Bounding extreme events in nonlinear dynamics using convex optimization

Abstract

We study a convex optimization framework for bounding extreme events in nonlinear dynamical systems governed by ordinary or partial differential equations (ODEs or PDEs). This framework bounds from above the largest value of an observable along trajectories that start from a chosen set and evolve over a finite or infinite time interval. The approach needs no explicit trajectories. Instead, it requires constructing suitably constrained auxiliary functions that depend on the state variables and possibly on time. Minimizing bounds over auxiliary functions is a convex problem dual to the non-convex maximization of the observable along trajectories. This duality is strong, meaning that auxiliary functions give arbitrarily sharp bounds, for sufficiently regular ODEs evolving over a finite time on a compact domain. When these conditions fail, strong duality may or may not hold; both situations are illustrated by examples. We also show that near-optimal auxiliary functions can be used to construct spacetime sets that localize trajectories leading to extreme events. Finally, in the case of polynomial ODEs and observables, we describe how polynomial auxiliary functions of fixed degree can be optimized numerically using polynomial optimization. The corresponding bounds become sharp as the polynomial degree is raised if strong duality and mild compactness assumptions hold. Analytical and computational ODE examples illustrate the construction of bounds and the identification of extreme trajectories, along with some limitations. As an analytical PDE example, we bound the maximum fractional enstrophy of solutions to the Burgers equation with fractional diffusion.

Revised according to comments by reviewers. Added references and rearranged introduction, conclusions, and proofs. 38 pages, 7 figures, 4 tables, 4 appendices, 87 references

References in corpus (7)

• An optimisation approach for analysing nonlinear stability with transition to turbulence in fluids as an exemplar
• A variational approach to probing extreme events in turbulent dynamical systems
• Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems
• Bounds on mean energy in the Kuramoto-Sivashinsky equation computed using semidefinite programming
• Fast ADMM for sum-of-squares programs using partial orthogonality
• Exploiting Sparsity in the Coefficient Matching Conditions in Sum-of-Squares Programming using ADMM
• Optimal mixing in two-dimensional stratified plane Poiseuille flow at finite Péclet and Richardson numbers

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• Maximum Amplification of Enstrophy in 3D Navier-Stokes Flows
• Systematic Search For Extreme and Singular Behavior in Some Fundamental Models of Fluid Mechanics
• Exploiting Term Sparsity in Moment-SOS hierarchy for Dynamical Systems
• A study of the double pendulum using polynomial optimization
• Minimum wave speeds in monostable reaction-diffusion equations: sharp bounds by polynomial optimization
• Uncertainty propagation for nonlinear dynamics: A polynomial optimization approach
• Peak Estimation of Time Delay Systems using Occupation Measures