## Multifidelity importance sampling

## Abstract

Estimating statistics of model outputs with the Monte Carlo method usually requires a large number of model evaluations. This leads to long runtimes if the model is expensive to evaluate. Importance sampling is one method 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 accuracymore »

- Authors:

- Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)

- Publication Date:

- Research Org.:
- Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

- OSTI Identifier:
- 1548302

- Alternate Identifier(s):
- OSTI ID: 1397335

- Grant/Contract Number:
- SC0009297; FG02-08ER2585

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Computer Methods in Applied Mechanics and Engineering

- Additional Journal Information:
- Journal Volume: 300; Journal Issue: C; Journal ID: ISSN 0045-7825

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- Monte Carlo method; Importance sampling; Surrogate modeling; Multifidelity methods

### Citation Formats

```
Peherstorfer, Benjamin, Cui, Tiangang, Marzouk, Youssef, and Willcox, Karen. Multifidelity importance sampling. United States: N. p., 2015.
Web. doi:10.1016/j.cma.2015.12.002.
```

```
Peherstorfer, Benjamin, Cui, Tiangang, Marzouk, Youssef, & Willcox, Karen. Multifidelity importance sampling. United States. doi:10.1016/j.cma.2015.12.002.
```

```
Peherstorfer, Benjamin, Cui, Tiangang, Marzouk, Youssef, and Willcox, Karen. Thu .
"Multifidelity importance sampling". United States. doi:10.1016/j.cma.2015.12.002. https://www.osti.gov/servlets/purl/1548302.
```

```
@article{osti_1548302,
```

title = {Multifidelity importance sampling},

author = {Peherstorfer, Benjamin and Cui, Tiangang and Marzouk, Youssef and Willcox, Karen},

abstractNote = {Estimating statistics of model outputs with the Monte Carlo method usually requires a large number of model evaluations. This leads to long runtimes if the model is expensive to evaluate. Importance sampling is one method 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. Moreover, 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.},

doi = {10.1016/j.cma.2015.12.002},

journal = {Computer Methods in Applied Mechanics and Engineering},

number = C,

volume = 300,

place = {United States},

year = {2015},

month = {12}

}

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