Parallel-in-time Multilevel Krylov Methods: A Prototype

TL;DR

A parallel-in-time multilevel iterative method for solving differential algebraic equation, arising from a discretization of linear time-dependent partial differential equation, suggesting the potential computational speed-up of up to 9 relative to the single-processor implementation and the speed-up of about 3 compared to the sequential $\theta$-scheme.

Abstract

This paper presents a parallel-in-time multilevel iterative method for solving differential algebraic equation, arising from a discretization of linear time-dependent partial differential equation. The core of the method is the multilevel Krylov method, introduced by Erlangga and Nabben~{\it [SIAM J. Sci. Comput., 30(2008), pp. 1572--1595]}. In the method, special time restriction and interpolation operators are proposed to coarsen the time grid and to map functions between fine and coarse time grids. The resulting Galerkin coarse-grid system can be interpreted as time integration of an equivalent differential algebraic equation associated with a larger time step and a modified $θ$-scheme. A perturbed coarse time-grid matrix is used on the coarsest level to decouple the coarsest-level system, allowing full parallelization of the method. Within this framework, spatial coarsening can be included in a natural way, reducing further the size of the coarsest grid problem to solve. Numerical results are presented for the 1- and 2-dimensional heat equation using {\it simulated} parallel implementation, suggesting the potential computational speed-up of up to 9 relative to the single-processor implementation and the speed-up of about 3 compared to the sequential $θ$-scheme.

Figures (13)

• Figure 1: Time coarsening, with $\Delta T = \nu \Delta t$, with $\nu = 3$.
• Figure 1: Performance comparison of multilevel Krylov with time coarsening and various numbers of processors used to solve the coarse-grid system in parallel, with $n_x = n_y = 128$, $n_t = 768$. Other than the coarse grid solve, 32 processors are used to perform all related matrix/vector operations.
• Figure 2: Agglomeration of 12 fine spatial gridpoints in $\Omega_j \subset \Omega \subset \mathbb{R}^2$ into 1 coarse spatial gridpoint.
• Figure 2: An example of time-space coarsening, resulting in 6 levels.
• Figure 3: 1D agglomeration using standard spatial coarsening. Left: the construction of $Z$. Right: the construction of $Y$.