Domain decomposition preconditioners of Neumann-Neumann type for hp‐approximations on boundary layer meshes in three dimensions

We develop and analyse Neumann-Neumann methods for hp finite‐element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in three dimensions. These are meshes that are highly anisotropic where the aspect ratio typically grows exponentially with the polynomial degree. The condition number of our preconditioners is shown to be independent of the aspect ratio of the mesh and of potentially large jumps of the coefficients. In addition, it only grows polylogarithmically with the polynomial degree, as in the case of p approximations on shape‐regular meshes. This work generalizes our previous one on two‐dimensional problems in Toselli & Vasseur (2003a, submitted to Numerische Mathematik, 2003c to appear in Comput. Methods Appl. Mech. Engng.) and the estimates derived here can be employed to prove condition number bounds for certain types of FETI method

Contribution to the study of efficient iterative methods for the numerical solution of partial differential equations

Multigrid and domain decomposition methods provide efficient algorithms for the numerical solution of partial differential equations arising in the modelling of many applications in Computational Science and Engineering. This manuscript covers certain aspects of modern iterative solution methods for the solution of large-scale problems issued from the discretization of partial differential equations. More specifically, we focus on geometric multigrid methods, non-overlapping substructuring methods and flexible Krylov subspace methods with a particular emphasis on their combination. Firstly, the combination of multigrid and Krylov subspace methods is investigated on a linear partial differential equation modelling wave propagation in heterogeneous media. Secondly, we focus on non-overlapping domain decomposition methods for a specific finite element discretization known as the hp finite element, where unrefinement/refinement is allowed both by decreasing/increasing the step size h or by decreasing/increasing the polynomial degree p of the approximation on each element. Results on condition number bounds for the domain decomposition preconditioned operators are given and illustrated by numerical results on academic problems in two and three dimensions. Thirdly, we review recent advances related to a class of Krylov subspace methods allowing variable preconditioning. We examine in detail flexible Krylov subspace methods including augmentation and/or spectral deflation, where deflation aims at capturing approximate invariant subspace information. We also present flexible Krylov subspace methods for the solution of linear systems with multiple right-hand sides given simultaneously. The efficiency of the numerical methods is demonstrated on challenging applications in seismics requiring the solution of huge linear systems of equations with multiple right-hand sides on parallel distributed memory computers. Finally, we expose current and future prospectives towards the design of efficient algorithms on extreme scale machines for the solution of problems coming from the discretization of partial differential equations

Deterministic algorithms for the low rank approximation of matrices

Cours sur invitation donné lors de l'Action Nationale de Formation CNRS intitulée: "Réduction de la dimension dans la fouille de données massives : enjeux, méthodes et outils pour le calcul.

Map/Reduce operations for scientific computing in Julia

We describe map and reduce techniques for applications related to scientific computing in Julia. Basic examples are introduced first. A more advanced application in linear algebra is finally discussed to show the benefits of map and reduce techniques

Limited memory preconditioners for nonsymmetric systems

This paper presents a class of limited memory preconditioners (LMPs) for solving linear systems of equations with multiple nonsymmetric matrices and multiple right-hand sides. These preconditioners based on limited memory quasi-Newton formulas require a small number k of linearly independent vectors. They may be used to improve an existing first-level preconditioner and are especially worth considering when the solution of a sequence of linear systems with slowly varying left-hand sides is addressed

A Parallel Evolution Strategy for Acoustic Full-Waveform Inversion

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

Stable time-parallel integration of advection dominated problems using Parareal with space coarsening

A common idea in the PinT community is that Parareal, one of the most popular time-parallel algorithm, is often numerically unstable when applied to hyperbolic problems, such as the advection equation. This has been both numerically observed and studied theoretically in the case of an implicit time integrator as a coarse solver (see, e.g, [1]). Such results could discourage the application of Parareal to Computational Fluid Dynamics (CFD) problems, especially for high Reynolds numbers (see, e.g, [2]). On the contrary, a recent application of Parareal with spatial coarsening to an explicit CFD solver [3] has shown that not only a stable numerical integration was obtained, but also that the Reynolds number played a minor role in the convergence behaviour, compared to other parameters of the parallel-in-time algorithm. Hence, in this talk, we present numerical experiments related to the application of Parareal with spatial coarsening to the one-dimensional advection- diffusion problem. We investigate the influence of several parameters on the convergence (importance of the diffusion term, spatial resolution, order of interpolation, regularity of the initial solution, time-slice length, nonlinearity,...). We advocate that "a high Reynolds number" is not a good enough reason for not using Parareal, and that a stable and efficient parallel in time integration can be made possible, even for highly advective problems, provided that important algorithmic components are carefully chosen

A new preconditioner update strategy for the solution of sequences of linear systems in structural mechanics: application to saddle point problems in elasticity

Many applications in structural mechanics require the numerical solution of sequences of linear systems typically issued from a finite element discretization of the governing equations on fine meshes. The method of Lagrange multipliers is often used to take into account mechanical constraints. The resulting matrices then exhibit a saddle point structure and the iterative solution of such preconditioned linear systems is considered as challenging. A popular strategy is then to combine preconditioning and deflation to yield an efficient method.We propose an alternative that is applicable to the general case and not only to matrices with a saddle point structure. In this approach, we consider to update an existing algebraic or application-based preconditioner, using specific available information exploiting the knowledge of an approximate invariant subspace or of matrix-vector products. The resulting preconditioner has the form of a limited memory quasi-Newton matrix and requires a small number of linearly independent vectors. Numerical experiments performed on three large-scale applications in elasticity highlight the relevance of the new approach. We show that the proposed method outperforms the deflation method when considering sequences of linear systems with varying matrices