Section: New Results

Parallelism and convergence in iterative linear solvers

Generation of Krylov subspace bases

Participant : Bernard Philippe.

This work was done in collaboration with L. Reichel, from University of Kent, USA (see 8.3.1 ).

It is published in a journal [19] .

Many problems in scientific computing involving a large sparse square matrix A are solved by Krylov subspace methods. This includes methods for the solution of large linear systems of equations with A, for the computation of a few eigenvalues and associated eigenvectors of A, and for the approximation of nonlinear matrix functions of A. When the matrix A is non-Hermitian, the Arnoldi process commonly is used to compute an orthonormal basis for a Krylov subspace associated with A. The Arnoldi process often is implemented with the aid of the modified Gram–Schmidt method. It is well known that the latter constitutes a bottleneck in parallel computing environments, and to some extent also on sequential computers. Several approaches to circumvent orthogonalization by the modified Gram–Schmidt method have been described in the literature, including the generation of Krylov subspace bases with the aid of suitably chosen Chebyshev or Newton polynomials. We review these schemes and describe new ones. Numerical examples are presented.

Parallel Adaptive Deflated GMRES

Participants : Jocelyne Erhel, Bernard Philippe.

This work was done in the context of the joint Inria/ NCSA laboratory on petascale computing (see 8.3.7 ), and the c2sexa project (see 8.1.3 ). Computations were done with GENCI supercomputers (see 8.1.6 ), using the software GPREMS, AGMRES, DGMRES (see 5.7 , 5.8 , 5.9 ).

It was presented at two conferences [30] [29] , is published in proceedings [39] and is submitted (in revision) to a journal [46] . The algorithms are implemented in the software DGMRES and AGMRES, which are freely available in the PETSC repository.

The GMRES iterative method is widely used as Krylov subspace accelerator for solving sparse linear systems when the coefficient matrix is nonsymmetric and indefinite. The Newton basis implementation has been proposed on distributed memory computers as an alternative to the classical approach with the Arnoldi process. The aim of our work here is to introduce a modification based on deflation and augmented techniques. This approach builds an augmented subspace or a preconditioning matrix in an adaptive way to accelerate the convergence of the restarted formulation. It can be combined with preconditioning methods based for example on domain decomposition. In our numerical experiments, we show the benefits of our method to solve large linear systems.

Memory efficient hybrid algebraic solvers for linear systems arising from compressible flows

Participants : Jocelyne Erhel, Bernard Philippe.

This work was done in collaboration with FLUOREM company, in the context of the joint Inria/ NCSA laboratory on petascale computing (see 8.3.7 ) and the C2S@EXA project (see 8.1.3 ). Computations were done with GENCI supercomputers (see 8.1.6 ), using the software GPREMS, AGMRES, DGMRES (see 5.7 , 5.8 , 5.9 ).

It has been published in a journal [18] .

This paper deals with the solution of large and sparse linear systems arising from design optimization in Computational Fluid Dynamics. From the algebraic decomposition of the input matrix, a hybrid robust direct/iterative solver is often defined with a Krylov subspace method as accelerator, a domain decomposition method as preconditioner and a direct method as subdomain solver. The goal of this paper is to reduce the memory requirements and indirectly the computational cost at different steps of this scheme. To this end, we use a grid-point induced block approach for the data storage and the partitioning part, a Krylov subspace method based on the restarted GMRES accelerated by deflation, a preconditioner formulated with the restricted additive Schwarz method and an aerodynamic/turbulent fields split at the subdomain level. Numerical results are presented with industrial test cases to show the benefits of these choices.

Efficient parallel implementation of the fully algebraic multiplicative Aitken-RAS preconditioning technique

Participant : Thomas Dufaud.

This work was done in collaboration with D. Tromeur-Dervout, from ICJ, University of Lyon and has been published in a journal [14] .

This paper details the software implementation of the ARAS preconditioning technique [48] , in the PETSc framework. Especially, the PETSc implementation of interface operators involved in ARAS and the introduction of a two level of parallelism in PETSc for the RAS are described. The numerical and parallel implementation performances are studied on academic and industrial problems, and compared with the RAS preconditioning. For saving computational time on industrial problems, the Aitken's acceleration operator is approximated from the singular values decomposition technique of the RAS iterate solutions.

An algebraic multilevel preconditioning framework based on information of a Richardson process

Participant : Thomas Dufaud.

This work was done in the context of the C2S@EXA project (see 8.1.3 ).

It has been presented at a conference [23] and submitted to the proceedings.

A fully algebraic framework for constructing coarse spaces for multilevel preconditioning techniques is proposed. Multilevel techniques are known to be robust for scalar elliptic Partial Differential Equations with standard discretization and to enhance the scalability of domain decomposition method such as RAS preconditioning techniques. An issue is their application to linear system encountered in industrial applications which can be derived from non-elliptic PDEs. Moreover, the building of coarse levels algebraically becomes an issue since the only known information is contained in the operator to inverse. Considering that a coarse space can be seen as a space to represent an approximated solution of a smaller dimension than the leading dimension of the system, it is possible to build a coarse level based on a coarse representation of the solution. Drawing our inspiration from the Aitken-SVD methodology, dedicated to Schwarz methods, we proposed to construct an approximation space by computing the Singular Value Decomposition of a set of iterated solutions of the Richardson process associated to a given preconditioner. This technique does not involve the knowledge of the underlying equations and can be applied to build coarse levels for several preconditioners. Numerical results are provided on both academic and industrial problems, using two-level additive preconditioners built with this methodology.