### Abstract

In failure probability estimation, importance sampling constructs a biasing distribution that targets the failure event such that a small number of model evaluations is sufficient to achieve a Monte Carlo estimate of the failure probability with an acceptable accuracy; however, the construction of the biasing distribution often requires a large number of model evaluations, which can become computationally expensive. We present a mixed multifidelity importance sampling (MMFIS) approach that leverages computationally cheap but erroneous surrogate models for the construction of the biasing distribution and that uses the original high-fidelity model to guarantee unbiased estimates of the failure probability. The key property of our MMFIS estimator is that it can leverage multiple surrogate models for the construction of the biasing distribution, instead of a single surrogate model alone. We show that our MMFIS estimator has a mean-squared error that is up to a constant lower than the mean-squared errors of the corresponding estimators that uses any of the given surrogate models alone—even in settings where no information about the approximation qualities of the surrogate models is available. In particular, our MMFIS approach avoids the problem of selecting the surrogate model that leads to the estimator with the lowest mean-squared error, which is challenging if the approximation quality of the surrogate models is unknown. We demonstrate our MMFIS approach on numerical examples, where we achieve orders of magnitude speedups compared to using the high-fidelity model only.

Original language | English (US) |
---|---|

Pages (from-to) | 61-75 |

Number of pages | 15 |

Journal | Journal of Computational Physics |

Volume | 341 |

DOIs | |

State | Published - Jul 15 2017 |

### Fingerprint

### Keywords

- Failure probability estimation
- Model reduction
- Multifidelity
- Rare event simulation
- Surrogate modeling
- Uncertainty quantification

### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)
- Computer Science Applications

### Cite this

*Journal of Computational Physics*,

*341*, 61-75. https://doi.org/10.1016/j.jcp.2017.04.012

**Combining multiple surrogate models to accelerate failure probability estimation with expensive high-fidelity models.** / Peherstorfer, Benjamin; Kramer, Boris; Willcox, Karen.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 341, pp. 61-75. https://doi.org/10.1016/j.jcp.2017.04.012

}

TY - JOUR

T1 - Combining multiple surrogate models to accelerate failure probability estimation with expensive high-fidelity models

AU - Peherstorfer, Benjamin

AU - Kramer, Boris

AU - Willcox, Karen

PY - 2017/7/15

Y1 - 2017/7/15

N2 - In failure probability estimation, importance sampling constructs a biasing distribution that targets the failure event such that a small number of model evaluations is sufficient to achieve a Monte Carlo estimate of the failure probability with an acceptable accuracy; however, the construction of the biasing distribution often requires a large number of model evaluations, which can become computationally expensive. We present a mixed multifidelity importance sampling (MMFIS) approach that leverages computationally cheap but erroneous surrogate models for the construction of the biasing distribution and that uses the original high-fidelity model to guarantee unbiased estimates of the failure probability. The key property of our MMFIS estimator is that it can leverage multiple surrogate models for the construction of the biasing distribution, instead of a single surrogate model alone. We show that our MMFIS estimator has a mean-squared error that is up to a constant lower than the mean-squared errors of the corresponding estimators that uses any of the given surrogate models alone—even in settings where no information about the approximation qualities of the surrogate models is available. In particular, our MMFIS approach avoids the problem of selecting the surrogate model that leads to the estimator with the lowest mean-squared error, which is challenging if the approximation quality of the surrogate models is unknown. We demonstrate our MMFIS approach on numerical examples, where we achieve orders of magnitude speedups compared to using the high-fidelity model only.

AB - In failure probability estimation, importance sampling constructs a biasing distribution that targets the failure event such that a small number of model evaluations is sufficient to achieve a Monte Carlo estimate of the failure probability with an acceptable accuracy; however, the construction of the biasing distribution often requires a large number of model evaluations, which can become computationally expensive. We present a mixed multifidelity importance sampling (MMFIS) approach that leverages computationally cheap but erroneous surrogate models for the construction of the biasing distribution and that uses the original high-fidelity model to guarantee unbiased estimates of the failure probability. The key property of our MMFIS estimator is that it can leverage multiple surrogate models for the construction of the biasing distribution, instead of a single surrogate model alone. We show that our MMFIS estimator has a mean-squared error that is up to a constant lower than the mean-squared errors of the corresponding estimators that uses any of the given surrogate models alone—even in settings where no information about the approximation qualities of the surrogate models is available. In particular, our MMFIS approach avoids the problem of selecting the surrogate model that leads to the estimator with the lowest mean-squared error, which is challenging if the approximation quality of the surrogate models is unknown. We demonstrate our MMFIS approach on numerical examples, where we achieve orders of magnitude speedups compared to using the high-fidelity model only.

KW - Failure probability estimation

KW - Model reduction

KW - Multifidelity

KW - Rare event simulation

KW - Surrogate modeling

KW - Uncertainty quantification

UR - http://www.scopus.com/inward/record.url?scp=85017470466&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85017470466&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2017.04.012

DO - 10.1016/j.jcp.2017.04.012

M3 - Article

VL - 341

SP - 61

EP - 75

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -