Travel
Grants
Description
We have secured, in particular through the Jülich
Supercomputing Centre, some travel support for PhD students
working in the field of parallel-in-time integration methods. The
selected students will receive up to EUR 1.500 each for covering parts
of their travel expenses. The Scientific Committee of the PinT
Workshop Series will select students based on applications. These
should include :
- a description of their thesis
topic, stating the connection to parallel-in-time methods (1-2
pages)
- a letter of motivation, containing
a short CV (1-2 pages)
- a short description of the group
and institute they are affiliated to (up to 1 page)
- the abstract of their
presentation
Students traveling to the meeting via this grant are expected to
present their work either on a poster or during a talk. Applications
should be sent to workshop_PINT18@ann.jussieu.fr
Deadline
for submission of applications is March 23, 2018
Previous Grantees
6th
Conference
on Parallel-in-Time Integration, Ascona, Switzerland
Ben S. Southworth,
University of Colorado at Boulder, USA
Talk: “Solving Space-time Discretizations of the Wave Equation with
Algebraic Multigrid”
Algebraic multigrid (AMG) is an iterative solver for large sparse
linear systems that, for many applications, scales linearly in
complexity with the number of degrees of freedom. It also scales in
parallel to hundreds of thousands of cores, making it a key component
of many high-performance simulation codes. For the symmetric positive
definite case, often resulting from the discretization of elliptic
partial differential equations (PDEs), convergence of AMG is well
motivated and AMG is among the fastest numerical solvers available.
However, highly nonsymmetric matrices pose difficulties for AMG in
theory and in practice. Although a number of efforts have been made to
develop AMG for nonsymmetric linear systems, few have considered
highly nonsymmetric cases, and those that have demonstrated success on
nonsymmetric systems typically use a multilevel strategy that does not
scale well in parallel (K-cycles or W-cycles). Each of these issues
are likely part of the reason that AMG has yet to be demonstrated as
an effective parallel-in-time solver. Although a number of works have
looked at geometric multigrid and parabolic PDEs, there is a large gap
in the parallel-intime literature in terms of algebraic solvers and/or
space-time discretizations of hyperbolic PDEs.
Although hyperbolic PDEs arise frequently in
physical simulation, their solution is often constrained to
sequential, explicit time stepping schemes, or implicit schemes with
relatively slow (not O(n)) linear solvers. Recently, we developed a
reduction-based algebraic multigrid (AMG) solver, AMGir, and its
generalization LAIR, to solve upwind discretizations of the steady-
state transport equation. AMGir/LAIR proved to be a fast and robust
solver for the steady-state transport equation, even on unstructured
meshes and with high-order finite elements. Due to the success of
AMGir/LAIR applied to upwind discretizations of gradients, a natural
research direction to consider is further developing the method with a
focus on parallel-in-time applications. Full spacetime discretizations
often use some form of upwind or semi-upwind discretization in time,
which leads to a global space-time matrix amenable to AMGir/LAIR.
Here, we consider a discretization of the 2d-space,
1d-time, wave equation written in first-order form, corresponding to a
system with two variables. A modification of AMGir/LAIR is developed
to account for the system structure, and the method proves very
effective. In particular, the reduction aspect of AMGir/LAIR applied
to the wave equation is striking – initial iterations appear to
diverge, but convergence factors then plummet to ρ ~ 1E−10. AMGir/LAIR
is able to seamlessly handle the system structure and solve the
space-time wave equation, a well-known challenging parallel-in-time
problem, showing promise as a robust framework for parallel-in-time.
Federico Danieli,
Mathematical Institute, University of Oxford, UK
Title: “An Alternative to the Coarse Solver for the Parareal
Algorithm”
The Parareal algorithm is one of the simplest and most widely spread
techniques to achieve parallelisation in the computation of the
solution of ODEs and PDEs by splitting their time domain. However,
ensuring its stability can be a challenging task, which for the
largest part revolves around the choice of the most apt pair of fine
and coarse solvers for the problem at hand. Stability is also an issue
in the case of advection-dominated equations, where the algorithm has
often been shown to perform poorly. In the attempt to overcome these
problems, an alternative formulation of Parareal is presented.
Starting from an interpretation of the algorithm as a Newton method,
we notice how the sensitivity of the solution from the application of
the fine solver, with respect to variations on the initial conditions,
appears in the update formula. Rather than resorting to the
application of a coarse solver in order to approximate this term and
consequently propagate the update along the time domain, we aim to
estimate this sensitivity in a direct manner. The approach chosen is
suitable for systems of ODEs of small size and some simple PDEs, and
extensions to general cases are not trivial and still remain object of
further work. However, the first experiments on model problems show
the potential of this method to overcome some of the limitations of
Parareal, as well as to boost its convergence speed.
