41 research outputs found
String Method for Generalized Gradient Flows: Computation of Rare Events in Reversible Stochastic Processes
Rare transitions in stochastic processes can often 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. This in turn suggests a numerical algorithm to
compute the transition trajectories in configuration space efficiently, based
on the string method traditionally restricted only to gradient diffusions
Lagrangian and geometric analysis of finite-time Euler singularities
We present a numerical method of analyzing possibly singular incompressible
3D Euler flows using massively parallel high-resolution adaptively refined
numerical simulations up to 8192^3 mesh points. Geometrical properties of
Lagrangian vortex line segments are used in combination with analytical
non-blowup criteria by Deng et al [Commun. PDE 31 (2006)] to reliably
distinguish between singular and near-singular flow evolution. We then apply
the presented technique to a class of high-symmetry initial conditions and
present numerical evidence against the formation of a finite-time singularity
in this case.Comment: arXiv admin note: text overlap with arXiv:1210.253
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
Non-equilibrium transitions in multiscale systems with a bifurcating slow manifold
Noise-induced transitions between metastable fixed points in systems evolving
on multiple time scales are analyzed in situations where the time scale
separation gives rise to a slow manifold with bifurcation. This analysis is
performed within the realm of large deviation theory. It is shown that these
non-equilibrium transitions make use of a reaction channel created by the
bifurcation structure of the slow manifold, leading to vastly increased
transition rates. Several examples are used to illustrate these findings,
including an insect outbreak model, a system modeling phase separation in the
presence of evaporation, and a system modeling transitions in active matter
self-assembly. The last example involves a spatially extended system modeled by
a stochastic partial differential equation
Extreme event quantification in dynamical systems with random components
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 Schr\"odinger equation
(NLSE) with random initial conditions
Instanton filtering for the stochastic Burgers equation
We address the question whether one can identify instantons in direct
numerical simulations of the stochastically driven Burgers equation. For this
purpose, we first solve the instanton equations using the Chernykh-Stepanov
method [Phys. Rev. E 64, 026306 (2001)]. These results are then compared to
direct numerical simulations by introducing a filtering technique to extract
prescribed rare events from massive data sets of realizations. Using this
approach we can extract the entire time history of the instanton evolution
which allows us to identify the different phases predicted by the direct method
of Chernykh and Stepanov with remarkable agreement