A unified analysis framework for iterative parallel-in-time algorithms

Abstract

Parallel-in-time integration has been the focus of intensive research efforts over the past two decades due to the advent of massively parallel computer architectures and the scaling limits of purely spatial parallelization. Various iterative parallel-in-time (PinT) algorithms have been proposed, like Parareal, PFASST, MGRIT, and Space-Time Multi-Grid (STMG). These methods have been described using different notations and the convergence estimates that are available for some of them are difficult to compare. We describe Parareal, PFASST, MGRIT and STMG for the Dahlquist model problem using a common notation and give precise convergence estimates using generating functions. This allows us, for the first time, to directly compare their convergence. We prove that all four methods eventually converge super-linearly and compare them directly numerically. Our framework also allows us to find new methods.

Low-rank Parareal: a low-rank parallel-in-time integrator

“…The bounds above depend on α = e h C r ,q and β = e h C q , where C q and C r ,q are the Lipschitz constants of T q and T r ,q respectively; see (16). While it seems difficult to give a priori results on the size C q and C r ,q , we can bound them up to first order in the theorem below.…”

“…This idea of solving in parallel an evolution problem as a nonlinear (discretized) system also appears in related methods like PFASST [8], MGRIT [11] and Space-Time Multi-Grid [17]. Theoretical results and numerical studies on a large numbers of cores show that these parallel-in-time methods can have good parallel performance for parabolic problems; see, e.g., [16,21,37]. So far, these methods did not incorporate a low-rank compression of the space dimension, which is the main topic of this work.…”

Improved ParaDiag via low-rank updates and interpolation

“…The poles of the rational function (19) are not nested, that is, the poles of p r j pzq are different from the ones of p r j`1 pzq. This is a disadvantage when using the poles of p r j pzq in an iterative method (such as ADI or RKSM) where usually the number of steps (the parameter j) is not known a priori.…”

“…We assume that RKSM converges after iterations to fixed accuracy and that the shift parameters are all different, as it happens, for instance, in EDS. If A is symmetric positive definite and its condition number grows polynomially with n (as, e.g., for finite difference or finite element discretizations of elliptic operators) then the bound (19) predicts " Oplog nq. RKSM needs to solve Op q linear systems at cost C sys each for generating the left factor W of the approximate solution δX « W Y Z ˚.…”

“…The Parareal algorithm [20,32] avoids this limitation by combining, iteratively, coarse (cheap) time steps that are performed sequentially with fine (expensive) time steps that are performed in parallel. There have been many developments and modifications of Parareal during the last two decades; we refer to [13,19] for overviews and a historical perspective. Our work relates to the so called ParaDiag algorithms [18], which proceed somewhat differently from Parareal by explicitly exploiting the structure of the matrix equation (1).…”