François-Xavier Roux

An unconventional domain decomposition method for an efficient parallel solution of large-scale finite element systems

A domain decomposition algorithm based on a hybrid variational principle is developed for the parallel finite element solution of selfadjoint elliptic partial differential equations. The spatial domain is partitioned into a set of totally disconnected subdomains, each assigned to an individual processor. Lagrange multipliers are introduced to enforce compatibility at the interface points. Within each subdomain, the singularity due to the disconnection is resolved in a two-step procedure. First, the null space component of each local operator is eliminated from the local problem. Next, its contribution to the local solution is related to the Lagrange multipliers through an orthogonality condition. Finally, a conjugate projected gradient algorithm is developed for the solution of the coupled system of local null space components and Lagrange multipliers. When implemented on local memory multiprocessors, the proposed hybrid method requires fewer interprocessor communications than conventional Schur methods. ...

Two-level domain decomposition methods with Lagrange multipliers for the fast iterative solution of acoustic scattering problems

We present two different but related Lagrange multiplier based domain decomposition (DD) methods for solving iteratively large-scale systems of equations arising from the finite element discretization of high-frequency exterior Helmholtz problems. The proposed methods are essentially two distinct extensions of the regularized finite element tearing and interconnecting (FETI) method to indefinite or complex problems. The first method employs a single Lagrange multiplier field to glue the local solutions at the subdomain interface boundaries. The second method employs two Lagrange multiplier fields for that purpose. The key ingredients of both of these FETI methods are the regularization of each subdomain matrix by a complex lumped mass matrix defined on the subdomain interface boundary, and the preconditioning of the global interface problem by a coarse second-level problem constructed with planar waves. We show numerically that both methods are scalable with respect to the mesh size, the subdomain size, and the wavenumber, but that the FETI method with a single Lagrange multiplier field – labeled FETI-H (H for Helmholtz) in this paper – delivers superior computational performances. We apply the FETI-H method to the parallel solution on a 24-processor Origin 2000 of an acoustic scattering problem with a submarine shaped obstacle, and report performance results that highlight the unique efficiency of this DD method for the solution of high frequency acoustic scattering problems.

Optimal Discrete Transmission Conditions for a Nonoverlapping Domain Decomposition Method for the Helmholtz Equation

This paper is dedicated to recent developments of a two-Lagrange multipliers domain decomposition method for the Helmholtz equation [C. Farhat et al., Comput. Methods Appl. Mech. Engrg., 184 (2000), pp. 213--240; M. J. Gander, F. Magoules, and F. Nataf, SIAM J. Sci. Comput., 24 (2002), pp. 38--60] involving an additional augmented operator along the interface between the subdomains. Most methods for optimizing the augmented interface operator are based on the discretization of approximations of the continuous transparent operator [B. Despres, Proceedings of the Second International Conference on Mathematical and Numerical Aspects of Wave Propagation, R. Kleinman et al., eds., SIAM, Philadelphia, 1993, pp. 197--206; J.-D. Benamou and B. Despres, J. Comput. Phys., 136 (1997), pp. 68--82; P. Chevalier and F. Nataf, Domain Decomposition Methods 10, AMS, Providence, RI, 1998, pp. 400--407; M. J. Gander, Proceedings of the 12th International Conference on Domain Decomposition Methods, (Chiba, Japan), ddm.org, 2000, pp. 247--254; M. J. Gander, F. Magoules, and F. Nataf, SIAM J. Sci. Comput., 24 (2002), pp. 38--60]. At the discrete level, the optimal operator can be proved to be equal to the Schur complement of the outer domain. This Schur complement can be directly approximated using purely algebraic techniques like sparse approximate inverse methods or incomplete factorization. The main advantage of such an algebraic approach is that it is much easier to implement in existing code without any information on the geometry of the interface and the finite element formulation used. Convergence results and parallel efficiency of several algebraic optimization techniques of an interface operator for acoustic analysis applications will be presented.