Marc Olm,
Technical University of Catalonia & CIMNE, Spain
Title: “Nonlinear parallel-in-time multilevel Schur complement
solvers for ordinary differential equations”
In this work, we propose a parallel-in-time solver for linear and
nonlinear ordinary differential equations. The approach is based on an
efficient multilevel solver of the Schur complement related to a
multilevel time partition. For linear problems, the scheme leads to a
fast direct method. Next, two different strategies for solving
nonlinear ODEs are proposed. First, we consider a Newton method over
the global nonlinear ODE, using the multilevel Schur complement solver
at every nonlinear iteration. Second, we state the global nonlinear
problem in terms of the nonlinear Schur complement (at an arbitrary
level), and perform nonlinear iterations over it. Numerical
experiments show that the proposed schemes are weakly scalable, i.e.,
we can efficiently exploit increasing computational resources to solve
for more time steps the same problem.
5th Workshop on Parallel-in-Time
Integration, Banff, Canada
Thibaut Lunet,
ISAE-Supaero & CERFACS, Toulouse, France
Talk: “On
the
time-parallelization
of
the
solution
of
Navier-Stokes equations using Parareal”
Unsteady turbulent flow simulations using the Navier Stokes
equations require larger and larger problem sizes. On an other side,
new supercomputer architectures will be available in the next
decade, with computational power based on a larger number of cores
rather than significantly increased CPU frequency. Hence most of the
current generation CFD software will face critical efficiency issues
if bounded to massive spatial parallelization and we consider time
parallelization as an attractive alternative to enhance efficiency
on multi-cores architectures. Several algorithms developed in the
last decades (Parareal, PFASST) may be straightforwardly applied to
the Navier-Stokes equations, but the Parareal algorithm remains one
of the simplest solutions in the case of explicit time stepping,
compressible flow Based on an optimized implementation of Parareal
we modelize the speed-up obtained when combining both space and time
parallelizations. This modelization takes into account the speedup
of an actual structured, massively parallel CFD solver and the cost
of time communications, both measured on two different
supercomputers. Some preliminary requirements for a worthy
time-parallel integration will be then derived, in terms of both
Parareal iteration count and size of the time subdomain window. We
then study within this framework, possible enhancements of the
well-known convergence difficulties for Parareal encountered for
advection dominated problems. The proposed approach is based on the
representation of Parareal as an algebraic system of nonlinear
equations solved by a preconditioned Newton’s method. The new
formulation targets the reduction of the degree of non-normality of
its Jacobian by slightly modifying the Parareal iteration.
Performance on examples related to canonical linear problems, like
the Dahlquist and the one-dimensional advection equation, is
analysed. To conclude we comment on the extension of this method to
nonlinear problems.
Stephanie Günther,
TU Kaiserslautern, AG Scientific Computing, Kaiserslautern, Germany
Talk: “Adjoint
Sensitivity
Computation
for
the
Parallel
Multigrid
Reduction in Time Software Library XBraid”
In this paper we present an adjoint solver for the multigrid in
time software library XBraid. XBraid provides a non-intrusive
approach for simulating unsteady dynamics on multiple processors
while parallelizing not only in space but also in the time domain.
It applies an iterative multigrid reduction in time algorithm to
existing spatially parallel classical time propagators and computes
the unsteady solution parallel in time. However, in many engineering
applications not only the primal unsteady flow computation is of
interest but also the ability to compute sensitivities that
determine the influence of design changes to some output quantity.
In recent years, adjoint solvers have widely been developed which
propagate sensitivity information backwards through the time domain.
We develop an adjoint solver for XBraid that enhances the primal
iterations by an iteration for computing adjoint sensitivities. In
each iteration, the adjoint code runs backwards through the primal
XBraid actions and computes the consistent discrete adjoint
sensitivities parallel in time. It is highly non-intrusive as
existing adjoint time propagators can easily be integrated through
the adjoint interface. We validate the adjoint code by applying it
to an unsteady partial differential equation that mimics the
behavior of separated flows past bluff bodies. In our 1D model, the
near wake is mimicked by a nonlinear ODE, namely the Lorenz
attractor which exhibits self-excited oscillations. The far wake is
modeled by an advection - diffusion equation whose upstream boundary
condition is determined by the ODE mimicking the near wake. We
demonstrate the integration of a serial time stepping algorithm,
that solves the PDE forward in time, into the parallel-in-time
XBraid framework as well as the development of the corresponding
adjoint interface. The resulting sensitivities are in good agreement
with those computed from finite differences. Nevertheless, there is
still great potential for optimizing the performance of the adjoint
code using advanced algorithmic differentiation techniques such as
reverse accumulation and checkpointing. Due to the iterative nature
of the primal and the adjoint flow computation, the method is very
well suited for simultaneous optimization algorithms like the
One-shot method which solve the optimization problem in the full
space. They have proven to be very efficient for optimization with
steady-state PDE constraints while its application to unsteady PDE
is still under development. The non-intrusive adjoint XBraid solver
is therefore highly desirable and will extend its application range
from pure simulation to optimizatio
