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A Parareal Algorithm with Low-Rank Coarse Solvers

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Pith's one-line read This paper proposes a Parareal algorithm whose coarse solver is the truncated singular value decomposition of the fine solver's transfer operator, and proves the error after k iterations is governed by the largest discarded singular value.

desk verdict First practical spectral coarse solver for Parareal, with rigorous convergence analysis and broad numerical evidence; the main limitation is that the practical gains depend on fast singular-value decay, which is demonstrated but not proven for general non-normal parabolic operators. read the letter →

arxiv 2508.08873 v2 pith:SGP6NGYI submitted 2025-08-12 math.NA cs.NA

Martin J. Gander , Mario Ohlberger , Stephan Rave This is my paper · ORCID
classification math.NAcs.NA MSC 65M2265Y05
keywords Pararealparallel-in-timespectralcoarsesolvertransferoperatorrandomizedSVDsingularvaluedecayparabolicPDEreducedbasismethod
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 aims to make the parallel-in-time Parareal algorithm faster for linear parabolic PDEs by using a low-rank spectral coarse solver: on each time interval, the coarse propagator is a truncated singular value decomposition of the fine solver's transfer operator instead of a cheaper time stepper. The central result is that after k Parareal iterations the error is bounded by $\varepsilon^k$ times a combinatorial factor, where $\varepsilon$ is the largest singular value discarded by the truncation. Because parabolic transfer operators have rapidly decaying singular values, a small truncation rank should already give fast convergence. The coarse solvers are built beforehand by randomized SVDs in an embarrassingly parallel fashion, exchanging expensive global iterations for independent local solves. Experiments on 1D, 2D, and 3D heat problems show this approach beating Parareal with single backward Euler coarse solvers both in iteration count and in measured speedup.

What carries the argument

The central object is the spectral coarse solver $G_n$, defined as the rank-$R_n$ truncated SVD of the fine solver's linear transfer operator $F'_n$, plus the same affine shift $b_n$ as $F_n$. Its role is to make the fine-coarse difference exactly equal to the tail of the singular value expansion: $\|F_n - G_n\| = \sigma_{n,R_n+1}$. The randomized SVD construction (random initial values, applied to $F'_n$ and its adjoint) is what makes computing these spectral approximations embarrassingly parallel and cheap.

What would settle it

Take a linear parabolic test case where the transfer operators are expected to have slow singular value decay, for instance a heat equation with very small diffusivity or very short Parareal intervals, compute the singular values of $F'_n$, and run the algorithm with rank $R$. If the largest discarded singular value $\varepsilon$ stays close to 1 for any affordable $R$, or the observed error does not follow the predicted $\varepsilon^k$ decay, the paper's characterization of parabolic problems and the practical claim fail.

Watch

Extended reading notes

Core claim

The paper's claim is that the coarse solver in Parareal can be a spectral approximation of the fine solver itself. Writing the fine solver $F_n$ as affine with linear part $F'_n$, the coarse solver $G_n$ keeps only the leading $R_n$ singular modes of $F'_n$, so the fine-coarse mismatch has operator norm exactly $\|F'_n - G'_n\| = \sigma_{n,R_n+1}$. The paper proves that the Parareal error satisfies $\|e^k_n\| \le \varepsilon^k \sum_m \binom{n-m}{k-1} \delta^{n-m-k} \|e^0_m\|$, with $\varepsilon = \max_n \sigma_{n,R_n+1}$ and $\delta = \max_n \sigma_{n,1}$, and a simpler $\varepsilon^k$ bound when $\delta \le 1$ or $\delta < 1$. For a time-independent self-adjoint fine solver the estimate sha

Load-bearing premise

The practical speedup rests on the assumption that the fine solver's transfer operators have rapidly decaying singular values, a property demonstrated numerically for parabolic heat problems but not proven in general for the discretized operators used here.

