Rogue waves — enigmatic giants of the sea — were thought to be caused by two different mechanisms. But a new idea that borrows from the hinterlands of probability theory has the potential to predict them all.
A theory for rogue waves based on instantons—a mathematical concept developed in quantum chromodynamics—has been successfully tested in controlled laboratory experiments.
G. Dematteis, T. Grafke, and E. Vanden-Eijnden, Phys. Rev. X 9 (2019), 041057
A statistical theory of rogue waves is proposed and tested against experimental data collected in a long water tank where random waves with different degrees of nonlinearity are mechanically generated and free to propagate along the flume. Strong evidence is given that the rogue waves observed in the tank are hydrodynamic instantons, that is, saddle point configurations of the action associated with the stochastic model of the wave system. As shown here, these hydrodynamic instantons are complex spatio-temporal wave field configurations, which can be defined using the mathematical framework of Large Deviation Theory and calculated via tailored numerical methods. These results indicate that the instantons describe equally well rogue waves that originate from a simple linear superposition mechanism (in weakly nonlinear conditions) or from a nonlinear focusing one (in strongly nonlinear conditions), paving the way for the development of a unified explanation to rogue wave formation.
G. Dematteis, T. Grafke, and E. Vanden-Eijnden, J. Uncertainty Quantification 7 (3), (2019), 1029
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 (NLSE) with random initial conditions.
T. Grafke, and E. Vanden-Eijnden, Chaos 29 (2019), 063118
An overview of rare events algorithms based on large deviation theory (LDT) is presented. It covers a range of numerical schemes to compute the large deviation minimizer in various setups, and discusses best practices, common pitfalls, and implementation trade-offs. Generalizations, extensions, and improvements of the minimum action methods are proposed. These algorithms are tested on example problems which illustrate several common difficulties which arise e.g. when the forcing is degenerate or multiplicative, or the systems are infinite-dimensional. Generalizations to processes driven by non-Gaussian noises or random initial data and parameters are also discussed, along with the connection between the LDT-based approach reviewed here and other methods, such as stochastic field theory and optimal control. Finally, the integration of this approach in importance sampling methods using e.g. genealogical algorithms is explored.
T. Grafke, J. Stat. Mech 2019/4 (2019) 043206
Rare transitions in stochastic processes often can be rigorously described via an underlying large deviation principle. Recent breakthroughs in the classification of reversible stochastic processes as gradient flows have led to a connection of large deviation principles to a generalized gradient structure. Here, we show that, as a consequence, metastable transitions in these reversible processes can be interpreted as heteroclinic orbits of the generalized gradient flow. As a consequence, this suggests a numerical algorithm to compute the transition trajectories in configuration space efficiently, based on the string method traditionally restricted only to gradient diffusions.
G. Dematteis, T. Grafke, and E. Vanden-Eijnden, Proc. Natl. Acad. Sci., 115 (2018), 855-860
The appearance of rogue waves in deep sea is investigated using the modified nonlinear Schrödinger (MNLS) equation with random initial conditions that are assumed to be Gaussian distributed, with a spectrum approximating the JONSWAP spectrum obtained from observations of the North Sea. It is shown that by supplementing the incomplete information contained in the JONSWAP spectrum with the MNLS dynamics one can reliably estimate the probability distribution of the sea surface elevation far in the tail at later times. Our results indicate that rogue waves occur when the system hit small pockets of wave configurations hidden in the core of their distribution that trigger large disturbances of the surface height via modulational instability. The rogue wave precursors in these pockets are wave patterns of regular height but with a very specific shape that is identified explicitly, thereby allowing for early detection. The method proposed here builds on tools from large deviation theory that reduce the calculation of the most likely rogue wave precursors to an optimization problem that can be solved efficiently.
In this project, Rogue Waves in deep sea are investigated using the modified nonlinear Schrödinger (MNLS) equation in one spatial-dimension with random initial conditions. The initial conditions approximate realistic conditions of a uni-directional sea state, taken as the JONSWAP spectrum. It is shown that one can use the incomplete information contained in this spectrum as prior and supplement this information with the MNLS dynamics to reliably estimate the probability distribution of the sea surface elevation far in the tail at later times. The Rogue Wave events encountered in numerical simulations, as well as in experiments in a 270m wave channel in Norway, agree with their most likely (instanton) configuration obtained from the theory.
T. Grafke, M. Cates, and E. Vanden-Eijnden, Phys. Rev. Lett., 119 (2017), 188003
We model an enclosed system of bacteria, whose motility-induced phase separation is coupled to slow population dynamics. Without noise, the system shows both static phase separation and a limit cycle, in which a rising global population causes a dense bacterial colony to form, which then declines by local cell death, before dispersing to re-initiate the cycle. Adding fluctuations, we find that static colonies are now metastable, moving between spatial locations via rare and strongly nonequilibrium pathways, whereas the limit cycle becomes quasi-periodic such that after each redispersion event the next colony forms in a random location. These results, which resemble some aspects of the biofilm-planktonic life cycle, can be explained by combining tools from large deviation theory with a bifurcation analysis in which the global population density plays the role of control parameter.
Active materials can self-organize in many more ways than their equilibrium counterparts. For example, self-propelled particles whose velocity decreases with their density can display motility-induced phase separation (MIPS), a phenomenon building on a positive feedback loop in which patterns emerge in locations where the particles slow down. Here, we investigate the effects of intrinsic fluctuations in the system's dynamics on MIPS. We show that these fluctuations can lead to transitions between metastable patterns. The pathway and rate of these transitions is analyzed within the realm of large deviation theory, and they are shown to proceed in a very different way than one would predict from arguments based on detailed-balance and microscopic reversibility.