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

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## Abstract

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.

## Keywords

Multifidelity Surrogates Symmetries Linear regression Kriging Co-Kriging## Nomenclature

*δ*(**x**)Discrepancy function

- \(\hat {\delta }(\mathbf {x})\)
Discrepancy function surrogate, also known as additive correction

*ρ*Constant scaling factor

*y*_{HF}(**x**)High-fidelity simulation

- \(\hat {y}_{HF}(\mathbf {x})\)
High-fidelity surrogate

*y*_{LF}(**x**)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

## Notes

### Acknowledgments

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

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