12 research outputs found
Multilevel convergence analysis of multigrid-reduction-in-time
This paper presents a multilevel convergence framework for
multigrid-reduction-in-time (MGRIT) as a generalization of previous two-grid
estimates. The framework provides a priori upper bounds on the convergence of
MGRIT V- and F-cycles, with different relaxation schemes, by deriving the
respective residual and error propagation operators. The residual and error
operators are functions of the time stepping operator, analyzed directly and
bounded in norm, both numerically and analytically. We present various upper
bounds of different computational cost and varying sharpness. These upper
bounds are complemented by proposing analytic formulae for the approximate
convergence factor of V-cycle algorithms that take the number of fine grid time
points, the temporal coarsening factors, and the eigenvalues of the time
stepping operator as parameters.
The paper concludes with supporting numerical investigations of parabolic
(anisotropic diffusion) and hyperbolic (wave equation) model problems. We
assess the sharpness of the bounds and the quality of the approximate
convergence factors. Observations from these numerical investigations
demonstrate the value of the proposed multilevel convergence framework for
estimating MGRIT convergence a priori and for the design of a convergent
algorithm. We further highlight that observations in the literature are
captured by the theory, including that two-level Parareal and multilevel MGRIT
with F-relaxation do not yield scalable algorithms and the benefit of a
stronger relaxation scheme. An important observation is that with increasing
numbers of levels MGRIT convergence deteriorates for the hyperbolic model
problem, while constant convergence factors can be achieved for the diffusion
equation. The theory also indicates that L-stable Runge-Kutta schemes are more
amendable to multilevel parallel-in-time integration with MGRIT than A-stable
Runge-Kutta schemes.Comment: 26 pages; 17 pages Supplementary Material
Numerical studies of a multigrid version of the parareal algorithm
In this work, a parallel-in-time method is combined with a multigrid algorithm
and further on with a spatial coarsening strategy. The most famous parallel-in-time
method is the parareal algorithm. Depending on two different operators, it enables the
parallelism of time-dependent problems. The operator with huge effort is carried out
in parallel. But despite parallelization this can lead to long run times for long-term
problems. Since the parareal algorithm has a two-level structure and the time-parallel
multigrid methods are also widespread in the area of parallel time integration, we
combine these approaches. We use the parareal algorithm as a smoothing operator in
the basic framework of a geometrical multigrid method, where we apply a coarsening
strategy in time. So we get a multigrid in time method which is strongly parallelizable.
For partial differential equations we add an extra spatial coarsening strategy to our
multigrid parareal version. All in all we get a method, which has a high parallel
efficiency and converges fast due to the multigrid framework, which is shown in the
numerical studies of this work. So we will get a highly accurate solution and can greatly
reduce the parallel complexity, which is especially important for long-term problems
with a limited number of processors
Parallel-in-time integration of the shallow water equations on the rotating sphere using Parareal and MGRIT
Despite the growing interest in parallel-in-time methods as an approach to
accelerate numerical simulations in atmospheric modelling, improving their
stability and convergence remains a substantial challenge for their application
to operational models. In this work, we study the temporal parallelization of
the shallow water equations on the rotating sphere combined with time-stepping
schemes commonly used in atmospheric modelling due to their stability
properties, namely an Eulerian implicit-explicit (IMEX) method and a
semi-Lagrangian semi-implicit method (SL-SI-SETTLS). The main goal is to
investigate the performance of parallel-in-time methods, namely Parareal and
Multigrid Reduction in Time (MGRIT), when these well-established schemes are
used on the coarse discretization levels and provide insights on how they can
be improved for better performance. We begin by performing an analytical
stability study of Parareal and MGRIT applied to a linearized ordinary
differential equation depending on the choice of a coarse scheme. Next, we
perform numerical simulations of two standard tests to evaluate the stability,
convergence and speedup provided by the parallel-in-time methods compared to a
fine reference solution computed serially. We also conduct a detailed
investigation on the influence of artificial viscosity and hyperviscosity
approaches, applied on the coarse discretization levels, on the performance of
the temporal parallelization. Both the analytical stability study and the
numerical simulations indicate a poorer stability behaviour when SL-SI-SETTLS
is used on the coarse levels, compared to the IMEX scheme. With the IMEX
scheme, a better trade-off between convergence, stability and speedup compared
to serial simulations can be obtained under proper parameters and artificial
viscosity choices, opening the perspective of the potential competitiveness for
realistic models.Comment: 35 pages, 23 figure
Multigrid reduction-in-time convergence for advection problems: A Fourier analysis perspective
A long-standing issue in the parallel-in-time community is the poor
convergence of standard iterative parallel-in-time methods for hyperbolic
partial differential equations (PDEs), and for advection-dominated PDEs more
broadly. Here, a local Fourier analysis (LFA) convergence theory is derived for
the two-level variant of the iterative parallel-in-time method of multigrid
reduction-in-time (MGRIT). This closed-form theory allows for new insights into
the poor convergence of MGRIT for advection-dominated PDEs when using the
standard approach of rediscretizing the fine-grid problem on the coarse grid.
Specifically, we show that this poor convergence arises, at least in part, from
inadequate coarse-grid correction of certain smooth Fourier modes known as
characteristic components, which was previously identified as causing poor
convergence of classical spatial multigrid on steady-state advection-dominated
PDEs. We apply this convergence theory to show that, for certain
semi-Lagrangian discretizations of advection problems, MGRIT convergence using
rediscretized coarse-grid operators cannot be robust with respect to CFL number
or coarsening factor. A consequence of this analysis is that techniques
developed for improving convergence in the spatial multigrid context can be
re-purposed in the MGRIT context to develop more robust parallel-in-time
solvers. This strategy has been used in recent work to great effect; here, we
provide further theoretical evidence supporting the effectiveness of this
approach
Recommended from our members
Space-time Methods for Time-dependent Partial Differential Equations
Modern discretizations of time-dependent PDEs consider the full problem in the space-time cylinder and aim to overcome limitations of classical approaches such as the method of lines (first discretize in space and then solve the resulting ODE) and the Rothe method (first discretize in time and then solve the PDE). A main advantage of a holistic space-time method is the direct access to space-time adaptivity and to the backward problem (required for the dual problem in optimization or error control). Moreover, this allows for parallel solution strategies simultaneously in time and space.
Several space-time concepts where proposed (different conforming and nonconforming space-time finite elements, the parareal method, wavefront relaxation etc.) but this topic has become a rapidly growing field in numerical analysis and scientific computing. In this workshop the focus is the development of adaptive and flexible space-time discretization methods for solving parabolic and hyperbolic space-time partial differential equations