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Article overview
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High level implementation of geometric multigrid solvers for finite element problems: applications in atmospheric modelling | | Lawrence Mitchell
; Eike Hermann Müller
; | | Date: |
2 May 2016 | | Abstract: | The implementation of efficient multigrid preconditioners for elliptic
partial differential equations (PDEs) is a challenge due to the complexity of
the resulting algorithms and corresponding computer code. For sophisticated
finite element discretisations on unstructured grids an efficient
implementation can be very time consuming and requires the programmer to have
in-depth knowledge of the mathematical theory, parallel computing and
optimisation techniques on manycore CPUs. In this paper we show how the
development of bespoke multigrid preconditioners can be simplified
significantly by using a framework which allows the expression of the each
component of the algorithm at the correct abstraction level. Our approach (1)
allows the expression of the finite element problem in a language which is
close to the mathematical formulation of the problem, (2) guarantees the
automatic generation and efficient execution of parallel optimised low-level
computer code and (3) is flexible enough to support different abstraction
levels and give the programmer control over details of the preconditioner. We
use the composable abstractions of the Firedrake/PyOP2 package to demonstrate
the efficiency of this approach for the solution of strongly anisotropic PDEs
in atmospheric modelling. The weak formulation of the PDE is expressed in
Unified Form Language (UFL) and the lower PyOP2 abstraction layer allows the
manual design of computational kernels for a bespoke geometric multigrid
preconditioner. We compare the performance of this preconditioner to a
single-level method and hypre’s BoomerAMG algorithm. The Firedrake/PyOP2 code
is inherently parallel and we present a detailled performance analysis for a
single node (24 cores) on the ARCHER supercomputer. Our implementation utilises
a significant fraction of the available memory bandwidth and shows very good
weak scaling on up to 6,144 compute cores. | | Source: | arXiv, 1605.0492 | | Services: | Forum | Review | PDF | Favorites | |
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