6,786 research outputs found
First order algorithms in variational image processing
Variational methods in imaging are nowadays developing towards a quite
universal and flexible tool, allowing for highly successful approaches on tasks
like denoising, deblurring, inpainting, segmentation, super-resolution,
disparity, and optical flow estimation. The overall structure of such
approaches is of the form ; where the functional is a data fidelity term also
depending on some input data and measuring the deviation of from such
and is a regularization functional. Moreover is a (often linear)
forward operator modeling the dependence of data on an underlying image, and
is a positive regularization parameter. While is often
smooth and (strictly) convex, the current practice almost exclusively uses
nonsmooth regularization functionals. The majority of successful techniques is
using nonsmooth and convex functionals like the total variation and
generalizations thereof or -norms of coefficients arising from scalar
products with some frame system. The efficient solution of such variational
problems in imaging demands for appropriate algorithms. Taking into account the
specific structure as a sum of two very different terms to be minimized,
splitting algorithms are a quite canonical choice. Consequently this field has
revived the interest in techniques like operator splittings or augmented
Lagrangians. Here we shall provide an overview of methods currently developed
and recent results as well as some computational studies providing a comparison
of different methods and also illustrating their success in applications.Comment: 60 pages, 33 figure
Ab-Initio Molecular Dynamics
Computer simulation methods, such as Monte Carlo or Molecular Dynamics, are
very powerful computational techniques that provide detailed and essentially
exact information on classical many-body problems. With the advent of ab-initio
molecular dynamics, where the forces are computed on-the-fly by accurate
electronic structure calculations, the scope of either method has been greatly
extended. This new approach, which unifies Newton's and Schr\"odinger's
equations, allows for complex simulations without relying on any adjustable
parameter. This review is intended to outline the basic principles as well as a
survey of the field. Beginning with the derivation of Born-Oppenheimer
molecular dynamics, the Car-Parrinello method and the recently devised
efficient and accurate Car-Parrinello-like approach to Born-Oppenheimer
molecular dynamics, which unifies best of both schemes are discussed. The
predictive power of this novel second-generation Car-Parrinello approach is
demonstrated by a series of applications ranging from liquid metals, to
semiconductors and water. This development allows for ab-initio molecular
dynamics simulations on much larger length and time scales than previously
thought feasible.Comment: 13 pages, 3 figure
Error correction based on partial information
We consider the decoding of linear and array codes from errors when we are
only allowed to download a part of the codeword. More specifically, suppose
that we have encoded data symbols using an code with code length
and dimension During storage, some of the codeword coordinates might
be corrupted by errors. We aim to recover the original data by reading the
corrupted codeword with a limit on the transmitting bandwidth, namely, we can
only download an proportion of the corrupted codeword. For a given
our objective is to design a code and a decoding scheme such that we
can recover the original data from the largest possible number of errors. A
naive scheme is to read coordinates of the codeword. This method
used in conjunction with MDS codes guarantees recovery from any errors. In this paper we show that we can instead read an
proportion from each of the codeword's coordinates. For a
well-designed MDS code, this method can guarantee recovery from errors, which is times more than the naive
method, and is also the maximum number of errors that an code can
correct by downloading only an proportion of the codeword. We present
two families of such optimal constructions and decoding schemes. One is a
Reed-Solomon code with evaluation points in a subfield and the other is based
on Folded Reed-Solomon codes. We further show that both code constructions
attain asymptotically optimal list decoding radius when downloading only a part
of the corrupted codeword. We also construct an ensemble of random codes that
with high probability approaches the upper bound on the number of correctable
errors when the decoder downloads an proportion of the corrupted
codeword.Comment: Extended version of the conference paper in ISIT 201
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio
Structure and Dynamics of the Instantaneous Water/Vapor Interface Revisited by Path-Integral and Ab-Initio Molecular Dynamics Simulations
The structure and dynamics of the water/vapor interface is revisited by means
of path-integral and second-generation Car-Parrinello ab-initio molecular
dynamics simulations in conjunction with an instantaneous surface definition
[A. P. Willard and D. Chandler, J. Phys. Chem. B 114, 1954 (2010)]. In
agreement with previous studies, we find that one of the OH bonds of the water
molecules in the topmost layer is pointing out of the water into the vapor
phase, while the orientation of the underlying layer is reversed. Therebetween,
an additional water layer is detected, where the molecules are aligned parallel
to the instantaneous water surface.Comment: 9 pages, 5 figure
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