Algebraic approach to absorbing boundary conditions for the Helmholtz equation

Recent work has shown that designing absorbing boundary conditions through algebraic approaches may be a nice alternative to the continuous approaches based on a Fourier analysis. In this paper, an original algebraic technique based on the computation of small patches is presented for the Helmholtz equation. This new technique is not directly linked to the continuous equations of the problem, nor to the numerical scheme. These properties make this technique very convenient to implement in a domain decomposition context. The proposed algebraic absorbing boundary conditions are used in a non-overlapping domain decomposition method and are defined on the interface between the subdomains. An additional coarse grid correction is then applied to ensure full scalability of the domain decomposition method upon the number of subdomains. This coarse grid correction involves trigonometric functions defined on the interface between the subdomains. Numerical experiments are presented and illustrate the robustness and parallel efficiency of the proposed method for acoustics applications.

Analysis of Patch Substructuring Methods

Analysis of Patch Substructuring Methods Patch substructuring methods are non-overlapping domain decomposition methods like classical substructuring methods, but they use information from geometric patches reaching into neighboring subdomains, condensated on the interfaces, to enhance the performance of the method, while keeping it non-overlapping. These methods are very convenient to use in practice, but their convergence properties have not been studied yet. We analyze geometric patch substructuring methods for the special case of one patch per interface. We show that this method is equivalent to an overlapping Schwarz method using Neumann transmission conditions. This equivalence is obtained by first studying a new, algebraic patch method, which is equivalent to the classical Schwarz method with Dirichlet transmission conditions and an overlap corresponding to the size of the patches. Our results motivate a new method, the Robin patch method, which is a linear combination of the algebraic and the geometric one, and can be interpreted as an optimized Schwarz method with Robin transmission conditions. This new method has a significantly faster convergence rate than both the algebraic and the geometric one. We complement our results by numerical experiments.

The dual Schur complement method with well-posed local Neumann problems: regularization with a perturbed Lagrangian formulation

The dual Schur complement (DSC) domain decomposition (DD) method introduced by Farhat and Roux is an efficient and practical algorithm for the parallel solution of self-adjoint elliptic partial differential equations. A given spatial domain is partitioned into disconnected subdomains where an incomplete solution for the primary field is first evaluated using a direct method. Next, intersubdomain field continuity is enforced via a combination of discrete, polynomial, and/or piece-wise polynomial Lagrange multipliers, applied at the subdomain interfaces. This leads to a smaller size symmetric dual problem where the unknowns are the “gluing” Lagrange multipliers, and which is best solved with a preconditioned conjugate gradient (PCG) algorithm. However, for time-independent elasticity problems, every floating subdomain is associated with a singular stiffness matrix, so that the dual interface operator is in general indefinite. Previously, we have dealt with this issue by filtering out at each iteration of th...

A Parallel Ocean Model for High Resolution Studies

Domain Decomposition Methods is used for the parallelization of the LODYC ocean general circulation model. The local dependencies problem is solved by using a pencil splitting and an overlapping strategy. Two different parallel solvers of the surface pressure gradient, a preconditioned conjugate gradient method and a Dual Schur Complement method, have been implemented. The code is now used for the high resolution study of the Atlantic circulation by the CLIPPER research project and is one of the components of the operational oceanography MERCATOR project.

First Considerations about Modelling the Ocean General Circulation on MIMD Machines by Domain Decomposition Method

In this paper, we talk about an action of parallelization in Geosciences field. As a candidate we have chosen the OPA7 code, a three-dimensional primitive-equation finitedifference ocean general circulation model.

Implementation of a Parallel Sparse Direct Solver on Vector Architecture

Linear systems with large sparse matrices are solved in finite element analysis of elasticity and/or fluid problems. Thanks to development of graph partitioning software, it becomes feasible to extract dense sub-matrices efficiently with minimizing fill-in during factorization. By analyzing task dependency of block factorization of dense matrix, multi-cores of CPUs which share the main memory are used in parallel and asynchronously. The tasks in dense sub-matrices consist of BLAS level 3 kernels which efficiently use arithmetic capabilities of modern super-scalar CPU with large cache memory and also of modern vector CPU. BLAS level 3 kernels can also efficiently use vector architecture, without writing any directives for explicit vectorization in the code. Nevertheless, the sparse part still remains in factorization process. Although it is only a small fraction of the whole process and almost negligible on the super-scalar CPU, its optimization is important on vector architecture due to short vector loop.