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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 |
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Pages (from-to) | 93-116 |
Number of pages | 24 |
Journal | SIAM/ASA Journal on Uncertainty Quantification |
Volume | 7 |
Issue number | 1 |
Early online date | 15 Jan 2019 |
DOIs | |
Publication status | Published - 31 Dec 2019 |
Bibliographical note
22 pages, 4 figuresKeywords
- 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
Fingerprint
Dive into the research topics of 'Continuous Level Monte Carlo and Sample-Adaptive Model Hierarchies'. Together they form a unique fingerprint.Projects
- 2 Finished
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Multiscale Modelling of Aerospace Composites
Butler, R. (PI) & Scheichl, R. (CoI)
Engineering and Physical Sciences Research Council
6/01/14 → 5/02/18
Project: Research council
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Multilevel Monte Carlo Methods for Elliptic Problems
Scheichl, R. (PI)
Engineering and Physical Sciences Research Council
1/07/11 → 30/06/14
Project: Research council