PARAEXP: a parallel integrator for linear initial-value problems. A novel parallel algorithm for the integration of linear initial-value problems is proposed. This algorithm is based on the simple observation that homogeneous problems can typically be integrated much faster than inhomogeneous problems. An overlapping time-domain decomposition is utilized to obtain decoupled inhomogeneous and homogeneous subproblems, and a near-optimal Krylov method is used for the fast exponential integration of the homogeneous subproblems. We present an error analysis and discuss the parallel scaling of our algorithm. The efficiency of this approach is demonstrated with numerical examples.

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  1. Hahne, Jens; Friedhoff, Stephanie; Bolten, Matthias: Algorithm 1016. PyMGRIT: a python package for the parallel-in-time method MGRIT (2021)
  2. Botchev, M. A.: A nested Schur complement solver with mesh-independent convergence for the time domain photonics modeling (2020)
  3. Gander, Martin J.; Kwok, Felix; Salomon, Julien: PARAOPT: a parareal algorithm for optimality systems (2020)
  4. Gander, Martin J.; Wu, Shu-Lin: A diagonalization-based parareal algorithm for dissipative and wave propagation problems (2020)
  5. Masetti, G.; Robol, L.: Computing performability measures in Markov chains by means of matrix functions (2020)
  6. Gander, Martin J.; Halpern, Laurence; Rannou, Johann; Ryan, Juliette: A direct time parallel solver by diagonalization for the wave equation (2019)
  7. Gander, Martin J.; Jiang, Yao-Lin; Song, Bo: A superlinear convergence estimate for the parareal Schwarz waveform relaxation algorithm (2019)
  8. Gander, Martin J.; Wu, Shu-Lin: Convergence analysis of a \textitperiodic-like waveform relaxation method for initial-value problems via the diagonalization technique (2019)
  9. Götschel, Sebastian; Minion, Michael L.: An efficient parallel-in-time method for optimization with parabolic PDEs (2019)
  10. Kwok, Felix; Ong, Benjamin W.: Schwarz waveform relaxation with adaptive pipelining (2019)
  11. Neumüller, Martin; Smears, Iain: Time-parallel iterative solvers for parabolic evolution equations (2019)
  12. Speck, Robert: Algorithm 997: pySDC -- prototyping spectral deferred corrections (2019)
  13. Wu, Shu-Lin; Zhou, Tao: Acceleration of the two-level MGRIT algorithm via the diagonalization technique (2019)
  14. Badia, Santiago; Olm, Marc: Nonlinear parallel-in-time Schur complement solvers for ordinary differential equations (2018)
  15. Botchev, M. A.; Hanse, A. M.; Uppu, R.: Exponential Krylov time integration for modeling multi-frequency optical response with monochromatic sources (2018)
  16. Gaudreault, Stéphane; Rainwater, Greg; Tokman, Mayya: KIOPS: a fast adaptive Krylov subspace solver for exponential integrators (2018)
  17. Kooij, Gijs L.; Botchev, Mike A.; Geurts, Bernard J.: An exponential time integrator for the incompressible Navier-Stokes equation (2018)
  18. Wu, Shu-Lin: Toward parallel coarse grid correction for the parareal algorithm (2018)
  19. Zhu, Shuai; Weng, Shilie: A parallel spectral deferred correction method for first-order evolution problems (2018)
  20. Gander, Martin J.; Halpern, Laurence: Time parallelization for nonlinear problems based on diagonalization (2017)

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