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Quasi-Monte Carlo and multilevel Monte Carlo methods for computing posterior expectations in elliptic inverse problems
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Scheichl, R., Stuart, A. M. and Teckentrup, Aretha (2017) Quasi-Monte Carlo and multilevel Monte Carlo methods for computing posterior expectations in elliptic inverse problems. SIAM/ASA Journal on Uncertainty Quantification, 5 (1). pp. 493-518. doi:10.1137/16M1061692 ISSN 2166-2525.
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WRAP-quasi-Monte-Carlo-multilevel-methods-posterior-elliptic-Stuart-2017.pdf - Accepted Version - Requires a PDF viewer. Download (677Kb) | Preview |
Official URL: http://dx.doi.org/10.1137/16M1061692
Abstract
We are interested in computing the expectation of a functional of a PDE solution under a Bayesian posterior distribution. Using Bayes's rule, we reduce the problem to estimating the ratio of two related prior expectations. For a model elliptic problem, we provide a full convergence and complexity analysis of the ratio estimator in the case where Monte Carlo, quasi-Monte Carlo, or multilevel Monte Carlo methods are used as estimators for the two prior expectations. We show that the computational complexity of the ratio estimator to achieve a given accuracy is the same as the corresponding complexity of the individual estimators for the numerator and the denominator. We also include numerical simulations, in the context of the model elliptic problem, which demonstrate the effectiveness of the approach.
| Item Type: | Journal Article | ||||||
|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
| Library of Congress Subject Headings (LCSH): | Statistical functionals, Differential equations, Partial, Monte Carlo method | ||||||
| Journal or Publication Title: | SIAM/ASA Journal on Uncertainty Quantification | ||||||
| Publisher: | SIAM | ||||||
| ISSN: | 2166-2525 | ||||||
| Official Date: | 27 April 2017 | ||||||
| Dates: |
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| Volume: | 5 | ||||||
| Number: | 1 | ||||||
| Page Range: | pp. 493-518 | ||||||
| DOI: | 10.1137/16M1061692 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||
| Date of first compliant deposit: | 8 December 2017 | ||||||
| Date of first compliant Open Access: | 8 December 2017 |
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