physics , 458:111089, 2022

    2022

  5. [13]

    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(1):A232–A263, 2024

    2024

  6. [14]

    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(3):61, 2024

    2024

  7. [15]

    Halko, P.-G

    N. Halko, P.-G. Martinsson, and J. A. Tropp. Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Review, 53(2):217–288, 2011

    2011

  8. [16]

    Koch and C

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

    2007

  9. [17]

    Lions and B

    P.-L. Lions and B. Mercier. Splitting algorithms for the sum of two nonlinear operators. SIAM Journal on Numerical Analysis , 16(6):964–979, 1979

    1979

  10. [18]

    C. Liu, D. Milesis, C.-W. Shu, and X. Zhang. Efficient optimization-based invariant-domain- preserving limiters in solving gas dynamics equations. Journal of Computational Physics , 558:114839, 2026

    2026

  11. [19]

    D. C. Liu and J. Nocedal. On the limited memory BFGS method for large scale optimization. Mathematical Programming, 45(1–3):503–528, 1989

    1989

  12. [20]

    Lubich and I

    C. Lubich and I. V. Oseledets. A projector-splitting integrator for dynamical low-rank approx- imation. BIT Numerical Mathematics , 54(1):171–188, 2014

    2014

  13. [21]

    Y. E. Nesterov. A method of solving a convex programming problem with convergence rate o( 1 k2 ). Doklady Akademii Nauk SSSR , 269(3):543–547, 1983

    1983

  14. [22]

    Y. E. Nesterov. Gradient methods for minimizing composite functions. Mathematical Pro- gramming, 140(1):125–161, Aug 2013

    2013

  15. [23]

    O’Donoghue and E

    B. O’Donoghue and E. J. Candès. Adaptive restart for accelerated gradient schemes. Founda- tions of Computational Mathematics , 15(3):715–732, Jun 2015. 37

    2015

  16. [24]

    Perthame and C.-W

    B. Perthame and C.-W. Shu. On positivity preserving finite volume schemes for Euler equa- tions. Numerische Mathematik , 73(1):119–130, 1996

    1996

  17. [25]

    M. J. D. Powell. Convergence properties of algorithms for nonlinear optimization. SIAM Review, 28(4):487–500, 1986

    1986

  18. [26]

    G.-J. Song, M. K. Ng, and T.-X. Jiang. Tangent space based alternating projections for nonnegative low rank matrix approximation. IEEE Transactions on Knowledge and Data Engineering, 35(11):11917–11934, 2023

    2023

  19. [27]

    Sun and J

    J. Sun and J. Zhang. Global convergence of conjugate gradient methods without line search. Annals of Operations Research , 103(1–4):161–173, Mar 2001

    2001

  20. [28]

    X. Tang, R. Dwaraknath, and L. Ying. Variational inference and density estimation with non-negative tensor train. arXiv preprint arXiv:2507.21519 , 2025

    2025 arXiv

  21. [29]

    Vandereycken

    B. Vandereycken. Low-rank matrix completion by Riemannian optimization. SIAM Journal on Optimization , 23(2):1214–1236, 2013

    2013

  22. [30]

    K. Wu, X. Zhang, and C.-W. Shu. High order numerical methods preserving invariant domain for hyperbolic and related systems. arXiv preprint arXiv:2512.09116 , 2025

    2025

  23. [31]

    Ye and N

    E. Ye and N. F. G. Loureiro. Quantized tensor networks for solving the Vlasov–Maxwell equations. Journal of Plasma Physics , 90(3):805900301, 2024

    2024

  24. [32]

    Zhang and C.-W

    X. Zhang and C.-W. Shu. On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. Journal of Computational Physics , 229(23):8918–8934, 2010

    2010

  25. [33]

    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 Computational Physics, 532:113970, 2025

    2025

  26. [34]

    Zheng, W

    N. Zheng, W. A. Sands, D. Hayes, A. J. Christlieb, and J.-M. Qiu. A semi-Lagrangian adaptive rank (SLAR) method for high-dimensional Vlasov dynamics. SIAM Journal on Scientific Computing, page accepted, 2026

    2026

  27. [35]

    Zheng, W

    S. Zheng, W. Huang, B. Vandereycken, and X. Zhang. Riemannian optimization using three different metrics for Hermitian PSD fixed-rank constraints. Computational Optimization and Applications, 91(3):1135–1184, 2025. 38

    2025

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.