Editorial extensions

If this is right

  • For stable fine solvers ($\delta \le 1$), the maximum error after $k$ iterations is at most $\binom{N}{k}\varepsilon^k$, so the iteration count needed for a fixed tolerance is governed directly by the truncation tail $\varepsilon$.
  • For a time-independent self-adjoint fine operator, the convergence rate is exactly $\sigma_{R+1}^k$, so the singular value spectrum tells the user in advance which rank gives which speed.
  • Because the affine part is the only data-dependent piece of the coarse solver, the same spectral coarse solvers can be reused for new source terms without further setup work.
  • Raising the truncation rank $R_n$ converts global Parareal iterations into independent local fine solves in the setup phase, so the method offers a knob to trade communication for parallelism.
  • On the 3D heat-sink test case, rank-2 spectral coarse solvers reach an error around $10^{-13}$ with roughly fourfold speedup over sequential integration, while Euler coarse solvers give almost no speedup at that accuracy.

Reading between the lines

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

  • The paper fixes one truncation rank for all intervals; the a posteriori bound it derives suggests a natural extension where each interval's rank is increased adaptively, based on the current Parareal update norms, rather than chosen up front.
  • Although the analysis is for linear PDEs, the transfer-operator viewpoint carries over to linearizations of nonlinear flows, so the same spectral coarse solver could be applied to Jacobians in Newton-type integrators; the paper does not explore this.
  • The setup phase's cost profile (parallel solves for random initial values, then block solves for many right-hand sides) points toward node-local or GPU batching as the main implementation lever for scaling beyond the direct-solver experiments shown.
  • If a target problem has slowly decaying singular values, the paper's own theorem predicts the gains will vanish; a power-iteration or block-Krylov variant would be the natural repair, since the paper notes power iterations exist but does not test them.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper proposes a Parareal algorithm in which the coarse propagator G_n is a truncated SVD approximation of the fine transfer operator F_n, including its affine part. The SVD is computed by randomized sampling, so the setup phase is embarrassingly parallel. The authors prove a priori error bounds (Theorems 1 and 2, Corollaries 1–2) showing that the Parareal error after k iterations is controlled by powers of ε = max_n σ_{n,R_n+1}, the largest truncated singular value of the fine-solver transfer operators, and an a posteriori bound (Theorem 3) expressed in terms of the observed Parareal updates. Numerical experiments on one-, two-, and three-dimensional parabolic problems compare exact and randomized SVD coarse solvers with single-step Euler coarse solvers, reporting faster convergence, reduced numbers of fine solves, and speedup over sequential time stepping. Code and data are made available.

Significance. If the results hold, this is a valuable contribution to parallel-in-time methods: it replaces an ad hoc coarse propagator with a low-rank approximation that has a transparent convergence theory, and the construction is embarrassingly parallel. The convergence analysis in Section 3 is rigorous, standard, and self-contained, extending the Gander–Hairer framework to this class of coarse solvers. The numerical experiments are substantial, including a 3D heat-sink problem with 444,693 degrees of freedom, and the paper ships reproducible code and detailed experimental configuration, which is a strength. The paper is also transparent about limitations: the randomized SVD construction is justified mainly numerically, the a posteriori estimator can be inaccurate near machine precision, and the practical speedup relies on rapid singular-value decay of the transfer operators, which is demonstrated empirically but not proven for the whole class of parabolic problems. These caveats are acknowledged in the text, but they should be reflected more carefully in the statements of the theoretical and practical claims.

