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We consider numerical methods for the incompressible Reynolds averaged Navier-Stokes equations discretized by finite difference techniques on non-staggered grids in body-fitted coordinates. A segregated approach is used to solve the pressure-velocity coupling problem. We present detailed efficiency comparisons between iterative pressure linear solvers including Krylov subspace and multigrid methods. Three-dimensional numerical experiments carried out on scalar and vector machines and performed on different fluid flow problems show that a combination of multigrid and Krylov subspace methods is a robust and efficient pressure solver.

Preliminary version of the book chapter entitled "Deflation and augmentation techniques in Krylov linear solvers" published in "Developments in Parallel, Distributed, Grid and Cloud Computing for Engineering", ed. Topping, B.H.V and Ivanyi, P., Saxe-Coburg Publications, Kippen, Stirlingshire, United Kingdom, ISBN 978-1-874672-62-3, p. 249-275, 2013; In this paper we present deflation and augmentation techniques that have been designed to accelerate the convergence of Krylov subspace methods for the solution of linear systems of equations. We review numerical approaches both for linear systems with a non-Hermitian coefficient matrix, mainly within the Arnoldi framework, and for Hermitian positive definite problems with the conjugate gradient method.; Dans ce rapport nous présentons des techniques de déflation et d'augmentation qui ont été développées pour accélérer la convergence des méthodes de Krylov pour la solution de systémes d'équations linéaires. Nous passons en revue des approches pour des systémes linéaires dont les matrices sont non-hermitiennes, principalement dans le contexte de la méthode d'Arnoldi, et pour des matrices hermitiennes définies positives avec la méthode du gradient conjugué.

Algebraic multigrid (AMG) represents a class of efficient preconditioners for large and sparse linear systems arising in particular from discretized elliptic partial differential equations. In the context of reservoir simulation, a standard preconditioner for the linear solver is the so-called CPR-AMG in which AMG is applied to an approximated pressure subsystem. In this method, AMG is the bottleneck for scalability because it involves a lot of communications across processors in particular for the setup phase, which constructs the coarse levels according to the coefficients of the Jacobian matrix. The goal of this work is to decrease the overall cost of the CPR-AMG in parallel by combining two different coarsening strategies: the first levels are computed using an aggregation scheme whereas the coarsest levels are treated using a classical point-wise Ruge-Stüben (RS) scheme. AMG preconditioners constructed using classical coarsening schemes are able to achieve good convergence rates of the preconditioned iterative method. The complexities of the multigrid hierarchies can be quite high and thus the classical AMG may be expensive in terms of memory requirements and computational times. Aggregation AMG methods, on the other hand, provide better means of complexity control and consequently the setup time required to construct the preconditioner can be considerably lower. The efficiency of the aggregation AMG methods, however, deteriorates with the increasing problem size although the two-level convergence rates can be very good. To take advantage of both the aggregation and classical AMG, we consider combining both approaches within one hierarchy in order to decrease the setup time of the AMG preconditioner while retaining the convergence properties of the two-level aggregation method.

In this work, we propose another alternative to find an initial velocity model for the acoustic FWI without any physical knowledge. Motivated by the recent growth of high performance computing (HPC), we tackle the high non-linearity of the problem to minimize, using global optimization methods which are easy to parallelize, in particular, evolution strategies. The first contribution adapt evolution strategies to the FWI setting where the cost function evaluation is the most expensive part. The second contribution is the parameterization of the regarded problem, by being able to represent the model, as faithfully as possible, while limiting the number of parameters needed, since each additional parameter is an additional dimension to explore. The last contribution is to propose a highly parallel evolution strategy adapted to the FWI setting. The initial results on the Salt Dome velocity model using low frequency range, show that great improvement can be done to automate the FWI.