A primer on noise-induced transitions in applied dynamical systems

Noise plays a fundamental role in a wide variety of physical and biological dynamical systems. It can arise from an external forcing or due to random dynamics internal to the system. It is well established that even weak noise can result in large behavioral changes such as transitions between or escapes from quasi-stable states. These transitions can correspond to critical events such as failures or extinctions that make them essential phenomena to understand and quantify, despite the fact that their occurrence is rare. This article will provide an overview of the theory underlying the dynamics of rare events for stochastic models along with some example applications

Numerical computation of rare events via large deviation theory

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

The instanton method and its numerical implementation in fluid mechanics

A precise characterization of structures occurring in turbulent fluid flows at high Reynolds numbers is one of the last open problems of classical physics. In this review we discuss recent developments related to the application of instanton methods to turbulence. Instantons are saddle point configurations of the underlying path integrals. They are equivalent to minimizers of the related Freidlin-Wentzell action and known to be able to characterize rare events in such systems. While there is an impressive body of work concerning their analytical description, this review focuses on the question on how to compute these minimizers numerically. In a short introduction we present the relevant mathematical and physical background before we discuss the stochastic Burgers equation in detail. We present algorithms to compute instantons numerically by an efficient solution of the corresponding Euler-Lagrange equations. A second focus is the discussion of a recently developed numerical filtering technique that allows to extract instantons from direct numerical simulations. In the following we present modifications of the algorithms to make them efficient when applied to two- or three-dimensional fluid dynamical problems. We illustrate these ideas using the two-dimensional Burgers equation and the three-dimensional Navier-Stokes equations

On the optimal design of wall-to-wall heat transport

We consider the problem of optimizing heat transport through an incompressible fluid layer. Modeling passive scalar transport by advection-diffusion, we maximize the mean rate of total transport by a divergence-free velocity field. Subject to various boundary conditions and intensity constraints, we prove that the maximal rate of transport scales linearly in the r.m.s. kinetic energy and, up to possible logarithmic corrections, as the $1/3$rd power of the mean enstrophy in the advective regime. This makes rigorous a previous prediction on the near optimality of convection rolls for energy-constrained transport. Optimal designs for enstrophy-constrained transport are significantly more difficult to describe: we introduce a "branching" flow design with an unbounded number of degrees of freedom and prove it achieves nearly optimal transport. The main technical tool behind these results is a variational principle for evaluating the transport of candidate designs. The principle admits dual formulations for bounding transport from above and below. While the upper bound is closely related to the "background method", the lower bound reveals a connection between the optimal design problems considered herein and other apparently related model problems from mathematical materials science. These connections serve to motivate designs.Comment: Minor revisions from review. To appear in Comm. Pure Appl. Mat

Burgers Turbulence

The last decades witnessed a renewal of interest in the Burgers equation. Much activities focused on extensions of the original one-dimensional pressureless model introduced in the thirties by the Dutch scientist J.M. Burgers, and more precisely on the problem of Burgers turbulence, that is the study of the solutions to the one- or multi-dimensional Burgers equation with random initial conditions or random forcing. Such work was frequently motivated by new emerging applications of Burgers model to statistical physics, cosmology, and fluid dynamics. Also Burgers turbulence appeared as one of the simplest instances of a nonlinear system out of equilibrium. The study of random Lagrangian systems, of stochastic partial differential equations and their invariant measures, the theory of dynamical systems, the applications of field theory to the understanding of dissipative anomalies and of multiscaling in hydrodynamic turbulence have benefited significantly from progress in Burgers turbulence. The aim of this review is to give a unified view of selected work stemming from these rather diverse disciplines.Comment: Review Article, 49 pages, 43 figure

Higher Order Conditions in Nonlinear Optimal Control

The most widely used tool for the solution of optimal control problems is the Pontryagin Maximum Principle. But the Maximum Principle is, in general, only a necessary condition for optimality. It is therefore desirable to have supplementary conditions, for example second order sufficient conditions, which confirm optimality (at least locally) of an extremal arc, meaning one that satisfies the Maximum Principle. Standard second order sufficient conditions for optimality, when they apply, yield the information not only that the extremal is locally minimizing, but that it is also locally unique. There are problems of interest, however, where minimizers are not locally unique, owing to the fact that the cost is invariant under small perturbations of the extremal of a particular structure (translations, rotations or time-shifting). For such problems the standard second order conditions can never apply. The first contribution of this thesis is to develop new second order conditions for optimality of extremals which are applicable in some cases of interest when minimizers are not locally unique. The new conditions can, for example, be applied to problems with periodic boundary conditions when the cost is invariant under time translations. The second order conditions investigated here apply to normal extremals. These extremals satisfy the conditions of the Maximum Principle in normal form (with the cost multiplier taken to be 1). It is, therefore, of interest to know when the Maximum Principle applies in normal form. This issue is also addressed in this thesis, for optimal control problems that can be expressed as calculus of variations problems. Normality of the Maximum Principle follows from the fact that, under the regularity conditions developed, the highest time derivative of an extremal arc is essentially bounded. The thesis concludes with a brief account of possible future research directions