25 research outputs found
The Parallel Full Approximation Scheme in Space and Time for a Parabolic Finite Element Problem
The parallel full approximation scheme in space and time (PFASST) is a
parallel-in-time integrator that allows to integrate multiple time-steps
simultaneously. It has been shown to extend scaling limits of spatial
parallelization strategies when coupled with finite differences, spectral
discretizations, or particle methods. In this paper we show how to use PFASST
together with a finite element discretization in space. While seemingly
straightforward, the appearance of the mass matrix and the need to restrict
iterates as well as residuals in space makes this task slightly more intricate.
We derive the PFASST algorithm with mass matrices and appropriate prolongation
and restriction operators and show numerically that PFASST can, after some
initial iterations, gain two orders of accuracy per iteration
On the convergence of spectral deferred correction methods
In this work we analyze the convergence properties of the Spectral Deferred
Correction (SDC) method originally proposed by Dutt et al. [BIT, 40 (2000), pp.
241--266]. The framework for this high-order ordinary differential equation
(ODE) solver is typically described wherein a low-order approximation (such as
forward or backward Euler) is lifted to higher order accuracy by applying the
same low-order method to an error equation and then adding in the resulting
defect to correct the solution. Our focus is not on solving the error equation
to increase the order of accuracy, but on rewriting the solver as an iterative
Picard integral equation solver. In doing so, our chief finding is that it is
not the low-order solver that picks up the order of accuracy with each
correction, but it is the underlying quadrature rule of the right hand side
function that is solely responsible for picking up additional orders of
accuracy. Our proofs point to a total of three sources of errors that SDC
methods carry: the error at the current time point, the error from the previous
iterate, and the numerical integration error that comes from the total number
of quadrature nodes used for integration. The second of these two sources of
errors is what separates SDC methods from Picard integral equation methods; our
findings indicate that as long as difference between the current and previous
iterate always gets multiplied by at least a constant multiple of the time step
size, then high-order accuracy can be found even if the underlying "solver" is
inconsistent the underlying ODE. From this vantage, we solidify the prospects
of extending spectral deferred correction methods to a larger class of solvers
to which we present some examples.Comment: 29 page
A multigrid perspective on the parallel full approximation scheme in space and time
For the numerical solution of time-dependent partial differential equations,
time-parallel methods have recently shown to provide a promising way to extend
prevailing strong-scaling limits of numerical codes. One of the most complex
methods in this field is the "Parallel Full Approximation Scheme in Space and
Time" (PFASST). PFASST already shows promising results for many use cases and
many more is work in progress. However, a solid and reliable mathematical
foundation is still missing. We show that under certain assumptions the PFASST
algorithm can be conveniently and rigorously described as a multigrid-in-time
method. Following this equivalence, first steps towards a comprehensive
analysis of PFASST using block-wise local Fourier analysis are taken. The
theoretical results are applied to examples of diffusive and advective type
PFASST-ER: Combining the Parallel Full Approximation Scheme in Space and Time with parallelization across the method
To extend prevailing scaling limits when solving time-dependent partial
differential equations, the parallel full approximation scheme in space and
time (PFASST) has been shown to be a promising parallel-in-time integrator.
Similar to a space-time multigrid, PFASST is able to compute multiple
time-steps simultaneously and is therefore in particular suitable for
large-scale applications on high performance computing systems. In this work we
couple PFASST with a parallel spectral deferred correction (SDC) method,
forming an unprecedented doubly time-parallel integrator. While PFASST provides
global, large-scale "parallelization across the step", the inner parallel SDC
method allows to integrate each individual time-step "parallel across the
method" using a diagonalized local Quasi-Newton solver. This new method, which
we call "PFASST with Enhanced concuRrency" (PFASST-ER), therefore exposes even
more temporal parallelism. For two challenging nonlinear reaction-diffusion
problems, we show that PFASST-ER works more efficiently than the classical
variants of PFASST and can be used to run parallel-in-time beyond the number of
time-steps.Comment: 12 pages, 12 figures, CVS PinT Workshop Proceeding
High order eigenvalues for the Helmholtz equation in complicated non-tensor domains through Richardson extrapolation of second order finite differences
We apply second order finite differences to calculate the lowest eigenvalues of the Helmholtz equation, for complicated non-tensor domains in the plane, using different grids which sample exactly the border of the domain. We show that the results obtained applying Richardson and Padé-Richardson extrapolations to a set of finite difference eigenvalues corresponding to different grids allow us to obtain extremely precise values. When possible we have assessed the precision of our extrapolations comparing them with the highly precise results obtained using the method of particular solutions. Our empirical findings suggest an asymptotic nature of the FD series. In all the cases studied, we are able to report numerical results which are more precise than those available in the literature.Instituto de Investigaciones FisicoquÃmicas Teóricas y Aplicada
High order eigenvalues for the Helmholtz equation in complicated non-tensor domains through Richardson extrapolation of second order finite differences
We apply second order finite differences to calculate the lowest eigenvalues of the Helmholtz equation, for complicated non-tensor domains in the plane, using different grids which sample exactly the border of the domain. We show that the results obtained applying Richardson and Padé-Richardson extrapolations to a set of finite difference eigenvalues corresponding to different grids allow us to obtain extremely precise values. When possible we have assessed the precision of our extrapolations comparing them with the highly precise results obtained using the method of particular solutions. Our empirical findings suggest an asymptotic nature of the FD series. In all the cases studied, we are able to report numerical results which are more precise than those available in the literature.Instituto de Investigaciones FisicoquÃmicas Teóricas y Aplicada