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Multi angle  Krylov subspace solvers and preconditioners
Vuik, Kees (Auteur de la Conférence) | CIRM (Editeur )

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Multi angle  High performance computing with Feel++: applications and numerical methods
Prud'homme, Christophe (Auteur de la Conférence) | CIRM (Editeur )

I will review (some of) the HPC solution strategies developed in Feel++. We present our advances in developing a language specific to partial differential equations embedded in C++. We have been developing the Feel++ framework (Finite Element method Embedded Language in C++) to the point where it allows to use a very wide range of Galerkin methods and advanced numerical methods such as domain decomposition methods including mortar and three fields methods, fictitious domain methods or certified reduced basis. We shall present an overview of the various ingredients as well as some illustrations. The ingredients include a very expressive embedded language, seamless interpolation, mesh adaption, seamless parallelisation. As to the illustrations, they exercise the versatility of the framework either by allowing the development and/or numerical verification of (new) mathematical methods or the development of large multi-physics applications - e.g. fluid-structure interaction using either an Arbitrary Lagrangian Eulerian formulation or a levelset based one; high field magnets modeling which involves electro-thermal, magnetostatics, mechanical and thermo-hydraulics model; ... - The range of users span from mechanical engineers in industry, physicists in complex fluids, computer scientists in biomedical applications to applied mathematicians thanks to the shared common mathematical embedded language hiding linear algebra and computer science complexities. I will review (some of) the HPC solution strategies developed in Feel++. We present our advances in developing a language specific to partial differential equations embedded in C++. We have been developing the Feel++ framework (Finite Element method Embedded Language in C++) to the point where it allows to use a very wide range of Galerkin methods and advanced numerical methods such as domain decomposition methods including mortar and three ...

65N30 ; 65N55 ; 65Y05 ; 65Y15

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Multi angle  Efficient iterative solvers: FETI methods with multiple search directions
Roux, François-Xavier (Auteur de la Conférence) | CIRM (Editeur )

In domain decomposition methods, most of the computational cost lies in the successive solutions of the local problems in subdomains via forward-backward substitutions and in the orthogonalization of interface search directions. All these operations are performed, in the best case, via BLAS-1 or BLAS-2 routines which are inefficient on multicore systems with hierarchical memory. A way to improve the parallel efficiency of the method consists in working with several search directions, since multiple forward-backward substitutions and reorthogonalizations involve BLAS-3 routines. In the case of a problem with several right-hand-sides, using a block Krylov method is a straightforward way to work with multiple search directions. This will be illustrated with an application in electromagnetism using FETI-2LM method. For problems with a single right-hand-side, deriving several search directions that make sense from the optimal one constructed by the Krylov method is not so easy. The recently developed S-FETI method gives a very good approach that does not only improve parallel efficiency but can also reduce the global computational cost in the case of very heterogeneous problems. In domain decomposition methods, most of the computational cost lies in the successive solutions of the local problems in subdomains via forward-backward substitutions and in the orthogonalization of interface search directions. All these operations are performed, in the best case, via BLAS-1 or BLAS-2 routines which are inefficient on multicore systems with hierarchical memory. A way to improve the parallel efficiency of the method consists in ...

65N22 ; 65N30 ; 65N55 ; 65Y05 ; 65F10

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Multi angle  Domain decomposition, hybrid methods, coarse space corrections
Nataf, Frédéric (Auteur de la Conférence) | CIRM (Editeur )

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Multi angle  Algorithms for future emerging technologies
Dongarra, Jack (Auteur de la Conférence) | CIRM (Editeur )

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Multi angle  A posteriori error estimates and solver adaptivity in numerical simulations
Vohralík, Martin (Auteur de la Conférence) | CIRM (Editeur )

We review how to bound the error between the unknown weak solution of a PDE and its numerical approximation via a fully computable a posteriori estimate. We focus on approximations obtained at an arbitrary step of a linearization (Newton-Raphson, fixed point, ...) and algebraic solver (conjugate gradients, multigrid, domain decomposition, ...). Identifying the discretization, linearization, and algebraic error components, we design local stopping criteria which keep them in balance. This gives rise to a fully adaptive inexact Newton method. Numerical experiments are presented in confirmation of the theory. We review how to bound the error between the unknown weak solution of a PDE and its numerical approximation via a fully computable a posteriori estimate. We focus on approximations obtained at an arbitrary step of a linearization (Newton-Raphson, fixed point, ...) and algebraic solver (conjugate gradients, multigrid, domain decomposition, ...). Identifying the discretization, linearization, and algebraic error components, we design local ...

65N15 ; 65N22 ; 65Y05

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Multi angle  A gentle introduction to parallel programming using OpenMP
Broquedis, François (Auteur de la Conférence) | CIRM (Editeur )

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