Abstract

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 Schrödinger equation with random initial conditions.

Keywords

  1. large deviation theory
  2. extreme events
  3. optimal control
  4. nonlinear Schrödinger equation
  5. solitons

MSC codes

  1. 60F10
  2. 65K10
  3. 49J20
  4. 76B25

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Information & Authors

Information

Published In

cover image SIAM/ASA Journal on Uncertainty Quantification
SIAM/ASA Journal on Uncertainty Quantification
Pages: 1029 - 1059
ISSN (online): 2166-2525

History

Submitted: 4 September 2018
Accepted: 20 May 2019
Published online: 13 August 2019

Keywords

  1. large deviation theory
  2. extreme events
  3. optimal control
  4. nonlinear Schrödinger equation
  5. solitons

MSC codes

  1. 60F10
  2. 65K10
  3. 49J20
  4. 76B25

Authors

Affiliations

Funding Information

Ministero dell'Istruzione, dell'Università e della Ricerca https://doi.org/10.13039/501100003407 : 2018-2022
Division of Materials Research https://doi.org/10.13039/100000078 : DMR-1420073
National Science Foundation https://doi.org/10.13039/100000001 : DMS-1522767
Politecnico di Torino https://doi.org/10.13039/100013000

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