Pith. sign in

REVIEW 3 major objections 4 minor 45 references

Iterative thresholding low-rank time integration

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read This integrator guarantees accuracy with near-optimal ranks.

desk verdict First time-stepping low-rank integrator with quasi-optimal rank bounds, but a false equality in Section 4.3 breaks the proof as written. read the letter →

arxiv 2507.15848 v1 pith:IKUEU6FM submitted 2025-07-21 math.NA cs.NA

Markus Bachmayr , Matthieu Dolbeault , Polina Sachsenmaier This is my paper · ORCID
classification math.NAcs.NA MSC 65F5565M1265Y2065L70
keywords low-rankapproximationtimeintegrationsoftthresholdingSchrödingerequationparabolicPDEquasi-optimalranksGauss-Legendrecollocationspectraldeferredcorrection
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that time integration in low-rank matrix representation can be made both provably accurate and provably rank-efficient: for a prescribed accuracy $\eta$, the method produces approximate solutions whose representation rank stays within a constant factor of the best-approximation rank at that accuracy. The model problem is the linear time-dependent Schrödinger equation on a product domain, treated through Duhamel's formula in the twisted variable $v(t)=e^{-it\Delta}u(t)$, collocation at Gauss-Legendre points, and a fixed-point iteration (Picard, or spectral deferred correction) in which every update is soft-thresholded with a geometrically decaying threshold. The main theorem bounds the global error by a term of order $h^{J+1}$ (with $h^{2J}$ accuracy at subinterval boundaries) times an exponential in $T$, and bounds every computed rank by an explicit constant times the ranks of the exact solution's best low-rank approximations at the target accuracy. To the authors' knowledge, this is the first time-stepping scheme, outside of space-time variational formulations, that provably achieves both a guaranteed error and quasi-optimal ranks; the paper also constructs and tests numerically a parabolic variant.

What carries the argument

The load-bearing mechanism is a contractive fixed-point iteration whose every sweep is followed by soft thresholding of singular values. The twisted variable $v(t)=e^{-it\Delta}u(t)$ converts the stiff Schrödinger equation into $v'=F_t v$ with $F_t=-ie^{-it\Delta}Ve^{it\Delta}$, so the Picard map $\Phi(v)_j = v_0 + \sum_m \omega_{j,m}F_{t_m}v_m$ has contraction constant $\rho = h\Lambda_J C_V$ depending only on the bounded potential norm, not on the spatial mesh. Soft thresholding $S_\alpha$ shrinks every singular value by $\alpha$ and is non-expansive, and it satisfies $\|v-S_\alpha v\|^2 = \min_w (\|v-w\|^2 + \alpha^2\operatorname{rank}(w))$, which makes the soft-thresholding error the natural benchmark for quasi-optimal ranks. A comparison lemma, $\operatorname{rank}(S_{\alpha+\beta}u) \le \operatorname{rank}(S_\alpha v) + \|u-v\|^2/\beta^2$, turns the convergence of the iterates into rank bounds. The Gauss-Legendre nodes (roots of the Legendre polynomial on each subinterval) make the fixed point an implicit Gauss Runge-Kutta method of order $2J$ that preserves the $L^2$ norm at interval boundaries, so the local error accumulates only linearly in the number of intervals.

What would settle it

A decisive test is to run Algorithm 3.1 on the two-dimensional Schrödinger equation with a bounded finite-rank potential, initial data of full rank, and the paper's parameters (for instance $h=0.1$, $J=11$, $\eta=10^{-6}$), and check on every interval whether $\operatorname{rank}(\tilde u_n)$ is bounded by the explicit constant of Theorem 27 times $\max_{1\le i\le n} \eta^2/\alpha_i^2$; a single violation would refute the quantitative claim. The same experiment with an unbounded potential such as $V(x)=1/|x|$ falls outside the theorem's assumptions, so observing divergence or rank blow-up there would show the bounded-potential premise is essential.

