Journal
SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION
Volume 7, Issue 3, Pages 1029-1059Publisher
SIAM PUBLICATIONS
DOI: 10.1137/18M1211003
Keywords
large deviation theory; extreme events; optimal control; nonlinear Schrodinger equation; solitons
Funding
- joint Math PhD program of Politecnico di Torino
- joint Math PhD program of Universit di Torino
- MIUR grant Dipartimenti di Eccellenza 2018-2022
- National Science Foundation Materials Research Science and Engineering Center Program [DMR-1420073, DMS-1522767]
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A central problem in uncertainty quantification is how to characterize the impact that our incomplete knowledge about models has on the predictions we make from them. This question naturally lends itself to a probabilistic formulation, by making the unknown model parameters random with given statistics. Here this approach is used in concert with tools from large deviation theory (LDT) and optimal control to estimate the probability that some observables in a dynamical system go above a large threshold after some time, given the prior statistical information about the system's parameters and/or its initial conditions. Specifically, it is established under which conditions such extreme events occur in a predictable way, as the minimizer of the LDT action functional. It is also shown how this minimization can be numerically performed in an efficient way using tools from optimal control. These findings are illustrated on the examples of a rod with random elasticity pulled by a time-dependent force, and the nonlinear Schrodinger equation with random initial conditions.
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