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IMEX Runge-Kutta Parareal for Non-diffusive Equations

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 356)

Abstract

Parareal is a widely studied parallel-in-time method that can achieve meaningful speedup on certain problems. However, it is well known that the method typically performs poorly on non-diffusive equations. This paper analyzes linear stability and convergence for IMEX Runge-Kutta Parareal methods on non-diffusive equations. By combining standard linear stability analysis with a simple convergence analysis, we find that certain Parareal configurations can achieve parallel speedup on non-diffusive equations. These stable configurations possess low iteration counts, large block sizes, and a large number of processors. Numerical examples using the nonlinear Schrödinger equation demonstrate the analytical conclusions.

Keywords

  • Parareal
  • Parallel-in-time
  • Implicit-explicit
  • High-order
  • Dispersive equations

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Notes

  1. 1.

    The numerical experiments were performed on the Cray XC40 “Cori” at the National Energy Research Scientific Computing Center using four 32-core Intel “Haswell” processor nodes. The Parareal method is implemented as part of the open-source package LibPFASST available at https://github.com/libpfasst/LibPFASST.

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Acknowledgements

The work of Buvoli was funded by the National Science Foundation, Computational Mathematics Program DMS-2012875.

The work of Minion was supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under contract number DE-AC02005CH11231. Parts of the simulations were performed using resources of the National Energy Research Scientific Computing Center (NERSC), a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

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Appendices

Appendix 1: Infinity Norm of the Parareal Iteration Matrix E

Let \(\mathbf {A}(\gamma )\) be the lower bidiagonal matrix

$$\begin{aligned} \mathbf {A}(\gamma ) = \left[ \begin{array}{cccc} 1 &{} &{} &{} \\ \gamma &{} 1 &{} &{} \\ &{} \ddots &{} \ddots &{} \\ &{} &{} \gamma &{} 1 \end{array}. \right] \end{aligned}$$

Lemma 1

The inverse of \(\mathbf {A}(\gamma )\) is given by

$$\begin{aligned} \mathbf {A}^{-1}_{i,j}(\gamma ) = {\left\{ \begin{array}{ll} (-\gamma )^{i-j} &{} j \le i \\ 0 &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

Proof

For convenience, we temporarily drop the \(\gamma \) so that \(\mathbf {A}=\mathbf {A}(\gamma )\), then

$$\begin{aligned} \left( \mathbf {A} \mathbf {A}^{-1} \right) _{ij} = \sum _{k=1}^{{N_p}+1} \mathbf {A}_{ik}\mathbf {A}_{kj}^{-1}&= {\left\{ \begin{array}{ll} 0 &{} j> i\\ \mathbf {A}_{ii}\mathbf {A}_{ii}^{-1} &{} i=j \\ \mathbf {A}_{ii}\mathbf {A}_{ij}^{-1} + A_{i, i-1} A_{i-1,j}^{-1} &{} j<i \\ \end{array}\right. } \\&= {\left\{ \begin{array}{ll} 0 &{} j > i\\ 1 &{} i=j \\ (-\gamma )^{i-j} + \gamma (-\gamma )^{i-1-j} &{} j<i \end{array}\right. } \\&= {\left\{ \begin{array}{ll} 1 &{} i=j, \\ 0 &{} \text {otherwise} . \end{array}\right. } \end{aligned}$$

Lemma 2

The product of \(\mathbf {A}(\omega )\mathbf {A}^{-1}(\gamma )\) is

$$\begin{aligned} \left( \mathbf {A}(\omega ) \mathbf {A}^{-1}(\gamma ) \right) _{ij} = {\left\{ \begin{array}{ll} 0 &{} j > i \\ 1 &{} i = j \\ (-\gamma )^{i-j-1}(\omega - \gamma ) &{} j < i \end{array}\right. } \end{aligned}$$

Proof

$$\begin{aligned} \left( \mathbf {A}(\omega ) \mathbf {A}^{-1}(\gamma ) \right) _{ij}&= \sum _{k=1}^{{N_p}+1} \mathbf {A}_{ik}(\omega )\mathbf {A}_{kj}^{-1}(\gamma ) \\&= {\left\{ \begin{array}{ll} 0 &{} j> i\\ \mathbf {A}_{ii}(\omega ) \mathbf {A}_{ii}^{-1}(\gamma ) &{} i=j \\ \mathbf {A}_{ii}(\omega ) \mathbf {A}_{ij}^{-1} + \mathbf {A}_{i, i-1}(\omega ) \mathbf {A}_{i-1,j}^{-1} (\gamma ) &{} j<i \end{array}\right. } \\&= {\left\{ \begin{array}{ll} 0 &{} j > i\\ 1 &{} i=j \\ (-\gamma )^{i-j} + \omega (-\gamma )^{i-1-j} &{} j<i \end{array}\right. } \end{aligned}$$

Lemma 3

The infinity norm of the matrix \(M(\omega , \gamma ) {=} \mathbf {I} {-} \mathbf {A}(\omega )\mathbf {A}^{-1}(\gamma ) {\in } \mathbb {R}^{{N_p}+1, {N_p}+1}\) is

$$\begin{aligned} \Vert \mathbf {M}(\omega , \gamma ) \Vert _{\infty } = \frac{1 - |\gamma |^{{N_p}}}{1 - |\gamma |} |\gamma - \omega |. \end{aligned}$$

Proof

Using Lemma 2, the jth absolute column sum of \(M(\omega , \gamma )\) is

$$\begin{aligned} c_j = \sum _{k=j+1}^{{N_p}+1} |(-\gamma )^{k-j-1}(\omega - \gamma )| = \sum _{k=1}^{{N_p}- j} |(-\gamma )^{k}||(\omega - \gamma )| \end{aligned}$$

It follows that \(\max _j c_j = c_1\), which can be rewritten as

$$\begin{aligned} \frac{1 - |\gamma |^{{N_p}}}{1 - |\gamma |} |\gamma - \omega |. \end{aligned}$$

Appendix 2: Additional Stability and Convergence Overlay Plots

Figures 8, 9, and 10 show stability and convergence overlay plots for Parareal. The following three figures show stability and convergence overlay plots for Parareal configurations with: \({N_T}= 2048\), IMEX-RK4 as the fine integrator, and three different coarse integrators. These additional figures supplement Fig. 3 and show the effects of changing the course integrator.

Fig. 8
figure 8

Stability and convergence overlay plots for Parareal configurations with a block size of \({N_T}=2048\) and IMEX-RK1, IMEX-RK4 as the coarse and fine integrators

Fig. 9
figure 9

Stability and convergence overlay plots for Parareal configurations with a block size of \({N_T}=2048\) and IMEX-RK2, IMEX-RK4 as the coarse and fine integrators

Fig. 10
figure 10

Stability and convergence overlay plots for Parareal configurations with a block size of \({N_T}=2048\) and IMEX-RK4, IMEX-RK4 as the coarse and fine integrators

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Buvoli, T., Minion, M. (2021). IMEX Runge-Kutta Parareal for Non-diffusive Equations. In: Ong, B., Schroder, J., Shipton, J., Friedhoff, S. (eds) Parallel-in-Time Integration Methods. PinT 2020. Springer Proceedings in Mathematics & Statistics, vol 356. Springer, Cham. https://doi.org/10.1007/978-3-030-75933-9_5

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