Cerfacs Enter the world of high performance ...

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Abstract : We are interested in the Monte Carlo (MC) sampling of discretized elliptic partial differential equations (PDEs) with random variable coefficients. The dominant computational load of such applications consists of solving large numbers of linear systems with variable matrix and right-hand side. As a means to alleviate this effort, we review, develop, implement and analyze efficient and scalable methodologies for stochastic elliptic PDEs that make use of appropriate combinations of iterative solvers and preconditioners. Three preconditioning strategies are developed and investigated.First, state-of-the-art parallel preconditioners are kept constant and used to solve all the sampled linear systems of MC simulations. This straightforward strategy serves as a point of comparison for the two other methods to improve upon. Second, preconditioners based on the deflation of correlated linear systems are defined while sampling the random coefficient field by Markov chain Monte Carlo. Different projections and restarting methods of the eigen-search space are considered for the online approximation of spectral information. As opposed to harmonic Rayleigh-Ritz projections, Rayleigh-Ritz projections are shown to avoid preconditioner applications when recycling the Krylov subspaces so that they should be favored for a better performance.Both the thick-restart and even more so the locally optimal thick-restart restarting of the eigen-search space lead to significant decreases of the number of solver iterations, particularly for larger linear systems. The preconditioning strategy based on deflation, which is adapted for Bayesian inference, works particularly well when using preconditoners whose action results in well-separated eigenvalues at the extremities of the spectrum. This is the case of block Jacobi preconditioners as well as preconditioners based on domain decomposition, but not when preconditioning with algebraic multigrids in the case of isotropic equations. Third, we partition the latent stochastic space of the random coefficient field into Voronoi cells, each of which is represented by a centroidal coefficient field on the basis of which a distinct preconditioner is defined which is used to solve the sampled linear systems whose corresponding coefficient fields lie within the cell. As such, we adopt a compact representation of the random coefficient field referred to as a Voronoi quantizer. We consider different distributions of centroidal coefficient fields, and we investigate the properties of the underlying preconditioning strategies in terms of expected number of solver iterations for sequential simulations, and of load balancing for parallel simulations. One distribution in particular, which minimizes the average distance between the coefficient field and its compact representation, minimizes the expected number of solver iterations. This distribution yields the smallest number of expected number of solver iterations so that it is particularly adapted for sequential simulations. Another distribution is considered which leads to equal attribution frequencies for all the cells, i.e., approximately the same number of linear systems is solved with each preconditioner of the quantizer. Our experiments show that this strategy yields a smaller spread of the expected number of solver iterations among the preconditioners so that it is more adapted for parallel simulations. Finally, a distribution based on deterministic grids with a stochastic dimension which increases with the number of preconditioners is proposed. This last distribution allows to bypass preliminary computations necessary to determine the optimal dimension of the approximating stochastic space for a given number of preconditioners.

Jury :
M. Pietro CONGEDO – Inria, École Polytechnique – Examiner
M. Olivier COULAUD – Inria, Université de Bordeaux – Examiner
M. Luc GIRAUD – Inria, Université de Bordeaux – Co-advisor
M. Julien LANGOU – University of Colorado Denver – Referee
M. Olivier LE MAÎTRE – CNRS, École Polytechnique – Examiner
M. Paul MYCEK – Cerfacs – Co-advisor
M. Anthony NOUY – École Centrale Nantes – Referee
Mme Nicole SPILLANE – CNRS, École Polytechnique – Examiner