2,033 research outputs found
Atomistic simulations of rare events using gentlest ascent dynamics
The dynamics of complex systems often involve thermally activated barrier
crossing events that allow these systems to move from one basin of attraction
on the high dimensional energy surface to another. Such events are ubiquitous,
but challenging to simulate using conventional simulation tools, such as
molecular dynamics. Recently, Weinan E et al. [Nonlinearity, 24(6),1831(2011)]
proposed a set of dynamic equations, the gentlest ascent dynamics (GAD), to
describe the escape of a system from a basin of attraction and proved that
solutions of GAD converge to index-1 saddle points of the underlying energy. In
this paper, we extend GAD to enable finite temperature simulations in which the
system hops between different saddle points on the energy surface. An effective
strategy to use GAD to sample an ensemble of low barrier saddle points located
in the vicinity of a locally stable configuration on the high dimensional
energy surface is proposed. The utility of the method is demonstrated by
studying the low barrier saddle points associated with point defect activity on
a surface. This is done for two representative systems, namely, (a) a surface
vacancy and ad-atom pair and (b) a heptamer island on the (111) surface of
copper.Comment: total 30 page
Multiscale Adaptive Representation of Signals: I. The Basic Framework
We introduce a framework for designing multi-scale, adaptive, shift-invariant
frames and bi-frames for representing signals. The new framework, called
AdaFrame, improves over dictionary learning-based techniques in terms of
computational efficiency at inference time. It improves classical multi-scale
basis such as wavelet frames in terms of coding efficiency. It provides an
attractive alternative to dictionary learning-based techniques for low level
signal processing tasks, such as compression and denoising, as well as high
level tasks, such as feature extraction for object recognition. Connections
with deep convolutional networks are also discussed. In particular, the
proposed framework reveals a drawback in the commonly used approach for
visualizing the activations of the intermediate layers in convolutional
networks, and suggests a natural alternative
Generalized Flows, Intrinsic Stochasticity, and Turbulent Transport
The study of passive scalar transport in a turbulent velocity field leads
naturally to the notion of generalized flows which are families of probability
distributions on the space of solutions to the associated ODEs, which no longer
satisfy the uniqueness theorem for ODEs. Two most natural regularizations of
this problem, namely the regularization via adding small molecular diffusion
and the regularization via smoothing out the velocity field are considered.
White-in-time random velocity fields are used as an example to examine the
variety of phenomena that take place when the velocity field is not spatially
regular. Three different regimes characterized by their degrees of
compressibility are isolated in the parameter space. In the regime of
intermediate compressibility, the two different regularizations give rise to
two different scaling behavior for the structure functions of the passive
scalar. Physically this means that the scaling depends on Prandtl number. In
the other two regimes the two different regularizations give rise to the same
generalized flows even though the sense of convergence can be very different.
The ``one force, one solution'' principle and the existence and uniqueness of
an invariant measure are established for the scalar field in the weakly
compressible regime, and for the difference of the scalar in the strongly
compressible regime.Comment: revised version, 16 pages, no figure
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