Watch

Extended reading notes

Core claim

The paper's central claim is that rank adaptation in low-rank time integration can be certified. For the linear Schrödinger equation $i\partial_t u = -\Delta u + V u$ on a product domain, with $V$ a bounded potential of finite tensor rank, Algorithm 3.1 is shown to satisfy the global error bound $$\sup_{1\le n\le N} \|\tilde u_n - u(nh)\| \le \left(\eta + \kappa_J $h^{{J+1}}$ + \kappa_{2J} $h^{{2J}}$ T\right) \exp\!\left(8\Lambda_J C_V \frac{1+\rho}{(1-\rho)^3} T\right)$$ and the rank bound $$\operatorname{rank}(\tilde u_n) \le \frac{16}{\$rho^{3}$}\left(\frac{1+\rho}{1-\rho}\right)^2 \left(1 + J\,\operatorname{rank}(V) \max_{1\le i\le n} r_i\right), \qquad r_i = \frac{\$eta^{2}$}{\$alpha_i^{2}$},$$ where $\alpha_i$ is the soft-thresholding level at which the exact solution has soft-thresholding error $\eta$ on the $i$-th subinterval. In words, the computed ranks stay within a constant factor, depending on $h$, $J$, and the rank of the potential, of the ranks of the best low-rank approximations at the target accuracy. A parabolic version, using Radau-Legendre nodes and a dyadic secondary quadrature for the Duhamel integrals, is constructed and tested numerically with quasi-optimal rank behavior, but its analysis is explicitly left open.

Load-bearing premise

The load-bearing premise is that the potential $V$ is uniformly bounded, giving a contraction constant $\rho = h\Lambda_J C_V < 1$; the rest of the argument assumes $V$ has finite tensor rank and the exact solution is smooth with decaying singular values, so none of the conclusions of Theorem 27 are claimed for Coulomb-type singular potentials.

Editorial extensions

If this is right

  • For solutions whose best-approximation ranks grow like $\varepsilon^{-1/s}$ with accuracy $\varepsilon$, the theorem gives computed ranks bounded by $h^{-3}(1+J\operatorname{rank}(V))\eta^{-1/s}$ up to a constant, so algebraic singular-value decay is preserved in the computed ranks.
  • For exponentially decaying singular values, where the optimal rank behaves like $(1+|\log\varepsilon|)^\beta$, the same argument gives computed ranks of order $h^{-3}(1+J\operatorname{rank}(V))(1+|\log\eta|)^\beta$, so high-accuracy solutions remain cheap to represent.
  • The time step $h$ is limited only by the contraction condition $\rho = h\Lambda_J C_V < 1$ and by the $h^{-3}$ factor in the rank constant, not by the spatial mesh; accuracy is obtained by raising the number $J$ of collocation points.
  • The Gauss-Legendre scheme preserves the $L^2$ norm at subinterval boundaries, matching the unitary character of the Schrödinger flow, while the parabolic Radau-Legendre variant reproduces the damping of high frequencies; the numerical tests show the error accumulating essentially linearly in time.
  • The approach sidesteps the tangent-space projection step of dynamical low-rank approximation, so the error analysis does not require the projection error to be small; the paper presents this as the first such guarantee for a time-stepping method outside of space-time variational formulations.

Reading between the lines

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

  • Editorial inference: the same soft-thresholding-plus-contractive-fixed-point recipe should transfer to tensor-train and hierarchical tensor formats, since rank reduction there reduces to SVD truncation of matricizations; the paper says the scheme generalizes directly, but the dimension-dependence of the constants is not analyzed.
  • Editorial inference: the $h^{-3}$ factor in the rank bound suggests the method should be used with $h$ fixed and $J$ increased, and that taking $h$ too small actively worsens the rank guarantee; a natural testable extension is an adaptive strategy that raises $J$ per interval rather than shrinking $h$.
  • Editorial inference: the assumption that $V$ is uniformly bounded and of finite tensor rank excludes Coulomb-type many-body potentials, the physically central case; the paper only mentions Strichartz estimates as a future direction, so extending the theorem to singular potentials is the key open step.
  • Editorial inference: in the parabolic case, the secondary dyadic quadrature breaks the equivalence to a collocation/Runge-Kutta method, and the paper leaves the analysis open; the numerical evidence suggests a quasi-optimal rank bound may hold, but proving it likely requires tracking the additional quadrature error inside the fixed-point contraction.
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

3 major / 4 minor

Summary. The paper develops low-rank time integration schemes based on iterative Picard/SDC fixed-point iterations combined with soft thresholding of singular values. The central result, Theorem 27, claims for the Gauss-Legendre-Picard scheme a global error bound of order η + κ_J h^{J+1} + κ_{2J} h^{2J}T (times an exponential in T) and rank bounds proportional, up to constants, to benchmark ranks r_n = η²/α_n². The analysis rests on the contractivity of the Duhamel fixed-point iteration with step size independent of spatial resolution, a constant-factor threshold decrease, and a series of rank and quadrature lemmas. Numerical experiments for a Schrödinger equation and a parabolic equation with anisotropic diffusion illustrate the rank evolution and error behavior.

