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Surrogate-Based Ensemble Grouping Strategies for Embedded Sampling-Based Uncertainty Quantification

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Quantification of Uncertainty: Improving Efficiency and Technology

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 137 ))

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Abstract

The embedded ensemble propagation approach introduced in Phipps et al. (SIAM J. Sci. Comput. 39(2):C162, 2017) has been demonstrated to be a powerful means of reducing the computational cost of sampling-based uncertainty quantification methods, particularly on emerging computational architectures. A substantial challenge with this method however is ensemble-divergence, whereby different samples within an ensemble choose different code paths. This can reduce the effectiveness of the method and increase computational cost. Therefore grouping samples together to minimize this divergence is paramount in making the method effective for challenging computational simulations. In this work, a new grouping approach based on a surrogate for computational cost built up during the uncertainty propagation is developed and applied to model advection-diffusion problems where computational cost is driven by the number of (preconditioned) linear solver iterations. The approach is developed within the context of locally adaptive stochastic collocation methods, where a surrogate for the number of linear solver iterations, generated from previous levels of the adaptive grid generation, is used to predict iterations for subsequent samples, and group them based on similar numbers of iterations. The effectiveness of the method is demonstrated by applying it to highly anisotropic advection-dominated diffusion problems with a wide variation in solver iterations from sample to sample. It extends the parameter-based grouping approach developed in D’Elia et al. (SIAM/ASA J. Uncertain. Quantif. 6:87, 2017) to more general problems without requiring detailed knowledge of how the uncertain parameters affect the simulation’s cost, and is also less intrusive to the simulation code.

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Notes

  1. 1.

    Here, Ω is a set of realizations, \(\mathscr {F}\) is a σ-algebra of events and \(\mathbb {P}:\mathscr {F}\rightarrow [0,1]\) is a probability measure.

  2. 2.

    For details regarding the functional spaces and the well-posedness of problem (1) we refer to [33].

  3. 3.

    Note that here we allow the forcing term f to be sample dependent.

  4. 4.

    Note that at each level, the number of samples is not usually evenly divisible by the ensemble size. To use a uniform ensemble size for all ensembles, samples are added by replicating the last sample in the last ensemble. This results in larger R values when the number of samples is small and the ensemble size is large, as can be seen in the results for R 1.

  5. 5.

    Note that we do not include timing results in Table 4 since the ensemble implementation is currently not optimized for GMRES. A significant cost within GMRES is the orthogonalization of each new Krylov vector against the previous set of vectors, which in Trilinos is implemented through GEMV dense matrix-vector product BLAS routine. An optimized implementation of this routine for ensembles is currently being developed.

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Acknowledgements

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

This material is based upon work supported by the U.S. Department of Energy, Office of Science, and Office of Advanced Scientific Computing Research (ASCR), as well as the National Nuclear Security Administration, Advanced Technology Development and Mitigation program. This research used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility.

The authors would like to thank Dr. Miro Stoyanov for useful conversations and for providing great support with TASMANIAN.

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Authors and Affiliations

  1. Computational Science and Analysis, Sandia National Laboratories, Livermore, CA, USA

    M. D’Elia

  2. Center for Computing Research, Sandia National Laboratories, Albuquerque, NM, USA

    E. Phipps & M. S. Ebeida

  3. Northrop Grumman Corporation, Albuquerque, NM, USA

    A. Rushdi

Authors
  1. M. D’Elia
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  2. E. Phipps
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  3. A. Rushdi
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  4. M. S. Ebeida
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Corresponding author

Correspondence to M. D’Elia .

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Editors and Affiliations

  1. Computational Multiscale, Sandia National Laboratories, Albuquerque, NM, USA

    Marta D'Elia

  2. Dept of Scientific Computing, Florida State Univ, Tallahassee, FL, USA

    Max Gunzburger

  3. SISSA mathLab, Mathematics Area, International School for Advanced Studie, Trieste, Italy

    Gianluigi Rozza

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© 2020 National Technology & Engineering Solutions of Sandia, and E. Phipps, A. Rushdi, and M. S. Ebeida, under exclusive license to SIP AG, part of Springer Nature

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D’Elia, M., Phipps, E., Rushdi, A., Ebeida, M.S. (2020). Surrogate-Based Ensemble Grouping Strategies for Embedded Sampling-Based Uncertainty Quantification. In: D'Elia, M., Gunzburger, M., Rozza, G. (eds) Quantification of Uncertainty: Improving Efficiency and Technology. Lecture Notes in Computational Science and Engineering, vol 137 . Springer, Cham. https://doi.org/10.1007/978-3-030-48721-8_3

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