major comments (2)
  1. [§2.1, §3, Theorem 1, Eqs. (11)–(12)] Theorem 1 bounds the Parareal error in terms of ε = max_n σ_{n,R_n+1}, the exact truncation singular values of F'_n. However, the coarse solvers used in the numerical experiments are constructed by the randomized SVD of §2.1, so the computed G_n is not the exact rank-R SVD truncation of F'_n. Consequently, the key inequality ∥F'_n − G'_n∥ ≤ σ_{n,R_n+1} used in the proof of Theorem 1 does not hold verbatim for the implemented algorithm. The authors should either supply a probabilistic perturbation estimate for the randomized SVD error (for example using the results of [18]) or explicitly state that the convergence analysis applies to the exact-SVD idealization and that the randomized version is supported by numerical evidence. As written, the abstract's claim of bounds "in terms of the computed singular values" is stronger than what is proved.
  2. [§4, Fig. 8; §5] The paper concludes that spectral coarse solvers "can dramatically accelerate" Parareal for problems of parabolic type. This claim is supported by the specific examples in Section 4, but the efficiency of the method rests on rapid singular-value decay of the transfer operators F'_n. The decay is observed in Fig. 8 and has been proved for the exact 1D heat equation in prior work [14]; it is not established for variable-coefficient, non-normal, or short-interval cases. If σ_{n,R+1} decays only slowly, ε in (9) is not small for practical ranks, the a priori bounds are weak, and the speedup over Euler coarse solvers can be eroded. The paper should make this assumption explicit in the conclusions and add a short discussion of when such decay can be expected (e.g., diffusion dominance, dependence on interval length). This does not require a new theorem, but it would make the scope of the headl
minor comments (4)
  1. [Throughout, esp. §4] There are many incorrect cross-references: "top row of Section 3", "Section 4.1" in the text should refer to specific figure numbers. Please correct these to the appropriate Figure labels (e.g., Figure 1, Figure 2, etc.).
  2. [§4, Figs. 6, 9, 11] The randomized SVD experiments are reported for single realizations only. Since the randomized SVD is stochastic, please either add error bars over several trials or state explicitly that a single representative run is shown.
  3. [§4.5, Eq. (26), Fig. 13] The efficiency η_k is reported to exceed 1 in some runs, i.e., the a posteriori bound (19) underestimates the actual error. The text attributes this to floating point accuracy, but this means the word "bound" is only valid in exact arithmetic. Please add a sentence in Section 3 noting that (19) is an exact-arithmetic bound and may be violated in floating point computations.
  4. [§2.2, Table 1] The cost discussion in §2.2 would benefit from a concise summary in the conclusions: the setup phase costs R+1 fine solves and R adjoint solves per interval, and this overhead is part of the speedup comparison. This is already in Table 1, but it is not mentioned in Section 5.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence analysis derives the Parareal error bound from the SVD truncation error of the coarse solver, not from observed convergence or fitted constants.

full rationale

The central derivation (Lemma 1, Theorems 1–3) is a direct induction on the Parareal error recursion (8). The key quantities δ = max_n σ_{n,1} and ε = max_n σ_{n,R_n+1} are defined by the SVD of the fine-solver transfer operator and satisfy ‖F′_n − G′_n‖ = σ_{n,R_n+1} by (5), so the bounds (11)–(14) express the contraction of the iteration in terms of the approximation error of the coarse solver. This is a rigorous a priori/a posteriori statement, not a fit of observed Parareal errors. The practical claim that small ranks suffice relies on rapid singular-value decay; the paper supports this by a Fourier argument for the heat equation (Eq. (22)) and by numerical experiments (Fig. 8), and it is explicitly conditional. The self-citation [14] is used for supporting facts (exponential decay of singular values and an analogous proof in a special case), but the main theorem does not reduce to it, and the proof of Theorem 2 is included in the paper. No step in the derivation defines the prediction in terms of the output, fits a parameter to the target quantity, or imports a uniqueness theorem. The only weakness is the unproven but numerically observed singular-value decay assumption for general non-normal parabolic operators; that is a support/correctness concern, not circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central results rest on standard SVD theory, a domain assumption on the affine/compact structure of the fine solver, a reliance on known randomized SVD guarantees, and an empirical (unproved) fast singular value decay for parabolic problems. No new physical or mathematical entities are introduced.