Significance. Assuming the main theorem can be repaired as indicated below, this is a notable contribution: it appears to be the first time-stepping low-rank integrator that simultaneously provides a guaranteed accuracy bound and a rank bound in terms of natural best-approximation ranks, without requiring a space-time variational formulation. The paper is careful and self-contained, with explicit constants, a detailed induction proof, and clear statements of assumptions. The use of the independent soft-thresholding result [9] is appropriate, and the numerical experiments are informative. The main mathematical idea is sound and the flaws identified below are localized and repairable.

major comments (3)
  1. [Section 4.3 and Theorem 27] The displayed equality defining α_n, namely α_n = inf_t min_{∥u(t)−v∥≤η} σmax(v) = max{α>0 : ∀t, ∥u(t)−S_αu(t)∥≤η}, is false in general. For u with singular values (2,1) and η=0.5, the left-hand side is 1.5 (attained by v with singular values (1.5,1)), while the right-hand side is approximately 0.354, the threshold at which ∥u−S_αu∥=0.5. The claim 'the minimizer is of the form v=S_{α(t)}u(t)' is also false. Consequently, the initialization ũ_0=H_{α0}u(0) with α0=1.5 produces error 1>η and rank 1>r_0≈0.111, contradicting the asserted properties and invalidating the induction in the proof of Theorem 27. The flaw is repairable by defining α_n as the largest α such that ∥u(t)−S_αu(t)∥≤η for all t in the interval, and by initializing with H_{α0}u(0) (or S_{α0}u(0)) for that α0; then the error and rank bounds in the proof go through. As written, however, the theorem's definition and proof are not valid.
  2. [Theorem 27 proof and Algorithm 3.1] The proof of Theorem 27 states 'We initialise the scheme with ũ_0 = H_{α0}u(0)', but Algorithm 3.1 contains no such step; line 3 of the algorithm sets α0←σmax(v0) and v0←0, and the initial boundary value is never thresholded. The theorem therefore analyzes a variant of the scheme that differs from the presented algorithm. In addition, the assignment v0←0 in line 3 overwrites the left boundary value that is needed in line 11 to form the next boundary approximation; a separate variable should be used for the fixed-point initial guess. The paper should reconcile the algorithm and the theorem by adding the thresholded initialization to the algorithm and fixing the name conflict.
  3. [Section 3.2] The practical implementation described after Algorithms 3.1 and 3.2 applies an additional recompression R_δ with tolerance δ_rel·res after every addition of low-rank matrices, and in the SDC case a further smaller recompression for the residual. None of the lemmas in Section 4 account for these intermediate recompressions; the analysis covers only the final recompression at the subinterval boundary (e.g., Lemma 24 and Proposition 25). As a result, the numerical experiments test a heuristic variant rather than the exactly analyzed scheme. The authors should either extend the analysis to include these steps or explicitly state that the experiments demonstrate a practical variant whose analysis is open.
minor comments (4)
  1. [Section 2.4] There is a typo, 'appoximation', in the sentence on the stability function; it should read 'approximation'.
  2. [Section 6.1] There is a typo, 'mendtioned', in the paragraph on recompression tolerances; it should read 'mentioned'.
  3. [Section 3.2, after Algorithm 3.1] The sentence 'Here res denotes the current residual error estimator' is vague because res is updated in line 7 as ∥φ_k−v_k∥_J; please give a precise definition at first use.
  4. [Lemma 15] In the proof, the final equality '2 Re(−i Σ ∥e^{it_j∆}p(t_j)∥²)=0' is correct but could be made clearer by explicitly noting that the sum is real and the real part of a purely imaginary number vanishes.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the thresholds are chosen from error targets, rank benchmarks come from the exact solution, and the estimates are proved by contraction arguments.

