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.


  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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D. S. Agafontsev and V. E. Zakharov, Integrable turbulence and formation of rogue waves, Nonlinearity, 28 (2015), 2791.
N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, Recent progress in investigating optical rogue waves, J. Optics, 15 (2013), 060201.
H. Bailung, S. K. Sharma, and Y. Nakamura, Observation of Peregrine solitons in a multicomponent plasma with negative ions, Phys. Rev. Lett., 107 (2011), 255005.
M. Bertola and A. Tovbis, Universality for the focusing nonlinear Schrödinger equation at the gradient catastrophe point: rational breathers and poles of the tritronquée solution to Painlevé I, Comm. Pure Appl. Math., 66 (2013), pp. 678--752.
A. A. Borovkov and B. A. Rogozin, On the multi-dimensional central limit theorem, Theory Probab. Appl., 10 (1965), pp. 55--62, https://doi.org/10.1137/1110005.
A. Borz\`\i and V. Schulz, Computational Optimization of Systems Governed by Partial Differential Equations, Comput. Sci. Eng. 8, SIAM, Philadelphia, 2011.
M. Broniatowski and A. Fuchs, Tauberian theorems, Chernoff inequality, and the tail behavior of finite convolutions of distribution functions, Adv. Math., 116 (1995), pp. 12--33.
C. Bustamante, J. F. Marko, E. D. Siggia, and S. Smith, Entropic elasticity of lambda-phage DNA, Science, 265 (1994), pp. 1599--1599, https://doi.org/10.1126/science.8079175.
F. Cérou and A. Guyader, Adaptive multilevel splitting for rare event analysis, Stoch. Anal. Appl., 25 (2007), pp. 417--443.
P. Cluzel, A. Lebrun, C. Heller, R. Lavery, J.-L. Viovy, D. Chatenay, and F. Caron, DNA: an extensible molecule, Science, 271 (1996), pp. 792--794.
W. Cousins and T. P. Sapsis, Reduced-order precursors of rare events in unidirectional nonlinear water waves, J. Fluid Mech., 790 (2016), pp. 368--388, https://doi.org/10.1017/jfm.2016.13.
S. M. Cox and P. C. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys., 176 (2002), pp. 430--455, https://doi.org/10.1006/jcph.2002.6995.
G. Dematteis, T. Grafke, and E. Vanden-Eijnden, Rogue waves and large deviations in deep sea, Proc. Natl. Acad. Sci. USA, 115 (2018), pp. 855--860.
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Corrected Reprint of the Second Edition, Stoch. Model. Appl. Probab., 38, Springer, Berlin, 2010.
U. Einmahl and J. Kuelbs, Dominating points and large deviations for random vectors, Probab. Theory Related Fields, 105 (1996), pp. 529--543.
R. El Koussaifi, A. Tikan, A. Toffoli, S. Randoux, P. Suret, and M. Onorato, Spontaneous emergence of rogue waves in partially coherent waves: A quantitative experimental comparison between hydrodynamics and optics, Phys. Rev. E, 97 (2018), 012208.
M. Farazmand and T. P. Sapsis, A Variational Approach to Probing Extreme Events in Turbulent Dynamical Systems, preprint, https://arxiv.org/abs/1704.04116, 2017.
U. Frisch and D. Sornette, Extreme deviations and applications, J. Phys. I, 7 (1997), pp. 1155--1171.
C. Giardina, J. Kurchan, V. Lecomte, and J. Tailleur, Simulating rare events in dynamical processes, J. Stat. Phys., 145 (2011), pp. 787--811.
P. Glasserman, P. Heidelberger, P. Shahabuddin, and T. Zajic, Multilevel splitting for estimating rare event probabilities, Oper. Res., 47 (1999), pp. 585--600.
E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo Cimento, 20 (1961).
W. W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system, Numer. Math., 87 (2000), pp. 247--282.
M. Iltis, Sharp asymptotics of large deviations in $\mathbb{R}^d$, J. Theoret. Probab., 8 (1995), pp. 501--522.
M. Iltis, Sharp asymptotics of large deviations for general state-space Markov-additive chains in R-d, Statist. Probab. Lett., 47 (2000), pp. 365--380.