free parameters (2)
  • Truncation rank R_n = user-chosen (e.g., R=1..6 in experiments)
    Controls accuracy vs cost of the coarse solver; the convergence bounds depend on ε = max_n σ_{n,R_n+1}.
  • Oversampling parameter p = p=0 or 1 in experiments
    Number of extra random vectors in the randomized SVD; affects the accuracy of the estimated singular vectors.
assumptions (5)
  • domain assumption The fine solver F_n is affine linear, F_n v = F'_n v + b_n, with F'_n compact.
    Assumed in Section 2; satisfied by Runge-Kutta schemes on linear ODEs/PDEs, and the analysis uses the SVD of F'_n.
  • standard math The SVD of F'_n exists and the truncated SVD error is given by the next singular value σ_{n,R+1}.
    Classical SVD theory for compact operators on Hilbert spaces, used in (3)-(5).
  • domain assumption The randomized SVD gives accurate approximations of the leading singular values/vectors with high probability.
    Relies on Halko et al. [18] for finite-dimensional V; the paper does not re-prove this and simply uses p=0/1 in experiments.
  • domain assumption For linear parabolic PDEs, the transfer operators have rapidly decaying singular values.
    Assumed in the introduction and observed numerically (Section 4); essential for small R to be effective.
  • domain assumption Stability: δ = max_n σ_{n,1} ≤ 1 (or < 1) for the linear convergence bounds.
    Used in Theorem 1 (13)-(14); holds for heat equation with Dirichlet (δ<1) or Neumann (δ=1).

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Pith. "Pith review of A Parareal Algorithm with Low-Rank Coarse Solvers." pith.science (2026). https://pith.science/paper/SGP6NGYI

@misc{pith2026250808873,
  author       = {Pith},
  title        = {Pith review of: A Parareal Algorithm with Low-Rank Coarse Solvers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SGP6NGYI}},
  note         = {Machine review of arXiv:2508.08873}
}
read the original abstract

We consider a new class of Parareal algorithms, which use ideas from localized reduced basis methods to construct the coarse solver from truncated SVD approximations of the transfer operators mapping initial values for a given time interval to the solution at the end of the interval. By leveraging randomized singular value decompositions, these low-rank approximations are obtained embarrassingly parallel by computing local fine solutions for random initial values. We show a priori and a posteriori error bounds in terms of the computed singular values of the transfer operators. Our numerical experiments demonstrate that our approach can significantly outperform Parareal with single-step coarse solvers. At the same time, it permits to further increase parallelism in Parareal by trading global iterations for a larger number of independent local solves.

Figures

Figures reproduced from arXiv: 2508.08873 by the authors.

Figure 1
Figure 1. Experiment 1: Solutions of test problem (21) with different final times T and boundary conditions. Taking norms then leads to ∥e k n∥ ≤ ε∥u k n−1 − u k−1 n−1 ∥ + δ∥e k n−1∥. Applying this inequality repeatedly to its own right-hand side and noting that e k 0 = u k 0 −u k−1 0 = 0, we obtain (19). For δ < 1, we obtain (20) from (19) using Pn−1 m=1 δ n−m−1 ≤ 1/(1 − δ). As for the a priori bound (12), the quantities δ a… view at source ↗
Figure 2
Figure 2. Experiment 1 with homogeneous Dirichlet conditions: ℓ 2 -errors of Parareal approximations over time for an increasing number of Parareal iterations; Gn = 0, 10 Parareal subintervals of different length by varying the final time T. where ˆfm and uˆ0,m denote the coefficients of the Fourier sine series for f and u0. Dis￾cretizing using a uniform mesh with 100 elements in space and 100 · T time steps, we obtain the so… view at source ↗
Figure 3
Figure 3. Experiment 1 with homogeneous Neumann conditions: ℓ 2 -errors of Parareal approximations over time for an increasing number of Parareal iterations; left: Gn = 0, right: rank-1 Gn with constant Fourier mode. Returning to Dirichlet conditions, we finally study the effect of including higher￾order Fourier modes √ 2 sin(mπx) into the definition of G (again setting bn = 0). In the bottom-right plot of Section 4.1, we see… view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Experiment 1 with different boundary conditions and coarse solvers Gn. Maxi￾mum ℓ 2 -errors over time vs. number of Parareal iterations for different final times T = 2l (top left/right, bottom left) and ranks R of Gn (bottom right). The dashed line in the bottom-right …
Figure 5
Figure 5. Figure 5: Experiment 2: Solution of test problem (23) (left) and singular values of the fine solvers F ′ n (right). The singular values only differ slightly with respect to the chosen time interval. 0 2 4 6 8 10−16 10−13 10−10 10−7 10−4 10−1 102 ` 2 -`∞-error Fourier modes 0 2 4…
Figure 6
Figure 6. Figure 6: Experiment 2: Maximum ℓ 2 -errors over time vs. number of Parareal iterations for different choices and ranks of Gn. The dashed lines indicate the Parareal errors when the Gn are given by a single backward Euler step. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Experiment 3: Solution of test problem (24) at different times t. 0 5 10 15 20 25 30 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 r singular value Experiment 3 1 2 3 4 5 6 7 8 9 10 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 r Experiment 4 [PITH_FULL_IMAGE:figures/ful…
Figure 8
Figure 8. Figure 8: Singular values of the fine solvers F ′ n (left: Experiment 3, right: Experiment 4). 18 [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: Experiment 3: Maximum ℓ 2 -errors over time vs. number of Parareal iterations for different ranks R of Gn. The dashed lines indicate the Parareal errors when the Gn are given by a single backward Euler step. exponential and SVD-based coarse solvers of rank 1 are alread…
Figure 10
Figure 10. Figure 10: Experiment 4: Solution of (25) at times t = 0.25 (top left), t = 0.5 (top right), t = 0.75 (bottom left) and t = 1 (bottom right). 0 4 8 12 16 20 24 10−16 10−13 10−10 10−7 10−4 10−1 102 k ` 2 -`∞-error randomized SVD (p = 0) 0 4 8 12 16 20 24 10−16 10−13 10−10 10−7 10…
Figure 11
Figure 11. Figure 11: Experiment 4: Maximum ℓ 2 -errors over time vs. number of Parareal iterations for different ranks R of Gn. The dashed lines indicate the Parareal errors when the Gn are given by a single backward Euler step. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 12
Figure 12. Figure 12: Experiment 4: Maximum ℓ 2 -errors over time vs. speedup w.r.t. a sequential solution for different ranks R of Gn and different numbers of Parareal iterations. The marker indicates the Parareal speedup when the Gn are given by a single backward Euler step. where, as in…
Figure 13
Figure 13. Figure 13: Efficiencies η k (26) for the a posteriori error bound (19) vs. number of Parareal iterations for different ranks R of Gn, p = 1. The plots are drawn dashed when the maximum error falls below 10−13 . as we have seen in Experiment 4, spectral coarse solvers can even le…

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Reference graph

Works this paper leans on

34 extracted references · 23 canonical work pages · cited by 1 Pith paper

  1. [18]

    Halko, P

    N. Halko, P. G. Martinsson, and J. A. Tropp. Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev., 53(2):217–288, 2011. doi:10.1137/090771806

    doi:10.1137/090771806 2011

  2. [14]

    A Parareal algorithm without Coarse Propagator?

    Martin J. Gander, Mario Ohlberger, and Stephan Rave. A Parareal algorithm without Coarse Propagator?, 2024. arXiv:2409.02673. 25

    work page Pith review arXiv 2024

  3. [1]

    Multi-step variant of the parareal algorithm: convergence analysis and numerics

    Katia Ait-Ameur and Yvon Maday. Multi-step variant of the parareal algorithm: convergence analysis and numerics. ESAIM Math. Model. Numer. Anal. , 58(2):673– 694, 2024. doi:10.1051/m2an/2024014

    work page arXiv 2024

  4. [2]

    Model reduction of parametrized systems , volume 17 of MS&A

    Peter Benner, Mario Ohlberger, Anthony Patera, Gianluigi Rozza, and Karsten Urban, editors. Model reduction of parametrized systems , volume 17 of MS&A. Modeling, Simulation and Applications . Springer, Cham, 2017. Selected papers from the 3rd MoRePaS Conference held at the International School for Advanced Studies (SISSA), Trieste, October 13–16, 2015. d...

    work page doi:10.1007/978-3-319-58786-8 2017

  5. [3]

    Localized model reduction for parameterized problems

    Andreas Buhr, Laura Iapichino, Mario Ohlberger, Stephan Rave, Felix Schindler, and Kathrin Smetana. Localized model reduction for parameterized problems. In 24 Peter Benner, Stefano Grivet-Talocia, Alfio Quarteroni, Gianluigi Rozza, Wilhelmus Schilders, and Lu ´ ıs Miguel Silveira, editors,Model Order Reduction (Volume 2) . De Gruyter, Berlin, Boston, 202...