full rationale

The derivation is self-contained rather than circular. In Algorithms 3.1 and 3.2 the threshold parameters are either decreased deterministically (alpha_{k+1} = theta alpha_k with theta = sqrt(rho) in the analysis) or adaptively by comparing residuals, and the tolerances epsilon_n and delta_n in Theorem 27 are computed from the target eta and the previous induction bound xi_{n-1}; no parameter is fitted to the output and then re-labeled as a prediction. The rank benchmark r_n = eta^2/alpha_n^2 is defined from the exact solution's soft-thresholding error, and the proof bounds the produced ranks by this benchmark through Lemma 18 (iterate error controlled by ||u* - S_{alpha_k}u*||), Lemma 20 (rank transfer), Lemma 21 (conversion to bar-r), Lemma 22 (rank growth bounded by J rank(V)), and Lemmas 23-24 with Proposition 25 (recompression and induction). The step from this soft-threshold benchmark to 'best approximation ranks' rests on Proposition 11, proved in the paper, and Remark 12, which invokes the separate published result [9] (Bachmayr-Schneider 2017); although the first author overlaps with the present paper, it is an external, parameter-free mathematical statement and thus counts as independent evidence rather than a load-bearing self-citation. I also weighed the Section 4.3 objection: the asserted equality alpha_n = inf_t min_{||u(t)-v||<=eta} sigma_max(v) = max{alpha>0 : ||u(t)-S_alpha u(t)||<=eta} is false in general (for singular values (2,1) and eta=0.5 the left side is 1.5 while the right side is about 0.354), so the initialization tilde u_0 = H_{alpha_0}u(0) and the bound rank(tilde u_0) <= r_0 are not justified as written. This is a real correctness gap, but it is not circularity: the theorem does not define its benchmark as the algorithm's own output, and the gap is repairable by defining alpha_n solely through the soft-thresholding condition. No step in the claimed derivation reduces to its inputs by construction, so the circularity score is 0.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central theorem rests on problem-class assumptions (bounded finite-rank potential, smooth solution with decaying singular values) and standard numerical-analysis facts (SVD, quadrature error bounds, Mirsky inequality). No fitted constants are used in the proof; the method's tolerances are derived from error targets. The only hand-chosen numbers are algorithmic hyperparameters in the numerical experiments.

free parameters (2)
  • θ = 1/2 (constant decrease) or 1/5 (adaptive) in experiments
    Decrease factor for soft threshold α. Theory allows any θ ∈ [ρ,1); the experiments pick θ = 1/2 or 1/5 to balance iterations and accuracy. Hand-chosen, not fit to data.
  • δrel (intermediate recompression tolerance) = 1e-3 relative for Schrödinder; 1e-4/1e-6 in parabolic and SDC residual
    Additional recompressions used in the numerical implementation are not part of the analyzed algorithm; tolerances are chosen by hand.
assumptions (6)
  • domain assumption The potential V is a uniformly bounded multiplication operator with ∥V∥_{W(0,T) → W(0,T)} = C_V < ∞.
    Used to obtain a Lipschitz constant ρ = h ΛJ C_V independent of the spatial discretization (Section 2.1, Eq. 2.4). Fails for Coulomb singular potentials without additional estimates.
  • domain assumption V has finite tensor rank: V = Σ_{ℓ=1}^{rank(V)} V_1^{(ℓ)} ⊗ V_2^{(ℓ)}.
    Needed in Lemma 22 to bound rank(Φ u) ≤ rank(u0) + J rank(V) rank(u). Stated in Section 4.
  • domain assumption The solution u of (1.1) is smooth enough so that the constants κJ and κ2J in Proposition 13, Lemmas 16 and 17 are finite.
    High-order local error bounds require bounded derivatives (∂t - iΔ + iV)^{2J+1}u and ∂_t^J(e^{-itΔ}V u) on W(0,T).
  • domain assumption The singular values of the exact solution decay at a known rate, e.g., best η-approximation rank ≲ η^{-1/s} or (1+|log η|)^β.
    Translates the soft-threshold rank benchmark ∥u - S_α u∥²/α² into quasi-optimal ranks (Remark 12, Theorem 27).
  • domain assumption Time step h is chosen so that ρ = h ΛJ C_V < 1, with ρ ≤ 1/2 for Proposition 25.
    Ensures contractivity of the fixed-point iterations (Proposition 8) and allows the rank induction in Proposition 25.
  • standard math Standard numerical-analysis facts: Gauss quadrature error bounds, SVD properties, Mirsky inequality, and existence of the Schrödinger evolution.
    Used throughout; the paper supplies proofs for the quadrature bounds and Mirsky inequality in the appendix.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Iterative thresholding low-rank time integration." pith.science (2026). https://pith.science/paper/IKUEU6FM

