A well-conditioned direct PinT algorithm for first-and second-order evolutionary equations

Abstract

In this paper, we propose a direct parallel-in-time (PinT) algorithm for timedependent problems with first-or second-order derivative. We use a second-order boundary value method as the time integrator that leads to a tridiagonal time discretization matrix. Instead of solving the corresponding all-at-once system iteratively, we diagonalize the time discretization matrix, which yields a direct parallel implementation across all time levels. A crucial issue on this methodology is how the condition number of the eigenvector matrix V grows as n is increased, where n is the number of time levels. A large condition number leads to large roundoff error in the diagonalization procedure, which could seriously pollute the numerical accuracy. Based on a novel connection between the characteristic equation and the Chebyshev polynomials, we present explicit formulas for computing V and V −1 , by which we prove that Cond 2 (V ) = O(n 2 ). This implies that the diagonalization process is well-conditioned and the roundoff error only increases moderately as n grows and thus, compared to other direct PinT algorithms, a much larger n can be used to yield satisfactory parallelism. Numerical results on parallel machine are given to support our findings, where over 60 times speedup is achieved with 256 cores.

A Vanka-type multigrid solver for complex-shifted Laplacian systems from diagonalization-based parallel-in-time algorithms

“…It was shown in [3] that | | < ( + √ / √ 2) for all , which implies = (1/ℎ) since ℎ = = 1/ . Figure 1 shows our proposed Vanka smoother is about 3 times faster than the Jacobi smoother, where the observed uniform convergence rates match well with the LFA prediction in Table 1.…”

“…The diagonalization of can be guaranteed by using a boundary-value method time scheme [3] or its circulant-type approximation as preconditioners [4][5][6]. Extensive numerical results [2,7] reveal that the ParaDIAG algorithms have a very promising parallel efficiency for both parabolic [5] and hyperbolic PDEs [6,8].…”

“…In our first example, we test a PinT direct solver [3] for solving 2D heat equation, where the time discretization matrix (with a time step size = 1/ ) is given by…”