Slides Schedule & Book of Abstracts Sponsors Participants Simulations
wednesday - thursday - friday - saturday
Saturday
- 09:00-10:00 : Giovanni Samaey
Micro-macro parareal methods: low-dimensional, effective coarse propagators and matching of high-dimensional fine states. — AbstractGiovanni Samaey¹
¹ NUMA, Department of Computer Science, KU Leuven (Belgium)
Many systems of current interest exhibit behavior on a wide range of time scales, which cannot be simulated directly on long (macroscopic) time intervals. In such a setting, efficient time-parallel methods can be constructed that iteratively improve an approximate coarse-grained model by time-parallel fine-scale corrections. We present and analyse a micro-macro parareal method that couples the microscopic model, of which we have full knowledge, to a macroscopic level, of which we assume only limited information. The macroscopic model is described by a finite set of macroscopic state variables — averages over the microscopic distribution. A crucial part of such an algorithm is the appropriate inference of a fine-scale state that is consistent with an imposed macroscopic state. We call this step matching.
In this talk, we will present and discuss micro-macro parareal methods based on the above idea for a set of model problems of increasing complexity: a singularly perturbed ODE, a one-dimensional climate model, and a stochastic model representing molecular dynamics. In each case, we discuss the nature of the matching operator and its convergence properties.
This presentation is based on joint work with Frédéric Legoll, Tony Lelièvre, Keith Myerscough, Thomas Slawig, and Przemyslaw Zielinski.
References
[1] Frédéric Legoll, Tony Lelievre, and Giovanni Samaey. A micro-macro parareal algorithm: application to singularly perturbed ordinary differential equations. SIAM Journal on Scientific Computing, 35(4):A1951–A1986, 2013.
[2] Kristian Debrabant, Giovanni Samaey, and Przemysław Zieli?ski. A micro-macro acceleration method for the monte carlo simulation of stochastic differential equations. SIAM Journal on Numerical Analysis, 55(6):2745–2786, 2017.
[3] Tony Lelièvre, Giovanni Samaey, and Przemysław Zieliński. Analysis of a micro-macro acceleration method with minimum relative entropy moment matching. arXiv preprint arXiv:1801.01740, 2018.
- 10:30-11:00 : Florent Hédin
A new implementation of the Generalized Parallel Replica dynamics for the long time simulation of metastable biochemical systems. — AbstractFlorent Hédin¹, Tony Lelièvre¹
¹ CERMICS, École des Ponts–ParisTech, 6 et 8 avenue Blaise Pascal, Cité Descartes – Champs sur Marne, F-77455 Marne-la-Vallée Cedex 2, France
Molecular dynamics (MD) simulations are nowadays of a common use for simulating large and complex biological or chemical systems. However, a commonly encountered obstacle is the timescale separation between the fastest conformational changes — occurring at the femtoseconds (fs) level — and the slowest one, occurring from the nanosecond (ns) to second (or more) timescale. The existence of metastable regions in the phase space is the main origin of this separation, and the simulation time required for observing state-to-state transition quickly becomes intractable.
Among the methods developed to address this challenge, the “Parallel Replica”(ParRep)[1] method maps the original dynamics to a state-to-state dynamics, relying on quasi-stationary distributions (QSDs).[2] The later developed “Generalized Parallel Replica”[3] method uses advanced techniques in order to estimate on the fly if convergence to the QSD is obtained, making the method easier to use when investigating biochemical systems.
We will present a new implementation of the Generalized parallel replica method,[4] targeting frequently encountered metastable biochemical systems, such as conformational equilibria or dissociation of protein–ligand complexes. It will be shown that the method is mathematically accurate, that it can be used for studying a wide variety of processes, and that the time parallelization strategy is scalable, making the application ready for High Performance Computing (HPC) machines.
References
[1] A. F. Voter. Phys. Rev. B, 57:R13985–R13988, Jun 1998.
[2] C. Le Bris, T. Lelièvre, M. Luskin, and D. Perez. Monte Carlo Methods and Applications, 18(2):119–146, Jan 2012. arXiv: 1105.4636.
[3] A. Binder, T. Lelièvre, and G. Simpson. J. Comput. Phys., 284:595–616, Mar 2015. arXiv: 1404.6191.
[4] F. Hédin and T. Lelièvre. The Journal of Chemical Physics, to be published, 2018. - 11:00-11:30 : Michael Minion
The Performance of the PFASST Method on a Suite of Shallow Water Test Equations on the Rotating Sphere. — AbstractM. Minion¹, F. Hamon¹, M. Schreiber²
¹ Lawrence Berkeley National Lab
² Exeter University
This talk will report on recent work to assess the parallel in time performance of the PFASST algorithm applied to a popular suite of test equations from the atmospheric modeling community. For this study, the PFASST algorithm is paired with a spatial discretization based on spherical harmonics and applied to popular test cases governed by the shallow water equations on a rotating sphere. The implementation uses the libpfasst library and the SWEET (Shallow Water Equation Environment for Tests) package. High-order semi-implicit or IMEX parallel in time schemes are tested with a careful consideration of the effects of viscosity and resolution on the parallel performance. - 11:30-12:00 : Charles Murray
Full space-time multigrid using time stepping specific multigrid restriction and prolongation. — AbstractC. Murray¹, T. Weinzierl¹
¹ Durham University
Classic multigrid methods operate on a grid discretised in space and use a complementing method (time stepping) to advance a solution in time. This synchronises/sequentialises the time steps. A space-time grid can be used to discretise both space and time requiring a stencil extended in the time dimension. Traditional multigrid methods on space-time grids use multigrid exclusively in space and couple space and time on the finest level, while traditional space parallelisation couples whole time slices with each other.
We propose (i) to use multigid on the global space-time grid which is well-known to comprise both classic spatial multigrid and parallel-in-time methods. We propose (ii) to use classic Galerkin operator construction to come up with coarse time stepping rules. The time stepping is read as limit of a convection operator. We propose (ii) to extend the algebraic multigrid technique of Black Box Multigrid (BoxMG) to compute operator- and time stepping-specific multigrid restriction and prolongation. Such an approach is doomed to fail if the proposed realisation is not fast and scaling. Our approach relies on three pillars. First, we apply dynamically adaptive meshes in space in time and we use full multigrid cycles in space and time, this is combined it with Full Approximation Storage (FAS). The latter allows for a straightforward realisation of arbitrary dynamic mesh refinement (AMR), combined with the space-time solver this gives dynamic time-stepping effectively for free. Second, we observe that classic multigrid methods require all stencils, i.e. matrix entries, to be computed prior to the solve, at the same time, the first few iterations determine approximations of limited accuracy only. To be able to exploit massively parallel computers without a ramp up (assembly) phase we propose to kick off multigrid with very rough stencil approximations but to use idle cores to determine better and better stencil integrations on-the-fly. Finally, we note that multigrid suffers from limited arithmetic intensity and concurrency, while the gap between compute power and memory bandwidth is widening. It is thus important to read each unknown as rarely as possible. Additive multigrid variants are well-known to have advantageous scaling properties but tend to be less stable and efficient than their multiplicative counterparts. Asynchronous Fast Adaptive Composite (AFAC) candidates to stabilise additive multigrid as each correction term is damped by an additional multilevel coarse grid solve. We propose a pipelined, single-touch realisation of AFAC in space and time where each degree of freedom is loaded only once per multilevel smoothing step. It is memory-access optimal.
-Excursion on Batz island-