@misc{pith2026250715848,
  author       = {Pith},
  title        = {Pith review of: Iterative thresholding low-rank time integration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IKUEU6FM}},
  note         = {Machine review of arXiv:2507.15848}
}
read the original abstract

We develop time integration methods in low-rank representation that can adaptively adjust approximation ranks to achieve a prescribed accuracy, while ensuring that these ranks remain proportional to the corresponding best approximation ranks. Our approach relies on an iterative scheme combined with soft thresholding of the iterates. A model case of a time-dependent Schr\"odinger equation with low-rank matrix approximation is analyzed in detail, and the required modifications for second-order parabolic problems are described. Numerical tests illustrate the results for both cases.

Figures

Figures reproduced from arXiv: 2507.15848 by the authors.

Figure 1
Figure 1. Illustration of the soft thresholding operation applied to some element v ∈ L 2 (Ω1 × Ω2). 3.1. Rank reduction. For every v ∈ L 2 (Ω1 × Ω2), there exists a singular value decompo￾sition (3.1) v = X∞ k=1 σkv (1) k ⊗ v (2) k , with the singular values σ1 ≥ σ2 ≥ . . . ≥ 0 and orthonormal families of singular vectors (v (1) k )k∈N in L 2 (Ω1) and (v (2) k )k∈N in L 2 (Ω2). We have already noted the well-known fact that … view at source ↗
Figure 2
Figure 2. Visualization of Lemma 10: deviations between the singular val￾ues of some element v ∈ L 2 (Ω1 × Ω2) and the output of a soft thresholding operator applied to v. The black dots represent coinciding values of the red and blue dots. Proof. For v of the form (3.1), we have ∥v − Sα(v)∥ 2 = X k∈N min{α, σk} 2 , hence the result follows from the inequalities min{α, σk} ≤ min{β, σk} ≤ β α min{α, σk}. □ 3.2. Iterative schem… view at source ↗
Figure 3
Figure 3. Left: illustration of the fixed-point v ∗ of Φ, and of the modified version vα due to the soft thresholding operator Sα. Right: convergence of the fixed-point iteration coupled with soft thresholding, v k , when the parameter α is decreased every second iteration. for some c, θ ∈ (0, 1). This amounts to decrease α whenever v k gets too close to vα, compared to its distance with v ∗ , see also [PITH_FULL_IMAGE:figur… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Visualization of the decomposition of the local error in Proposition 13. Let u ∗ (t) be the exact solution of the collocation formulation (2.16) on [0, h] with initial data u ∗ 0 , that is, u ∗ (t) = e it∆u ∗ 0 − i Z t 0 X J m=1 e i(t−tm)∆V u∗ mℓm(s) ds, for t ∈ [0, h]…
Figure 5
Figure 5. Figure 5: Visualization of Lemma 23: the squares of the red singular values sum up to at least δ 2 , and the blue ones to at most ∥v∥ 2 − δ 2 , thus rδ cannot be too close to rank(v) [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]
Figure 6
Figure 6. Figure 6: Left: decrease of the residual error estimator over the iterations for varying values of θ on a single interval [t0, t0 + h]. Right: evolution of the ranks using different strategies for the decrease of α and different iteration tolerances (10−3 and 10−6 ) compared to …
Figure 7
Figure 7. Figure 7: Left: evolution of the absolute error with respect to a reference solution. Right: deviation of the norm of the computed approximations from ∥u0∥ = 1 over time. about twice as many iterations as the constant decrease of α. This can be explained by the following conside…
Figure 8
Figure 8. Figure 8: Left: evolution of the ranks using different fixed-point iterations and strategies for decreasing of α, with an iteration tolerance of 10−4 , com￾pared to the optimal ranks of a truncated reference solution. Right: corre￾sponding evolution of the absolute errors with r…
Figure 9
Figure 9. Figure 9: Comparison of the ranks of the computed approximation using Picard iterations and a constant decrease of α with θ = 0.5, its truncation to a tolerance of 10−3 , the arising intermediate ranks in the computation of the approximation and the truncated reference solution.…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

