Article (Scientific journals)
GMRES with embedded ensemble propagation for the efficient solution of parametric linear systems in uncertainty quantification of computational models
Liegeois, Kim; Boman, Romain; Phipps, Eric T. et al.
2020In Computer Methods in Applied Mechanics and Engineering, 369
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Keywords :
GMRES; Intrusive uncertainty quantification; Embedded ensemble propagation; BLAS; Control flow divergence
Abstract :
[en] In a previous work, embedded ensemble propagation was proposed to improve the efficiency of sampling-based uncertainty quantification methods of computational models on emerging computational architectures. It consists of simultaneously evaluating the model for a subset of samples together, instead of evaluating them individually. A first approach introduced to solve parametric linear systems with ensemble propagation is ensemble reduction. In Krylov methods for example, this reduction consists in coupling the samples together using an inner product that sums the sample contributions. Ensemble reduction has the advantages of being able to use optimized implementations of BLAS functions and having a stopping criterion which involves only one scalar. However, the reduction potentially decreases the rate of convergence due to the gathering of the spectra of the samples. In this paper, we investigate a second approach: ensemble propagation without ensemble reduction in the case of GMRES. This second approach solves each sample simultaneously but independently to improve the convergence compared to ensemble reduction. This raises two new issues which are solved in this paper: the fact that optimized implementations of BLAS functions cannot be used anymore and that ensemble divergence, whereby individual samples within an ensemble must follow different code execution paths, can occur. We tackle those issues by implementing a high-performing ensemble GEMV and by using masks. The proposed ensemble GEMV leads to a similar cost per GMRES iteration for both approaches, i.e. with and without reduction. For illustration, we study the performances of the new linear solver in the context of a mesh tying problem. This example demonstrates improved ensemble propagation speed-up without reduction.
Disciplines :
Mechanical engineering
Author, co-author :
Liegeois, Kim ;  Université de Liège - ULiège > Département d'aérospatiale et mécanique > Computational and stochastic modeling
Boman, Romain  ;  Université de Liège - ULiège > Département d'aérospatiale et mécanique > Département d'aérospatiale et mécanique
Phipps, Eric T.;  Sandia National Laboratories > Scalable Algorithms Department
Wiesner, Tobias A.;  Sandia National Laboratories > Scalable Algorithms Department
Arnst, Maarten ;  Université de Liège - ULiège > Département d'aérospatiale et mécanique > Computational and stochastic modeling
Language :
English
Title :
GMRES with embedded ensemble propagation for the efficient solution of parametric linear systems in uncertainty quantification of computational models
Publication date :
01 September 2020
Journal title :
Computer Methods in Applied Mechanics and Engineering
ISSN :
0045-7825
eISSN :
1879-2138
Publisher :
Elsevier, Amsterdam, Netherlands
Volume :
369
Peer reviewed :
Peer Reviewed verified by ORBi
Funders :
FRIA - Fonds pour la Formation à la Recherche dans l'Industrie et dans l'Agriculture [BE]
Available on ORBi :
since 08 June 2020

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