Structural and Multidisciplinary Optimization

, Volume 60, Issue 6, pp 2205–2220 | Cite as

Linear regression-based multifidelity surrogate for disturbance amplification in multiphase explosion

  • M. Giselle Fernández-Godino
  • Sylvain Dubreuil
  • Nathalie Bartoli
  • Christian Gogu
  • S. Balachandar
  • Raphael T. HaftkaEmail author
Research Paper


When simulations are very expensive and many are required, as for optimization or uncertainty quantification, a way to reduce cost is using surrogates. With multiple simulations to predict the quantity of interest, some being very expensive and accurate (high-fidelity simulations) and others cheaper but less accurate (low-fidelity simulations), it may be worthwhile to use multifidelity surrogates (MFSs). Moreover, if we can afford just a few high-fidelity simulations or experiments, MFS becomes necessary. Co-Kriging, which is probably the most popular MFS, replaces both low-fidelity and high-fidelity simulations by a single MFS. A recently proposed linear regression–based MFS (LR-MFS) offers the option to correct the LF simulations instead of correcting the LF surrogate in the MFS. When the low-fidelity simulation is cheap enough for use in an application, such as optimization, this may be an attractive option. In this paper, we explore the performance of LR-MFS using exact and surrogate-replaced low-fidelity simulations. The problem studied is a cylindrical dispersal of 100-μ m-diameter solid particles after detonation and the quantity of interest is a measure of the amplification of the departure from axisymmetry. We find very substantial accuracy improvements for this problem using the LR-MFS with exact low-fidelity simulations. Inspired by these results, we also compare the performance of co-Kriging to the use of Kriging to correct exact low-fidelity simulations and find a similar accuracy improvement when simulations are directly used. For this problem, further improvements in accuracy are achievable by taking advantage of inherent parametric symmetries. These results may alert users of MFSs to the possible advantages of using exact low-fidelity simulations when this is affordable.


Multifidelity Surrogates Symmetries Linear regression Kriging Co-Kriging 



Discrepancy function

\(\hat {\delta }(\mathbf {x})\)

Discrepancy function surrogate, also known as additive correction


Constant scaling factor


High-fidelity simulation

\(\hat {y}_{HF}(\mathbf {x})\)

High-fidelity surrogate


Low-fidelity simulation

\(\hat {y}_{LF}(\mathbf {x})\)

Low-fidelity surrogate

\(\hat {y}_{\hat {add}}(\mathbf {x})\)

Multifidelity surrogate that uses additive correction and where the prediction is performed using a low-fidelity surrogate

\(\hat {y}_{\hat {comp}}(\mathbf {x})\)

Multifidelity surrogate that uses comprehensive correction and where the prediction is performed using a low-fidelity surrogate

\(\hat {y}_{add}(\mathbf {x})\)

Multifidelity surrogate that uses additive correction and where the prediction is performed using low-fidelity simulations

\(\hat {y}_{comp}(\mathbf {x})\)

Multifidelity surrogate that uses comprehensive correction and where the prediction is performed using low-fidelity simulations



We gratefully acknowledge the contribution of a reviewer, Professor Andy Keane, who suggested trying the additive Kriging.

Funding sources

This work was partially supported by the Center for Compressible Multiphase Turbulence, the U.S. Department of Energy, National Nuclear Security Administration, Advanced Simulation and Computing Program, as a Cooperative Agreement under the Predictive Science Academic Alliance Program, under Contract No. DE-NA0002378.

This work was partially supported by the French National Research Agency (ANR) through the ReBReD project under grant ANR-16-CE10-0002 and by a ONERA internal project MUFIN dedicated about multi-fidelity.

This work was partially performed under U.S. Government contract 89233218CNA000001 for Los Alamos National Laboratory (LANL), which is operated by Triad National Security, LLC for the U.S. Department of Energy/National Nuclear Security Administration. Approved for public release LA-UR-19-22491.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material

158_2019_2387_MOESM1_ESM.tex (120 kb)
(TEX 119 KB)


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Los Alamos National LaboratoryLos AlamosUSA
  2. 2.ONERA/DTIS, Université de ToulouseToulouseFrance
  3. 3.Université de Toulouse, CNRS, UPS, INSA, ISAE, Mines Albi, Institut, Clément Ader (ICA)ToulouseFrance
  4. 4.University of FloridaGainesvilleUSA

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