45 extracted references · 42 canonical work pages

  1. [9]

    Iterative methods based on soft thresholding of hierarchical tensors

    Markus Bachmayr and Reinhold Schneider. Iterative methods based on soft thresholding of hierarchical tensors. Foundations of Computational Mathematics , 17:1037–1083, 2017

    work page 2017

  2. [1]

    Error analysis of a class of semi-discrete schemes for solving the Gross– Pitaevskii equation at low regularity

    Yvonne Alama Bronsard. Error analysis of a class of semi-discrete schemes for solving the Gross– Pitaevskii equation at low regularity. Journal of Computational and Applied Mathematics , 418:114632, 2023

    work page 2023

  3. [2]

    Borges, Duˇ san Lj

    Hessah Alqahtani, Carlos F. Borges, Duˇ san Lj. Djuki´ c, Rada M. Mutavdˇ zi´ c Djuki´ c, Lothar Reichel, and Miodrag M. Spalevi´ c. Computation of pairs of related Gauss-type quadrature rules. Applied Numerical Mathematics, 208:32–42, 2025

    work page 2025

  4. [3]

    Multilevel preconditioning and low-rank tensor iteration for space-time simultaneous discretizations of parabolic PDEs

    Roman Andreev and Christine Tobler. Multilevel preconditioning and low-rank tensor iteration for space-time simultaneous discretizations of parabolic PDEs. Numerical Linear Algebra with Applications , 22(2):317–337, 2015

    work page 2015

  5. [4]

    Robust implicit adaptive low rank time-stepping methods for matrix differential equations

    Daniel Appel¨ o and Yingda Cheng. Robust implicit adaptive low rank time-stepping methods for matrix differential equations. Journal of Scientific Computing , 102(3):81, 2025

    work page 2025

  6. [5]

    Aristoff, Joshua T

    Jeffrey M. Aristoff, Joshua T. Horwood, and Aubrey B. Poore. Orbit and uncertainty propagation: A comparison of Gauss–Legendre-, Dormand–Prince-, and Chebyshev–Picard-based approaches. Celestial Mechanics and Dynamical Astronomy , 118(1):13–28, 2014

    work page 2014

  7. [6]

    Low-rank tensor methods for partial differential equations

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

    2023

  8. [7]

    Adaptive near-optimal rank tensor approximation for high- dimensional operator equations

    Markus Bachmayr and Wolfgang Dahmen. Adaptive near-optimal rank tensor approximation for high- dimensional operator equations. Foundations of Computational Mathematics , 15(4):839–898, 2015. 32 ITERATIVE THRESHOLDING LOW-RANK TIME INTEGRATION

    work page 2015

Show all 45 references
  1. [8]

    A space-time adaptive low-rank method for high-dimensional parabolic partial differential equations

    Markus Bachmayr and Manfred Faldum. A space-time adaptive low-rank method for high-dimensional parabolic partial differential equations. Journal of Complexity , 82:Paper No. 101839, 70, 2024

    2024

  2. [10]

    Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods

    Thomas Boiveau, Virginie Ehrlacher, Alexandre Ern, and Anthony Nouy. Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods. ESAIM Math. Model. Numer. Anal. , 53(2):635–658, 2019

    2019

  3. [11]

    A class of exponential integrators based on spectral deferred correction

    Tommaso Buvoli. A class of exponential integrators based on spectral deferred correction. SIAM Journal on Scientific Computing , 42(1):A1–A27, 2020

    2020

  4. [12]

    Projected exponential methods for stiff dynamical low-rank approximation problems

    Benjamin Carrel and Bart Vandereycken. Projected exponential methods for stiff dynamical low-rank approximation problems. arXiv preprint arXiv:2312.00172, 2023

    2023 arXiv

  5. [13]

    On the convergence of spectral deferred correction methods

    Mathew Causley and David Seal. On the convergence of spectral deferred correction methods. Commu- nications in Applied Mathematics and Computational Science , 14(1):33–64, 2019

