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A parallel-in-time method based on the Parareal algorithm and High-Order Dynamic Mode Decomposition with applications to fluid simulations

Weifan Liu

TL;DR

This work introduces Parareal-HODMD, a parallel-in-time method that replaces the costly coarse propagator for stiff problems with a data-driven coarse step guided by High-Order Dynamic Mode Decomposition. By running two coarse solvers in parallel and using HODMD to predict corrections from their difference, the method delivers low-cost, accurate coarse updates within the Parareal iterations. Through fluid-structure interaction examples and lubrication-type thin-film flows, the approach achieves significant reductions in serial work and improved speedups over classic Parareal, particularly for long-time simulations. The results suggest a flexible, scalable pathway to accelerate stiff-time integrations on modern heterogeneous architectures, with future work focusing on scalable SVD techniques and expanded parallelism in the DMD computations.

Abstract

The high cost of sequential time integration is one major constraint that limits the speedup of a time-parallel algorithm like the Parareal algorithm due to the difficulty of coarsening time steps in a stiff numerical problem. To address this challenge, we develop a parallel-in-time approach based on the Parareal algorithm, in which we construct a novel coarse solver using a data-driven method based on Dynamic Mode Decomposition in place of a classic time marching scheme. The proposed solver computes an approximation of the solution using two numerical schemes of different accuracies in parallel, and apply High-Order Dynamic Mode Decomposition (HODMD) to reduce the cost of sequential computations. Compared to the original Parareal algorithm, the proposed approach allows for the construction of low-cost coarse solvers for many complicated stiff problems. We demonstrate through several numerical examples in fluid dynamics that the proposed method can effectively reduce the serial computation cost and improve the parallel speedup of long-time simulations which are hard to accelerate using the original Parareal algorithm.

A parallel-in-time method based on the Parareal algorithm and High-Order Dynamic Mode Decomposition with applications to fluid simulations

TL;DR

This work introduces Parareal-HODMD, a parallel-in-time method that replaces the costly coarse propagator for stiff problems with a data-driven coarse step guided by High-Order Dynamic Mode Decomposition. By running two coarse solvers in parallel and using HODMD to predict corrections from their difference, the method delivers low-cost, accurate coarse updates within the Parareal iterations. Through fluid-structure interaction examples and lubrication-type thin-film flows, the approach achieves significant reductions in serial work and improved speedups over classic Parareal, particularly for long-time simulations. The results suggest a flexible, scalable pathway to accelerate stiff-time integrations on modern heterogeneous architectures, with future work focusing on scalable SVD techniques and expanded parallelism in the DMD computations.

Abstract

The high cost of sequential time integration is one major constraint that limits the speedup of a time-parallel algorithm like the Parareal algorithm due to the difficulty of coarsening time steps in a stiff numerical problem. To address this challenge, we develop a parallel-in-time approach based on the Parareal algorithm, in which we construct a novel coarse solver using a data-driven method based on Dynamic Mode Decomposition in place of a classic time marching scheme. The proposed solver computes an approximation of the solution using two numerical schemes of different accuracies in parallel, and apply High-Order Dynamic Mode Decomposition (HODMD) to reduce the cost of sequential computations. Compared to the original Parareal algorithm, the proposed approach allows for the construction of low-cost coarse solvers for many complicated stiff problems. We demonstrate through several numerical examples in fluid dynamics that the proposed method can effectively reduce the serial computation cost and improve the parallel speedup of long-time simulations which are hard to accelerate using the original Parareal algorithm.

Paper Structure

This paper contains 14 sections, 25 equations, 5 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Parareal Algorithm
  3. Dynamic Mode Decomposition
  4. Novel Parareal algorithm based on High-Order Dynamic Mode Decomposition
  5. Initial serial sweep
  6. Serial correction sweep
  7. Algorithm
  8. Numerical examples
  9. Microswimmers in Stokes flow
  10. Single rod-like swimmer
  11. Elastic sphere in the shear flow
  12. Thin liquid film on a two-dimensional substrate
  13. Scaling of the algorithm
  14. Conclusion

Figures (5)

  • Figure 1: Configuration of the rod swimmer and the velocity along the rod at $t=1$
  • Figure 2: (a) Decay of true relative error $\eta^k$ as the number of iteration increases (b) Decay of relative increment $\tilde{\eta}^k$ as the number of iteration increases
  • Figure 3: (a) Three colored tracers marked on the surface of the sphere (b) The trajectory of the three tracers for $t\in[0, 100]$ (c) True relative error $\eta^k$ of the Parareal-HODMD and the original Parareal algorithm
  • Figure 4: (a) Thin film at $t=2$ (b) Plot of $A(x,y)$ in the x-y plane with $A_2>A_1$.
  • Figure 5: Strong scaling of Parareal-HODMD and the original Parareal algorithm