We introduce computationally efficient Monte Carlo methods for studying the statistics of stochastic differential equations in two distinct settings. In the first, we derive importance sampling methods for data assimilation when the noise in the model and observations are small. The methods are formulated in discrete time, where the "posterior" distribution we want to sample from can be analyzed in an accessible small noise expansion. We show that a "symmetrization" procedure akin to antithetic coupling can improve the order of accuracy of the sampling methods, which is illustrated with numerical examples. In the second setting, we develop "stochastic continuation" methods to estimate level sets for statistics of stochastic differential equations with respect to their parameters. We adapt Keller's Pseudo-Arclength continuation method to this setting using stochastic approximation, and generalized least squares regression. Furthermore, we show that the methods can be improved through the use of coupling methods to reduce the variance of the derivative estimates that are involved.
| Identifer | oai:union.ndltd.org:arizona.edu/oai:arizona.openrepository.com:10150/624570 |
| Date | January 2017 |
| Creators | Leach, Andrew Bradford, Leach, Andrew Bradford |
| Contributors | Lin, Kevin, Morzfeld, Matthias, Lin, Kevin, Morzfeld, Matthias, Lega, Joceline, Sethuraman, Sunder, Venkataramani, Shankar |
| Publisher | The University of Arizona. |
| Source Sets | University of Arizona |
| Language | en_US |
| Detected Language | English |
| Type | text, Electronic Dissertation |
| Rights | Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. |
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