### Abstract

In this paper, we present a generalization of the multilevel Monte Carlo (MLMC) method to a setting where the level parameter is a continuous variable. This continuous level Monte Carlo (CLMC) estimator provides a natural framework in PDE applications to adapt the model hierarchy to each sample. In addition, it can be made unbiased with respect to the expected value of the true quantity of interest provided the quantity of interest converges sufficiently fast. The practical implementation of the CLMC estimator is based on interpolating actual evaluations of the quantity of interest at a finite number of resolutions. As our new level parameter, we use the logarithm of a goal-oriented finite element error estimator for the accuracy of the quantity of interest. We prove the unbiasedness, as well as a complexity theorem that shows the same rate of complexity for CLMC as for MLMC. Finally, we provide some numerical evidence to support our theoretical results, by successfully testing CLMC on a standard PDE test problem. The numerical experiments demonstrate clear gains for samplewise adaptive refinement strategies over uniform refinements.

Original language | English |
---|---|

Pages (from-to) | 93-116 |

Number of pages | 24 |

Journal | SIAM/ASA Journal on Uncertainty Quantification |

Volume | 7 |

Issue number | 1 |

DOIs | |

Publication status | Published - 15 Jan 2019 |

### Keywords

- math.NA
- CLMC
- Adaptivity
- Complexity theorem
- Heterogeneous elliptic PDEs
- Unbiased estimator
- Multilevel Monte Carlo

### ASJC Scopus subject areas

- Applied Mathematics
- Discrete Mathematics and Combinatorics
- Statistics and Probability
- Statistics, Probability and Uncertainty
- Modelling and Simulation

### Cite this

**Continuous Level Monte Carlo and Sample-Adaptive Model Hierarchies.** / Detommaso, Gianluca; Dodwell, Tim; Scheichl, Robert.

Research output: Contribution to journal › Article

*SIAM/ASA Journal on Uncertainty Quantification*, vol. 7, no. 1, pp. 93-116. https://doi.org/10.1137/18M1172259

}

TY - JOUR

T1 - Continuous Level Monte Carlo and Sample-Adaptive Model Hierarchies

AU - Detommaso, Gianluca

AU - Dodwell, Tim

AU - Scheichl, Robert

N1 - 22 pages, 4 figures

PY - 2019/1/15

Y1 - 2019/1/15

N2 - In this paper, we present a generalization of the multilevel Monte Carlo (MLMC) method to a setting where the level parameter is a continuous variable. This continuous level Monte Carlo (CLMC) estimator provides a natural framework in PDE applications to adapt the model hierarchy to each sample. In addition, it can be made unbiased with respect to the expected value of the true quantity of interest provided the quantity of interest converges sufficiently fast. The practical implementation of the CLMC estimator is based on interpolating actual evaluations of the quantity of interest at a finite number of resolutions. As our new level parameter, we use the logarithm of a goal-oriented finite element error estimator for the accuracy of the quantity of interest. We prove the unbiasedness, as well as a complexity theorem that shows the same rate of complexity for CLMC as for MLMC. Finally, we provide some numerical evidence to support our theoretical results, by successfully testing CLMC on a standard PDE test problem. The numerical experiments demonstrate clear gains for samplewise adaptive refinement strategies over uniform refinements.

AB - In this paper, we present a generalization of the multilevel Monte Carlo (MLMC) method to a setting where the level parameter is a continuous variable. This continuous level Monte Carlo (CLMC) estimator provides a natural framework in PDE applications to adapt the model hierarchy to each sample. In addition, it can be made unbiased with respect to the expected value of the true quantity of interest provided the quantity of interest converges sufficiently fast. The practical implementation of the CLMC estimator is based on interpolating actual evaluations of the quantity of interest at a finite number of resolutions. As our new level parameter, we use the logarithm of a goal-oriented finite element error estimator for the accuracy of the quantity of interest. We prove the unbiasedness, as well as a complexity theorem that shows the same rate of complexity for CLMC as for MLMC. Finally, we provide some numerical evidence to support our theoretical results, by successfully testing CLMC on a standard PDE test problem. The numerical experiments demonstrate clear gains for samplewise adaptive refinement strategies over uniform refinements.

KW - math.NA

KW - CLMC

KW - Adaptivity

KW - Complexity theorem

KW - Heterogeneous elliptic PDEs

KW - Unbiased estimator

KW - Multilevel Monte Carlo

UR - http://www.scopus.com/inward/record.url?scp=85070685706&partnerID=8YFLogxK

U2 - 10.1137/18M1172259

DO - 10.1137/18M1172259

M3 - Article

VL - 7

SP - 93

EP - 116

JO - SIAM/ASA Journal on Uncertainty Quantification

JF - SIAM/ASA Journal on Uncertainty Quantification

SN - 2166-2525

IS - 1

ER -