    work page doi:10.1515/9783110671490-006 2021

  6. [4]

    Randomized local model order reduction

    Andreas Buhr and Kathrin Smetana. Randomized local model order reduction. SIAM J. Sci. Comput. , 40(4):A2120–A2151, 2018. doi:10.1137/17M1138480

    work page doi:10.1137/17m1138480 2018

  7. [5]

    Gander, and Bart Vandereycken

    Benjamin Carrel, Martin J. Gander, and Bart Vandereycken. Low-rank parareal: a low-rank parallel-in-time integrator. BIT Numerical Mathematics , 63(1):13, 2023. doi:10.1007/s10543-023-00953-3

    work page doi:10.1007/s10543-023-00953-3 2023

  8. [6]

    Hesthaven, and Xueyu Zhu

    Feng Chen, Jan S. Hesthaven, and Xueyu Zhu. On the use of reduced basis methods to accelerate and stabilize the parareal method. Reduced Order Methods for modeling and computational reduction , pages 187–214, 2014. doi:10.1007/ 978-3-319-02090-7_7

    work page 2014

Show all 34 references
  1. [7]

    Lisandro Dalcin and Yao-Lung L. Fang. Mpi4py: Status Update After 12 Years of Development. Computing in Science & Engineering , 23(4):47–54, 2021. doi: 10.1109/MCSE.2021.3083216

    2021

  2. [8]

    Time- parallel implicit integrators for the near-real-time prediction of linear structural dynamic responses

    Charbel Farhat, Julien Cortial, Clim` ene Dastillung, and Henri Bavestrello. Time- parallel implicit integrators for the near-real-time prediction of linear structural dynamic responses. International Journal for Numerical Methods in Engineering , 67(5):697–724, 2006. doi:10.1...

    2006 doi

  3. [9]

    Gander and M

    M. Gander and M. Petcu. Analysis of a Krylov subspace enhanced parareal algorithm for linear problems. ESAIM: Proceedings, 25:114–129, 2008. doi:10.1051/proc: 082508

    2008 doi

  4. [10]

    Gander and Ernst Hairer

    Martin J. Gander and Ernst Hairer. Nonlinear convergence analysis for the parareal algorithm. In Domain decomposition methods in science and engineering XVII , pages 45–56. Springer, 2008. doi:10.1007/978-3-540-75199-1_4

    2008 doi

  5. [11]

    Gander and Ernst Hairer

    Martin J. Gander and Ernst Hairer. Analysis for parareal algorithms applied to Hamiltonian differential equations. Journal of Computational and Applied Mathe- matics, 259:2–13, 2014. doi:10.1016/j.cam.2013.01.011

    2014 doi

  6. [12]

    Gander and Thibaut Lunet

    Martin J. Gander and Thibaut Lunet. Time Parallel Time Integration . SIAM, Philadelphia, PA, 2024. doi:10.1137/1.9781611978025

    2024 doi

  7. [13]

    Gander, Thibaut Lunet, Daniel Ruprecht, and Robert Speck

    Martin J. Gander, Thibaut Lunet, Daniel Ruprecht, and Robert Speck. A unified analysis framework for iterative parallel-in-time algorithms. SIAM J. Sci. Comput. , 45(5):A2275–A2303, 2023. doi:10.1137/22M1487163

    2023 doi

  8. [15]

    Gander and Stefan Vandewalle

    Martin J. Gander and Stefan Vandewalle. Analysis of the parareal time-parallel time-integration method. SIAM Journal on Scientific Computing , 29(2):556–578,

  9. [16]

    Hirstoaga, Van-Thanh Nguyen, and Julien Salomon

    Laura Grigori, Sever A. Hirstoaga, Van-Thanh Nguyen, and Julien Salomon. Re- duced model-based parareal simulations of oscillatory singularly perturbed ordi- nary differential equations. J. Comput. Phys. , 436:Paper No. 110282, 18, 2021. doi:10.1016/j.jcp.2021.110282

    2021

  10. [17]

    Tom Gustafsson and G. D. McBain. Scikit-fem: A Python package for finite element assembly. Journal of Open Source Software , 5(52):2369, 2020. doi:10.21105/joss. 02369

