### Abstract

Estimating statistics of model outputs with the Monte Carlo method often requires a large number of model evaluations. This leads to long runtimes if the model is expensive to evaluate. Importance sampling is one approach that can lead to a reduction in the number of model evaluations. Importance sampling uses a biasing distribution to sample the model more efficiently, but generating such a biasing distribution can be difficult and usually also requires model evaluations. A different strategy to speed up Monte Carlo sampling is to replace the computationally expensive high-fidelity model with a computationally cheap surrogate model; however, because the surrogate model outputs are only approximations of the high-fidelity model outputs, the estimate obtained using a surrogate model is in general biased with respect to the estimate obtained using the high-fidelity model. We introduce a multifidelity importance sampling (MFIS) method, which combines evaluations of both the high-fidelity and a surrogate model. It uses a surrogate model to facilitate the construction of the biasing distribution, but relies on a small number of evaluations of the high-fidelity model to derive an unbiased estimate of the statistics of interest. We prove that the MFIS estimate is unbiased even in the absence of accuracy guarantees on the surrogate model itself. The MFIS method can be used with any type of surrogate model, such as projection-based reduced-order models and data-fit models. Furthermore, the MFIS method is applicable to black-box models, i.e., where only inputs and the corresponding outputs of the high-fidelity and the surrogate model are available but not the details of the models themselves. We demonstrate on nonlinear and time-dependent problems that our MFIS method achieves speedups of up to several orders of magnitude compared to Monte Carlo with importance sampling that uses the high-fidelity model only.

Original language | English (US) |
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Pages (from-to) | 490-509 |

Number of pages | 20 |

Journal | Computer Methods in Applied Mechanics and Engineering |

Volume | 300 |

DOIs | |

State | Published - Mar 1 2016 |

### Fingerprint

### Keywords

- Importance sampling
- Monte Carlo method
- Multifidelity methods
- Surrogate modeling

### ASJC Scopus subject areas

- Computational Mechanics
- Mechanics of Materials
- Mechanical Engineering
- Physics and Astronomy(all)
- Computer Science Applications

### Cite this

*Computer Methods in Applied Mechanics and Engineering*,

*300*, 490-509. https://doi.org/10.1016/j.cma.2015.12.002

**Multifidelity importance sampling.** / Peherstorfer, Benjamin; Cui, Tiangang; Marzouk, Youssef; Willcox, Karen.

Research output: Contribution to journal › Article

*Computer Methods in Applied Mechanics and Engineering*, vol. 300, pp. 490-509. https://doi.org/10.1016/j.cma.2015.12.002

}

TY - JOUR

T1 - Multifidelity importance sampling

AU - Peherstorfer, Benjamin

AU - Cui, Tiangang

AU - Marzouk, Youssef

AU - Willcox, Karen

PY - 2016/3/1

Y1 - 2016/3/1

N2 - Estimating statistics of model outputs with the Monte Carlo method often requires a large number of model evaluations. This leads to long runtimes if the model is expensive to evaluate. Importance sampling is one approach that can lead to a reduction in the number of model evaluations. Importance sampling uses a biasing distribution to sample the model more efficiently, but generating such a biasing distribution can be difficult and usually also requires model evaluations. A different strategy to speed up Monte Carlo sampling is to replace the computationally expensive high-fidelity model with a computationally cheap surrogate model; however, because the surrogate model outputs are only approximations of the high-fidelity model outputs, the estimate obtained using a surrogate model is in general biased with respect to the estimate obtained using the high-fidelity model. We introduce a multifidelity importance sampling (MFIS) method, which combines evaluations of both the high-fidelity and a surrogate model. It uses a surrogate model to facilitate the construction of the biasing distribution, but relies on a small number of evaluations of the high-fidelity model to derive an unbiased estimate of the statistics of interest. We prove that the MFIS estimate is unbiased even in the absence of accuracy guarantees on the surrogate model itself. The MFIS method can be used with any type of surrogate model, such as projection-based reduced-order models and data-fit models. Furthermore, the MFIS method is applicable to black-box models, i.e., where only inputs and the corresponding outputs of the high-fidelity and the surrogate model are available but not the details of the models themselves. We demonstrate on nonlinear and time-dependent problems that our MFIS method achieves speedups of up to several orders of magnitude compared to Monte Carlo with importance sampling that uses the high-fidelity model only.

AB - Estimating statistics of model outputs with the Monte Carlo method often requires a large number of model evaluations. This leads to long runtimes if the model is expensive to evaluate. Importance sampling is one approach that can lead to a reduction in the number of model evaluations. Importance sampling uses a biasing distribution to sample the model more efficiently, but generating such a biasing distribution can be difficult and usually also requires model evaluations. A different strategy to speed up Monte Carlo sampling is to replace the computationally expensive high-fidelity model with a computationally cheap surrogate model; however, because the surrogate model outputs are only approximations of the high-fidelity model outputs, the estimate obtained using a surrogate model is in general biased with respect to the estimate obtained using the high-fidelity model. We introduce a multifidelity importance sampling (MFIS) method, which combines evaluations of both the high-fidelity and a surrogate model. It uses a surrogate model to facilitate the construction of the biasing distribution, but relies on a small number of evaluations of the high-fidelity model to derive an unbiased estimate of the statistics of interest. We prove that the MFIS estimate is unbiased even in the absence of accuracy guarantees on the surrogate model itself. The MFIS method can be used with any type of surrogate model, such as projection-based reduced-order models and data-fit models. Furthermore, the MFIS method is applicable to black-box models, i.e., where only inputs and the corresponding outputs of the high-fidelity and the surrogate model are available but not the details of the models themselves. We demonstrate on nonlinear and time-dependent problems that our MFIS method achieves speedups of up to several orders of magnitude compared to Monte Carlo with importance sampling that uses the high-fidelity model only.

KW - Importance sampling

KW - Monte Carlo method

KW - Multifidelity methods

KW - Surrogate modeling

UR - http://www.scopus.com/inward/record.url?scp=84949895033&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84949895033&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2015.12.002

DO - 10.1016/j.cma.2015.12.002

M3 - Article

VL - 300

SP - 490

EP - 509

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0374-2830

ER -