A probabilistic decomposition-synthesis method for the quantification of rare events due to internal instabilities
We consider the problem of the probabilistic quantification of dynamical systems that have heavy-tailed characteristics. These heavy-tailed features are associated with rare transient responses due to the occurrence of internal instabilities. Systems with these properties can be found in a variety o...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Elsevier BV,
2018-12-07T21:02:02Z.
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| Subjects: | |
| Online Access: | Get fulltext |
| Summary: | We consider the problem of the probabilistic quantification of dynamical systems that have heavy-tailed characteristics. These heavy-tailed features are associated with rare transient responses due to the occurrence of internal instabilities. Systems with these properties can be found in a variety of areas including mechanics, fluids, and waves. Here we develop a computational method, a probabilistic decomposition-synthesis technique, that takes into account the nature of internal instabilities to inexpensively determine the non-Gaussian probability density function for any arbitrary quantity of interest. Our approach relies on the decomposition of the statistics into a 'non-extreme core', typically Gaussian, and a heavy-tailed component. This decomposition is in full correspondence with a partition of the phase space into a 'stable' region where we have no internal instabilities, and a region where non-linear instabilities lead to rare transitions with high probability. We quantify the statistics in the stable region using a Gaussian approximation approach, while the non-Gaussian distribution associated with the intermittently unstable regions of phase space is inexpensively computed through order-reduction methods that take into account the strongly nonlinear character of the dynamics. The probabilistic information in the two domains is analytically synthesized through a total probability argument. The proposed approach allows for the accurate quantification of non-Gaussian tails at more than 10 standard deviations, at a fraction of the cost associated with the direct Monte-Carlo simulations. We demonstrate the probabilistic decomposition-synthesis method for rare events for two dynamical systems exhibiting extreme events: a two-degree-of-freedom system of nonlinearly coupled oscillators, and in a nonlinear envelope equation characterizing the propagation of unidirectional water waves. Keywords: intermittency, heavy-tails, rare events, stochastic dynamical systems, rogue waves, uncertainty quantification. United States. Office of Naval Research (Grant N00014-14-1-0520) United States. Office of Naval Research (Grant N00014-15-1-2381) United States. Air Force. Office of Scientific Research (Grant FA9550-16-1-0231) United States. Army Research Office (Grant 66710-EG-YIP) United States. Defense Advanced Research Projects Agency (Grant N66001-15-2-4055) |
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