This work proposes a data-driven method for enabling the efficient, stable time-parallel numerical solution of systems of ordinary differential equations (ODEs). The method assumes that low-dimensional bases that accurately capture the time evolution of the dynamical-system state are available. The method adopts the parareal framework for time parallelism, which is defined by an initialization method, a coarse propagator that advances solutions on a coarse time grid, and a fine propagator that operates on an underlying fine time grid. Rather than employing usual approaches for initialization and coarse propagation, we propose novel data-driven techniques that leverage the available time-evolution bases. The coarse propagator is defined by a forecast (proposed in [K. Carlberg, J. Ray, and B. van Bloemen Waanders, Comput. Methods Appl. Mech. Engrg., 289 (2015), pp. 79--103]) applied locally within each coarse time interval, which comprises the following steps: (1) apply the fine propagator for a small number of time steps, (2) approximate the state over the entire coarse time interval using gappy proper orthogonal decomposition (POD) with the local time-evolution bases, and (3) select the approximation at the end of the time interval as the propagated state. We also propose both local-forecast and global-forecast initialization techniques. The method is particularly well suited for POD-based reduced-order models (ROMs), as the time-evolution bases can be extracted from readily available data, i.e., the right singular vectors arising during POD computation. In addition to performing analyses related to the method's accuracy, speedup, stability, and convergence, we also numerically demonstrate the method's performance. Here, numerical experiments on ROMs for a nonlinear convection--reaction problem demonstrate the method's ability to realize near-ideal speedups.


  1. time-parallel
  2. parareal
  3. forecasting
  4. gappy proper orthogonal decomposition
  5. data-driven approximation
  6. model reduction

MSC codes

  1. 65B99
  2. 65D30
  3. 65L05A
  4. 65L06
  5. 65L20
  6. 65M12
  7. 65M55
  8. 65Y05

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Supplementary Material

PLEASE NOTE: These supplementary files have not been peer-reviewed.

Index of Supplementary Materials

Title of paper: Data-driven time parallelism via forecasting

Authors: K. Carlberg, L. Brencher, B. Haasdonk, A. Barth

File: M117436SupMat.pdf

Type: PDF

Contents: Proofs of the analytical results presented in the main manuscript (Section S1), a numerical investigation of the dependence of the quantities appearing in Lemma 4.9 on the time discretization (Section S2), theoretical performance improvements when forecasting is also employed for defining the initial guess in the Newton solver (Section S3), supporting convergence-analysis results (Section S4), an illustration in the particular case of of parameterized linear ODEs in which the proposed method yields an ideal predictive coarse propagator (Section S5), and figures that provide additional interpretation of the numerical experiments (Section S6). Additionally, Section S7 provides a list of notation of the main article file.


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Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: B466 - B496
ISSN (online): 1095-7197


Submitted: 13 March 2018
Accepted: 22 March 2019
Published online: 23 May 2019


  1. time-parallel
  2. parareal
  3. forecasting
  4. gappy proper orthogonal decomposition
  5. data-driven approximation
  6. model reduction

MSC codes

  1. 65B99
  2. 65D30
  3. 65L05A
  4. 65L06
  5. 65L20
  6. 65M12
  7. 65M55
  8. 65Y05



Funding Information

Sandia National Laboratories https://doi.org/10.13039/100006234 : DE-NA0003525

Funding Information

Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : EXC 310/2

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