    2020 doi

  11. [19]

    The reduced basis technique as a coarse solver for parareal in time simulations

    Liping He. The reduced basis technique as a coarse solver for parareal in time simulations. Journal of Computational Mathematics , pages 676–692, 2010. URL: https://www.jstor.org/stable/43693608

    2010

  12. [20]

    Hesthaven, Gianluigi Rozza, and Benjamin Stamm

    Jan S. Hesthaven, Gianluigi Rozza, and Benjamin Stamm. Certified reduced basis methods for parametrized partial differential equations. SpringerBriefs in Mathematics. Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao, 2016. BCAM SpringerBriefs. doi:10.1007/978...

    2016 doi

  13. [21]

    Optimizing coarse propagators in parareal algo- rithms

    Bangti Jin, Qingle Lin, and Zhi Zhou. Optimizing coarse propagators in parareal algo- rithms. SIAM J. Sci. Comput. , 47(2):A735–A761, 2025. doi:10.1137/23M1619733

    2025 doi

  14. [22]

    R. B. Lehoucq, D. C. Sorensen, and C. Yang. ARPACK Users’ Guide . Software, Environments, and Tools. Society for Industrial and Applied Mathematics, 1998. doi:10.1137/1.9780898719628

    1998 doi

  15. [23]

    R´ esolution d’edp par un sch´ ema en temps parar´ eel.Comptes Rendus de l’Acad´ emie des Sciences-Series I-Mathematics, 332(7):661–668, 2001

    Jacques-Louis Lions, Yvon Maday, and Gabriel Turinici. R´ esolution d’edp par un sch´ ema en temps parar´ eel.Comptes Rendus de l’Acad´ emie des Sciences-Series I-Mathematics, 332(7):661–668, 2001. doi:10.1016/S0764-4442(00)01793-6

    2001 doi

  16. [24]

    Per-Gunnar Martinsson and Joel A. Tropp. Randomized numerical linear algebra: Foundations and algorithms. Acta Numerica, 29:403–572, 2020. doi:10.1017/ S0962492920000021

    2020

  17. [25]

    R. Milk, S. Rave, and F. Schindler. pyMOR – Generic Algorithms and Interfaces for Model Order Reduction. SIAM Journal of Scientific Computing , 38(5):S194–S216,

  18. [26]

    Reduced basis methods for partial differential equations , volume 92 of Unitext

    Alfio Quarteroni, Andrea Manzoni, and Federico Negri. Reduced basis methods for partial differential equations , volume 92 of Unitext. Springer, Cham, 2016. doi:10.1007/978-3-319-15431-2 . 26

    2016 doi

  19. [27]

    A parareal algorithm with spectral coarse solver (supplement)

    Stephan Rave. A parareal algorithm with spectral coarse solver (supplement). Zenodo, 2025. doi:10.5281/zenodo.16780700

    2025 doi

  20. [28]

    Optimal local approximation spaces for parabolic problems

    Julia Schleuß and Kathrin Smetana. Optimal local approximation spaces for parabolic problems. Multiscale Model. Simul., 20(1):551–582, 2022. doi:10.1137/20M1384294

    2022 doi

  21. [29]

    Randomized quasi-optimal local approximation spaces in time

    Julia Schleuß, Kathrin Smetana, and Lukas Ter Maat. Randomized quasi-optimal local approximation spaces in time. SIAM J. Sci. Comput. , 45(3):A1066–A1096,

  22. [30]

    Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, St´ efan J

    Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, St´ efan J. van der Walt, Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew R. J. Nelson,...

    2020

  23. [31]

    Toward parallel coarse grid correction for the parareal algorithm

    Shu-Lin Wu. Toward parallel coarse grid correction for the parareal algorithm. SIAM J. Sci. Comput. , 40(3):A1446–A1472, 2018. doi:10.1137/17M1141102. 27

    2018 doi

  24. [2007]

    doi:10.1137/05064607X

    doi

  25. [2016]

    doi:10.1137/15M1026614

    doi

  26. [2023]

    doi:10.1137/22M1481002

    doi

Pith tools

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