    2019

  6. [14]

    A rank-adaptive robust integrator for dynamical low-rank approximation

    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

  7. [15]

    An unconventional robust integrator for dynamical low-rank approximation

    Gianluca Ceruti and Christian Lubich. An unconventional robust integrator for dynamical low-rank approximation. BIT Numerical Mathematics , 62(1):23–44, 2022

    2022

  8. [16]

    Clenshaw

    Charles W. Clenshaw. The numerical solution of linear differential equations in Chebyshev series. Math- ematical Proceedings of the Cambridge Philosophical Society , 53(1):134–149, 1957

    1957

  9. [17]

    Clenshaw and Harry J

    Charles W. Clenshaw and Harry J. Norton. The solution of nonlinear ordinary differential equations in Chebyshev series. The Computer Journal , 6(1):88–92, 1963

    1963

  10. [18]

    Rank-adaptive tensor methods for high-dimensional nonlinear PDEs

    Alec Dektor, Abram Rodgers, and Daniele Venturi. Rank-adaptive tensor methods for high-dimensional nonlinear PDEs. Journal of Scientific Computing , 88(2):36, 2021

    2021

  11. [19]

    A space-time variational formulation for the many-body electronic Schr¨ odinger evolution equation.arXiv preprint arXiv:2405.18094 , 2024

    Mi-Song Dupuy, Virginie Ehrlacher, and Cl´ ement Guillot. A space-time variational formulation for the many-body electronic Schr¨ odinger evolution equation.arXiv preprint arXiv:2405.18094 , 2024

    2024 arXiv

  12. [20]

    Spectral deferred correction methods for ordinary differential equations

    Alok Dutt, Leslie Greengard, and Vladimir Rokhlin. Spectral deferred correction methods for ordinary differential equations. BIT Numerical Mathematics , 40:241–266, 2000

    2000

  13. [21]

    Golub and John H

    Gene H. Golub and John H. Welsch. Calculation of Gauss quadrature rules. Math. Comp. , 23:221–230; addendum, ibid. 23 (1969), no. 106, loose microfiche suppl. A1–A10, 1969

    1969

  14. [22]

    A literature survey of low-rank tensor approxi- mation techniques

    Lars Grasedyck, Daniel Kressner, and Christine Tobler. A literature survey of low-rank tensor approxi- mation techniques. GAMM-Mitteilungen, 36(1):53–78, 2013

    2013

  15. [23]

    Tensor spaces and numerical tensor calculus , volume 56 of Springer Series in Computational Mathematics

    Wolfgang Hackbusch. Tensor spaces and numerical tensor calculus , volume 56 of Springer Series in Computational Mathematics . Springer, Cham, 2019. Second edition

    2019

  16. [24]

    Hager, Hongyan Hou, and Anil V

    William W. Hager, Hongyan Hou, and Anil V. Rao. Lebesgue constants arising in a class of collocation methods. IMA Journal of Numerical Analysis , 2016

    2016

  17. [25]

    On the spectral deferred correction of splitting methods for initial value problems

    Thomas Hagstrom and Ruhai Zhou. On the spectral deferred correction of splitting methods for initial value problems. Communications in Applied Mathematics and Computational Science , 1(1):169–205, 2007

    2007

  18. [26]

    Ernst Hairer, Gerhard Wanner, and Syvert P. Nørsett. Solving Ordinary Differential Equations I , vol- ume 8 of Springer Series in Computational Mathematics . Springer Berlin Heidelberg, Berlin, Heidelberg, 1993

    1993

  19. [27]

    On the convergence of Lawson methods for semilinear stiff problems

    Marlis Hochbruck, Jan Leibold, and Alexander Ostermann. On the convergence of Lawson methods for semilinear stiff problems. Numerische Mathematik , 145:553–580, 2020

    2020

  20. [28]

    Exponential integrators

    Marlis Hochbruck and Alexander Ostermann. Exponential integrators. Acta Numerica, 19:209–286, 2010

    2010

  21. [29]

    Accelerating the convergence of spectral deferred correc- tion methods

    Jingfang Huang, Jun Jia, and Michael Minion. Accelerating the convergence of spectral deferred correc- tion methods. Journal of Computational Physics , 214(2):633–656, 2006

    2006

  22. [30]

