Analysis of the parareal algorithm for linear parametric differential equations

References

• G. Bal, On the convergence and the stability of the parareal algorithm to solve partial differential equations, Domain Decomposition Methods in Science and Engineering, Springer, Berlin, 2005, pp. 425–432.
   Google Scholar
• M.A. Bazaz, M. Nabi, and S. Janardhanan, A review of parametric model order reduction techniques, IEEE International Conference on Signal Processing, Computing and Control, Solan, 2012, pp. 1–6.
   Google Scholar
• P. Benner, S. Gugercin, and K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev. 57(4) (2015), pp. 483–531.
   Web of Science ®Google Scholar
• C.E. Brehier and X. Wang, On parareal algorithms for semilinear parabolic stochastic PDEs, SIAM J. Numer. Anal 58(1) (2020), pp. 254–278.
   Web of Science ®Google Scholar
• X.Y. Dai and Y. Maday, Stable parareal in time method for first- and second-order hyperbolic systems, SIAM J. Sci. Comput. 35(35) (2011), pp. A52–A78.
   Google Scholar
• V. Druskin and V. Simoncini, Adaptive rational Krylov subspaces for large-scale dynamical systems, Syst. Control Lett. 60(8) (2011), pp. 546–560.
   Web of Science ®Google Scholar
• D.S. Dzhumabayev, Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation, USSR Comput. Math. Math. Phys. 29(1) (1989), pp. 34–46.
   Web of Science ®Google Scholar
• D.S. Dzhumabaev and S.M. Temesheva, A parametrization method for solving nonlinear two-point boundary value problems, Comput. Math. Math. Phys. 47(1) (2007), pp. 37–61.
   Google Scholar
• P.F. Fischer, F. Hecht, and Y. Maday, A parareal in time semi-implicit approximation of the Navier-Stokes equations, in R. Kornhuber, R.H.W. Hoppe, J. Peeriaux, O. Pironneau, O.B. Widlund, and J. Xu, eds., Proceedings of the 15th international domain decomposition conference, Lecture Notes in Computer Science and Engineering, Springer, Berlin, Vol. 40, 2003, pp. 433–440.
   Google Scholar
• M.J. Gander and E. Hairer, Nonlinear convergence analysis for the parareal algorithm, in Domain Decomposition Methods in Science and Engineering, Vol. XVII, Springer, Berlin, 2008, pp. 45–56.
   Google Scholar
• M.J. Gander and E. Hairer, Analysis for parareal algorithms applied to Hamiltonian differential equations, J. Comput. Appl. Math 259 (2014), pp. 2–13.
   Web of Science ®Google Scholar
• M.J. Gander, Y.L. Jiang, B. Song, and H. Zhang, Analysis of two parareal algorithms for time-periodic problems, SIAM J. Sci. Comput. 35(5) (2013), pp. A2393–A2415.
   Web of Science ®Google Scholar
• M.J. Gander, F. Kwok, and J. Salomon, PARAOPT: A parareal algorithm for optimality systems, SIAM J. Sci. Comput. 42(5) (2020), pp. A2773–A2802.
   Web of Science ®Google Scholar
• M.J. Gander and S. Vandewalle, Analysis of the parareal time-parallel time-integration method, SIAM J. Sci. Comput. 29(2) (2007), pp. 556–578.
   Web of Science ®Google Scholar
• M.J. Gander and S.L. Wu, A diagonalization-based parareal algorithm for dissipative and wave propagation problems, SIAM J. Numer. Anal. 58(5) (2020), pp. 2981–3009.
   Web of Science ®Google Scholar
• Y.L. Jiang, Periodic waveform relaxation of nonlinear dynamic equations, Appl. Math. Comput.135(2-3) (2003), pp. 219–226.
   Web of Science ®Google Scholar
• Y.L. Jiang, R.M.M. Chen, and O. Wing, Periodic waveform relaxation of nonlinear dynamic systems by quasi-linearization, IEEE Trans. Circuits Syst. I. 50(4) (2003), pp. 589–593.
   Google Scholar
• Y.L. Jiang and O. Wing, A note on the spectra and pseudospectra of waveform relaxation operators for linear differential-algebraic equations, SIAM J. Numer. Anal. 38(1) (2000), pp. 186–201.
   Web of Science ®Google Scholar
• Y.L. Jiang and O. Wing, A note on convergence conditions of waveform relaxation algorithms for nonlinear differential-algebraic equations, Appl. Numer. Math. 36(2) (2001), pp. 281–297.
   Web of Science ®Google Scholar
• E. Lelarasmee, A.E. Ruehli, and A.L. Sangiovanni-Vincentenlli, The waveform relaxation method for time-domain analysis of large integrated circuits, IEEE Trans. CAD IC Syst. 1(3) (1982), pp. 131–145.
   Google Scholar
• J.L. Lions, Y. Maday, and G. Turinici, A parareal in time discretization of PDE's, C. R. Acad. Sci. Paris Sér. I Math. 332(7) (2001), pp. 661–668.
   Google Scholar
• J. Liu and Y.L. Jiang, A parareal waveform relaxation algorithm for semi-linear parabolic partial differential equations, J. Comput. Appl. Math. 236 (2012), pp. 4245–4263.
   Web of Science ®Google Scholar
• J. Liu and Y.L. Jiang, A parareal algorithm based on waveform relaxation, Math. Comput. Simulat.82(11) (2012), pp. 2167–2181.
   Web of Science ®Google Scholar
• Z. Miao, Y.L. Jiang, and Y.B. Yang, Convergence analysis of a parareal-in-time algorithm for the incompressible non-isothermal flows, Int. J. Comput. Math. 96(7) (2019), pp. 1398–1415.
   Web of Science ®Google Scholar
• B. Minglibayeva and A.T. Assanova, An existence of an isolated solution to nonlinear two-point boundary value problem with parameter, Lobachevskii J. Math. 42(3) (2021), pp. 587–597.
   Web of Science ®Google Scholar
• B Song and Y.L. Jiang, Analysis of a new parareal algorithm based on waveform relaxation method for time-periodic problems, Numer. Algor. 67(3) (2014), pp. 599–622.
   Web of Science ®Google Scholar
• B Song and Y.L. Jiang, A new parareal waveform relaxation algorithm for time-periodic problems, Int. J. Comput. Math. 92(2) (2015), pp. 377–393.
   Web of Science ®Google Scholar
• S. Vandewalle and R. Piessens, On dynamic iteration methods for solving time-periodic differential equations, SIAM J. Numer. Anal. 30(1) (1993), pp. 286–303.
   Web of Science ®Google Scholar
• S.L. Wu, Three rapidly convergent parareal solvers with application to time-dependent PDEs with fractional Laplacian, Math. Method. Appl. Sci. 40(11) (2017), pp. 3912–3926.
   Web of Science ®Google Scholar
• L. Wu, P. Shi, H. Gao, and J. Wang, H∞ model reduction for linear parameter-varying systems with distributed delay, Int. J. Control. 82(3) (2009), pp. 408–422.
   Web of Science ®Google Scholar
• S.L. Wu and T. Zhou, Convergence analysis for three parareal solvers, SIAM J. Sci. Comput. 37(2) (2015), pp. A970–A992.
   Web of Science ®Google Scholar
• S.L. Wu and T. Zhou, Fast parareal iterations for fractional diffusion equations, J. Comput. Phys. 329 (2017), pp. 210–226.
   Web of Science ®Google Scholar
• S.L. Wu and T. Zhou, Parareal algorithms with local time-integrators for time fractional differential equations, J. Comput. Phys. 358 (2017), pp. 135–149.
   Web of Science ®Google Scholar