14,109 research outputs found
On Quasi-Newton Forward--Backward Splitting: Proximal Calculus and Convergence
We introduce a framework for quasi-Newton forward--backward splitting
algorithms (proximal quasi-Newton methods) with a metric induced by diagonal
rank- symmetric positive definite matrices. This special type of
metric allows for a highly efficient evaluation of the proximal mapping. The
key to this efficiency is a general proximal calculus in the new metric. By
using duality, formulas are derived that relate the proximal mapping in a
rank- modified metric to the original metric. We also describe efficient
implementations of the proximity calculation for a large class of functions;
the implementations exploit the piece-wise linear nature of the dual problem.
Then, we apply these results to acceleration of composite convex minimization
problems, which leads to elegant quasi-Newton methods for which we prove
convergence. The algorithm is tested on several numerical examples and compared
to a comprehensive list of alternatives in the literature. Our quasi-Newton
splitting algorithm with the prescribed metric compares favorably against
state-of-the-art. The algorithm has extensive applications including signal
processing, sparse recovery, machine learning and classification to name a few.Comment: arXiv admin note: text overlap with arXiv:1206.115
ATK-ForceField: A New Generation Molecular Dynamics Software Package
ATK-ForceField is a software package for atomistic simulations using
classical interatomic potentials. It is implemented as a part of the Atomistix
ToolKit (ATK), which is a Python programming environment that makes it easy to
create and analyze both standard and highly customized simulations. This paper
will focus on the atomic interaction potentials, molecular dynamics, and
geometry optimization features of the software, however, many more advanced
modeling features are available. The implementation details of these algorithms
and their computational performance will be shown. We present three
illustrative examples of the types of calculations that are possible with
ATK-ForceField: modeling thermal transport properties in a silicon germanium
crystal, vapor deposition of selenium molecules on a selenium surface, and a
simulation of creep in a copper polycrystal.Comment: 28 pages, 9 figure
Cosmological Parameters from Observations of Galaxy Clusters
Studies of galaxy clusters have proved crucial in helping to establish the
standard model of cosmology, with a universe dominated by dark matter and dark
energy. A theoretical basis that describes clusters as massive,
multi-component, quasi-equilibrium systems is growing in its capability to
interpret multi-wavelength observations of expanding scope and sensitivity. We
review current cosmological results, including contributions to fundamental
physics, obtained from observations of galaxy clusters. These results are
consistent with and complementary to those from other methods. We highlight
several areas of opportunity for the next few years, and emphasize the need for
accurate modeling of survey selection and sources of systematic error.
Capitalizing on these opportunities will require a multi-wavelength approach
and the application of rigorous statistical frameworks, utilizing the combined
strengths of observers, simulators and theorists.Comment: 53 pages, 21 figures. To appear in Annual Review of Astronomy &
Astrophysic
Limited-Memory Greedy Quasi-Newton Method with Non-asymptotic Superlinear Convergence Rate
Non-asymptotic convergence analysis of quasi-Newton methods has gained
attention with a landmark result establishing an explicit superlinear rate of
O. The methods that obtain this rate, however, exhibit a
well-known drawback: they require the storage of the previous Hessian
approximation matrix or instead storing all past curvature information to form
the current Hessian inverse approximation. Limited-memory variants of
quasi-Newton methods such as the celebrated L-BFGS alleviate this issue by
leveraging a limited window of past curvature information to construct the
Hessian inverse approximation. As a result, their per iteration complexity and
storage requirement is O where is the size of the window
and is the problem dimension reducing the O computational cost and
memory requirement of standard quasi-Newton methods. However, to the best of
our knowledge, there is no result showing a non-asymptotic superlinear
convergence rate for any limited-memory quasi-Newton method. In this work, we
close this gap by presenting a limited-memory greedy BFGS (LG-BFGS) method that
achieves an explicit non-asymptotic superlinear rate. We incorporate
displacement aggregation, i.e., decorrelating projection, in post-processing
gradient variations, together with a basis vector selection scheme on variable
variations, which greedily maximizes a progress measure of the Hessian estimate
to the true Hessian. Their combination allows past curvature information to
remain in a sparse subspace while yielding a valid representation of the full
history. Interestingly, our established non-asymptotic superlinear convergence
rate demonstrates a trade-off between the convergence speed and memory
requirement, which to our knowledge, is the first of its kind. Numerical
results corroborate our theoretical findings and demonstrate the effectiveness
of our method
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