## Section: New Results

### Parallelism and convergence in Krylov methods

Participants : Édouard Canot, Jocelyne Erhel, Désiré Nuentsa Wakam, Bernard Philippe.

This work is done in the context of the Cinemas2 and the Libraero contracts, 7.2 and 8.1.3 . It is also done in collaboration with the joint INRIA/ NCSA laboratory on petascale computing.

A Ph-D thesis was defended this year [12] .

#### Some properties of Krylov methods

Participant : Jocelyne Erhel.

A survey was presented at a conference and published in a book chapter [37] [24] .

Solving a linear system is at the heart of many scientific and engineering applications. Generally, this operations is the most time and memory consuming part of the simulation. This paper focuses on some properties of Krylov iterative methods. Iterative methods of Krylov type require less memory than direct methods, but the number of iterations increases rapidly with the size of the system. The convergence rate and the accuracy of the results depend on the condition number which can blow up at large scale. Therefore, it is essential to combine these methods with a preconditioner; the idea is to solve another system, close to the original one, but which is easier to solve; also, on parallel computers, it must be scalable. In Krylov iterative methods, the matrix is not transformed but the kernel operation is the matrix-vector product; thus it is possible to use matrix-free versions without storing the matrix. However, preconditioning will sometimes require the matrix. Krylov methods are described in many books. In this survey, we choose the framework of polynomial and projection methods. We first give general properties. Then, we study specific methods for the three different types of matrices: the case of SPD matrices is analyzed first, followed by the case of symmetric indefinite matrices. The general case of nonsymmetric matrices is studied with the description of several Krylov methods. Finally, some practical issues, preconditioning and parallelism are discussed.

#### Generation of Krylov subspace bases

Participant : Bernard Philippe.

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

It has been 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 preconditioned GMRES with Multiplicative Schwarz

Participants : Édouard Canot, Jocelyne Erhel, Désiré Nuentsa Wakam, Bernard Philippe.

This work was published in a journal [18] .

This paper presents a robust hybrid solver for linear systems that combines a Krylov subspace method as accelerator with a Schwarz-based preconditioner. This preconditioner uses an explicit formulation associated to one iteration of the multiplicative Schwarz method. The Newton-basis GMRES, which aim at expressing a good data parallelism between subdomains is used as accelerator. In the first part of this paper, we present the pipeline parallelism that is obtained when the multiplicative Schwarz preconditioner is used to build the Krylov basis for the GMRES method. This is referred as the first level of parallelism. In the second part, we introduce a second level of parallelism inside the subdomains. For Schwarz-based preconditioners, the number of subdommains is kept small to provide a robust solver. Therefore, the linear systems associated to subdomains are solved efficiently with this approach. Numerical experiments are performed on several problems to demonstrate the benefits of using these two levels of parallelism in the solver, mainly in terms of numerical robustness and global efficiency.

#### Adaptive deflation in preconditioned GMRES algorithm using a combined preconditioning

Participants : Jocelyne Erhel, Désiré Nuentsa Wakam, Bernard Philippe.

This work has been presented at a conference and a workshop [35] , [27] and submitted to the proceedings of DD20 [45] . The software module DGMRES is integrated in the Petsc distribution.

Many scientific libraries are currently based on the GMRES method as a Krylov subspace iterative method for solving large linear systems. The restarted formulation known as GMRES($m$) has been extensively studied and several approaches have been proposed to reduce the negative effects due to the restarting procedure. A common effect in GMRES($m$) is a slow convergence rate or a stagnation in the iterative process. In this situation, it is less attractive as a general solver in industrial applications. In this work, we propose an adaptive deflation strategy which retains useful information at the time of restart to avoid stagnation in GMRES($m$) and improve its convergence rate. We give a parallel implementation in the PETSc package. The provided numerical results show that this approach can be effectively used in the hybrid direct/iterative methods to solve large-scale systems.

#### Adaptive deflation in preconditioned GMRES algorithm using an augmented subspace

Participants : Jocelyne Erhel, Désiré Nuentsa Wakam, Bernard Philippe.

This work has been presented at a conference [31] and submitted to the journal ETNA [46] .

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 techniques. This approach builds an augmented subspace in an adaptive way to accelerate the convergence of the restarted formulation. In our numerical experiments, we show the benefits of using this implementation with hybrid direct/iterative methods to solve large linear systems.

#### Using deflated preconditioned GMRES for industrial CFD problems

Participant : Désiré Nuentsa Wakam.

This work has been submitted to the journal Computers and Fluids [47] .

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.