2,351 research outputs found
Hybrid PDE solver for data-driven problems and modern branching
The numerical solution of large-scale PDEs, such as those occurring in
data-driven applications, unavoidably require powerful parallel computers and
tailored parallel algorithms to make the best possible use of them. In fact,
considerations about the parallelization and scalability of realistic problems
are often critical enough to warrant acknowledgement in the modelling phase.
The purpose of this paper is to spread awareness of the Probabilistic Domain
Decomposition (PDD) method, a fresh approach to the parallelization of PDEs
with excellent scalability properties. The idea exploits the stochastic
representation of the PDE and its approximation via Monte Carlo in combination
with deterministic high-performance PDE solvers. We describe the ingredients of
PDD and its applicability in the scope of data science. In particular, we
highlight recent advances in stochastic representations for nonlinear PDEs
using branching diffusions, which have significantly broadened the scope of
PDD.
We envision this work as a dictionary giving large-scale PDE practitioners
references on the very latest algorithms and techniques of a non-standard, yet
highly parallelizable, methodology at the interface of deterministic and
probabilistic numerical methods. We close this work with an invitation to the
fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European
Journal of Applied Mathematics (EJAM
Efficient Multigrid Preconditioners for Atmospheric Flow Simulations at High Aspect Ratio
Many problems in fluid modelling require the efficient solution of highly
anisotropic elliptic partial differential equations (PDEs) in "flat" domains.
For example, in numerical weather- and climate-prediction an elliptic PDE for
the pressure correction has to be solved at every time step in a thin spherical
shell representing the global atmosphere. This elliptic solve can be one of the
computationally most demanding components in semi-implicit semi-Lagrangian time
stepping methods which are very popular as they allow for larger model time
steps and better overall performance. With increasing model resolution,
algorithmically efficient and scalable algorithms are essential to run the code
under tight operational time constraints. We discuss the theory and practical
application of bespoke geometric multigrid preconditioners for equations of
this type. The algorithms deal with the strong anisotropy in the vertical
direction by using the tensor-product approach originally analysed by B\"{o}rm
and Hiptmair [Numer. Algorithms, 26/3 (2001), pp. 219-234]. We extend the
analysis to three dimensions under slightly weakened assumptions, and
numerically demonstrate its efficiency for the solution of the elliptic PDE for
the global pressure correction in atmospheric forecast models. For this we
compare the performance of different multigrid preconditioners on a
tensor-product grid with a semi-structured and quasi-uniform horizontal mesh
and a one dimensional vertical grid. The code is implemented in the Distributed
and Unified Numerics Environment (DUNE), which provides an easy-to-use and
scalable environment for algorithms operating on tensor-product grids. Parallel
scalability of our solvers on up to 20,480 cores is demonstrated on the HECToR
supercomputer.Comment: 22 pages, 6 Figures, 2 Table
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