    Trefethen

    Aly-Khan Kassam and Lloyd N. Trefethen. Fourth-order time-stepping for stiff PDEs. SIAM Journal on Scientific Computing , 26(4):1214–1233, 2005

    2005

  23. [31]

    Dynamical low-rank approximation

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

    2007

  24. [32]

    Second order exponential splittings in the presence of unbounded and time-dependent operators: Construction and convergence

    Karolina Kropielnicka and Juan Carlos Del Valle. Second order exponential splittings in the presence of unbounded and time-dependent operators: Construction and convergence. SIAM Journal on Numerical Analysis, 63(1):288–305, 2025

    2025

  25. [33]

    Randomized low-rank Runge–Kutta methods

    Hei Yin Lam, Gianluca Ceruti, and Daniel Kressner. Randomized low-rank Runge–Kutta methods. SIAM Journal on Matrix Analysis and Applications , 46(2):1587–1615, 2025

    2025

  26. [34]

    Generalized Runge-Kutta processes for stable systems with large Lipschitz con- stants

    John Douglas Lawson. Generalized Runge-Kutta processes for stable systems with large Lipschitz con- stants. SIAM Journal on Numerical Analysis , 4(3):372–380, 1967

    1967

  27. [35]

    Layton and Michael L

    Anita T. Layton and Michael L. Minion. Implications of the choice of quadrature nodes for Picard integral deferred corrections methods for ordinary differential equations. BIT, 45(2):341–373, 2005

    2005

  28. [36]

    High-order implicit low-rank method with spectral deferred correction for matrix differential equations

    Shun Li, Yan Jiang, and Yingda Cheng. High-order implicit low-rank method with spectral deferred correction for matrix differential equations. arXiv preprint arXiv:2412.09400 , 2024

    2024 arXiv

  29. [37]

    Oseledets

    Christian Lubich and Ivan V. Oseledets. A projector-splitting integrator for dynamical low-rank approx- imation. BIT, 54(1):171–188, 2014. ITERATIVE THRESHOLDING LOW-RANK TIME INTEGRATION 33

    2014

  30. [38]

    The eigen- and singular values of the sum and product of linear operators

    Aleksandr Semenovich Markus. The eigen- and singular values of the sum and product of linear operators. Russian Mathematical Surveys , 19(4):91, 1964

    1964

  31. [39]

    Robust high-order low-rank BUG integrators based on explicit Runge-Kutta methods

    Fabio Nobile and S´ ebastien Riffaud. Robust high-order low-rank BUG integrators based on explicit Runge-Kutta methods. arXiv preprint arXiv:2502.07040 , 2025

    2025 arXiv

  32. [40]

    Implicit integration of nonlinear evolution equations on tensor manifolds

    Abram Rodgers and Daniele Venturi. Implicit integration of nonlinear evolution equations on tensor manifolds. Journal of Scientific Computing , 97(2):33, 2023

    2023

  33. [41]

    Introduction to Numerical Analysis

    Josef Stoer and Roland Bulirsch. Introduction to Numerical Analysis . Springer New York, 1980

    1980

  34. [42]

    Implicit low-rank Riemannian schemes for the time integration of stiff partial differential equations

    Marco Sutti and Bart Vandereycken. Implicit low-rank Riemannian schemes for the time integration of stiff partial differential equations. Journal of Scientific Computing , 101(1):3, 2024

    2024

  35. [43]

    Faster SDC convergence on non-equidistant grids by DIRK sweeps.BIT Numerical Math- ematics, 55(4):1219–1241, 2015

    Martin Weiser. Faster SDC convergence on non-equidistant grids by DIRK sweeps.BIT Numerical Math- ematics, 55(4):1219–1241, 2015

    2015

  36. [44]

    Mingru Yang and Steven R. White. Time-dependent variational principle with ancillary Krylov subspace. Physical Review B , 102(9):094315, 2020

    2020

  37. [45]

    Lemma 32 (Gauss-Legendre quadrature error)

    Appendix In the following, we include some auxiliary lemmas needed in the proof of Theorem 27, and recall their proofs for the sake of completeness. Lemma 32 (Gauss-Legendre quadrature error) . Let ωj and tj for 1 ≤ j ≤ J be the Gauss-Legendre quadrature weights and nodes on t...

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.