J. L. Jensen, Saddlepoint Approximations, Oxford Statist. Sci. Ser. 16, Oxford University Press, New York, 1995.
S. Juneja and P. Shahabuddin, Rare-event simulation techniques: An introduction and recent advances, Handbooks Oper. Res. Manag. Sci., 13 (2006), pp. 291--350.
A.-K. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff PDEs, SIAM J. Sci. Comput., 26 (2005), pp. 1214--1233, https://doi.org/10.1137/S1064827502410633.
B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, The Peregrine soliton in nonlinear fibre optics, Nature Phys., 6 (2010), pp. 790--795.
J. Kuelbs, Large deviation probabilities and dominating points for open convex sets: nonlogarithmic behavior, Ann. Probab., 28 (2000), pp. 1259--1279.
F. Lankaš, J. Šponer, P. Hobza, and J. Langowski, Sequence-dependent elastic properties of DNA, J. Mol. Biol., 299 (2000), pp. 695--709.
P. Ney, Dominating points and the asymptotics of large deviations for random walk on $\mathbb{R}^d$, Ann. Probab., 11 (1983), pp. 158--167.
M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. Arecchi, Rogue waves and their generating mechanisms in different physical contexts, Phys. Rep., 528 (2013), pp. 47--89, https://doi.org/10.1016/j.physrep.2013.03.001.
A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics, Phys. Rep., 542 (2014), pp. 1--132.
L. P. Pitaevsky, Vortex lines in an imperfect Bose gas, Soviet Phys. J. Exp. Theoret. Phys., 13 (1961).
R.-E. Plessix, A review of the adjoint-state method for computing the gradient of a functional with geophysical applications, Geophys. J. Int., 167 (2006), pp. 495--503.
F. Ragone, J. Wouters, and F. Bouchet, Computation of extreme heat waves in climate models using a large deviation algorithm, Proc. Natl. Acad. Sci. USA, 115 (2018), 201712645.
S. Randoux, P. Walczak, M. Onorato, and P. Suret, Intermittency in integrable turbulence, Phys. Rev. Lett., 113 (2014), 113902.
P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, Single-shot observation of optical rogue waves in integrable turbulence using time microscopy, Nat. Commun., 7 (2016).
J. Tailleur and J. Kurchan, Probing rare physical trajectories with Lyapunov weighted dynamics, Nat. Phys., 3 (2007), p. 203.
A. Tikan, C. Billet, G. El, A. Tovbis, M. Bertola, T. Sylvestre, F. Gustave, S. Randoux, G. Genty, P. Suret, and J. M. Dudley, Universality of the Peregrine soliton in the focusing dynamics of the cubic nonlinear Schrödinger equation, Phys. Rev. Lett., 119 (2017), 033901.
F. Tröltzsch, Optimal Control of Partial Differential Equations, Grad. Stud. Math. 112, AMS, Providence, RI, 2010.
E. Vanden-Eijnden and J. Weare, Rare Event Simulation of Small Noise Diffusions, Comm. Pure Appl. Math., 65 (2012), pp. 1770--1803.
S. R. S. Varadhan, Large deviations, Courant Lect. Notes Math. 27, AMS, 2016.
A. Walther, Automatic differentiation of explicit Runge-Kutta methods for optimal control, Comput. Optim. Appl., 36 (2007), pp. 83--108.
L. C. Wilcox, G. Stadler, T. Bui-Thanh, and O. Ghattas, Discretely exact derivatives for hyperbolic PDE-constrained optimization problems discretized by the discontinuous Galerkin method, J. Sci. Comput., 63 (2015), pp. 138--162.
S. Wright and J. Nocedal, Numerical Optimization, Springer Ser. Oper. Res., Springer, Berlin, 2000.
V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), pp. 190--194.
V. E. Zakharov, Turbulence in integrable systems, Stud. Appl. Math., 122 (2009), pp. 219--234.

Information & Authors


Published In

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


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


  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



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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