5,810 research outputs found
Pseudo-Exponential-Type Solutions of Wave Equations Depending on Several Variables
Using matrix identities, we construct explicit pseudo-exponential-type
solutions of linear Dirac, Loewner and Schr\"odinger equations depending on two
variables and of nonlinear wave equations depending on three variables
Variable Metric Random Pursuit
We consider unconstrained randomized optimization of smooth convex objective
functions in the gradient-free setting. We analyze Random Pursuit (RP)
algorithms with fixed (F-RP) and variable metric (V-RP). The algorithms only
use zeroth-order information about the objective function and compute an
approximate solution by repeated optimization over randomly chosen
one-dimensional subspaces. The distribution of search directions is dictated by
the chosen metric.
Variable Metric RP uses novel variants of a randomized zeroth-order Hessian
approximation scheme recently introduced by Leventhal and Lewis (D. Leventhal
and A. S. Lewis., Optimization 60(3), 329--245, 2011). We here present (i) a
refined analysis of the expected single step progress of RP algorithms and
their global convergence on (strictly) convex functions and (ii) novel
convergence bounds for V-RP on strongly convex functions. We also quantify how
well the employed metric needs to match the local geometry of the function in
order for the RP algorithms to converge with the best possible rate.
Our theoretical results are accompanied by numerical experiments, comparing
V-RP with the derivative-free schemes CMA-ES, Implicit Filtering, Nelder-Mead,
NEWUOA, Pattern-Search and Nesterov's gradient-free algorithms.Comment: 42 pages, 6 figures, 15 tables, submitted to journal, Version 3:
majorly revised second part, i.e. Section 5 and Appendi
Two New Bounds on the Random-Edge Simplex Algorithm
We prove that the Random-Edge simplex algorithm requires an expected number
of at most 13n/sqrt(d) pivot steps on any simple d-polytope with n vertices.
This is the first nontrivial upper bound for general polytopes. We also
describe a refined analysis that potentially yields much better bounds for
specific classes of polytopes. As one application, we show that for
combinatorial d-cubes, the trivial upper bound of 2^d on the performance of
Random-Edge can asymptotically be improved by any desired polynomial factor in
d.Comment: 10 page
Cellular properties of convulsant-treated rat neo-cortical neurons during postnatal development
Quantum-noise quenching in atomic tweezers
The efficiency of extracting single atoms or molecules from an ultracold
bosonic reservoir is theoretically investigated for a protocol based on lasers,
coupling the hyperfine state in which the atoms form a condensate to another
stable state, in which the atom experiences a tight potential in the regime of
collisional blockade, the quantum tweezers. The transfer efficiency into the
single-atom ground state of the tight trap is fundamentally limited by the
collective modes of the condensate, which are thermally and dynamically
excited. The noise due to these excitations can be quenched for sufficiently
long laser pulses, thereby achieving high efficiencies. These results show that
this protocol can be applied for initializing a quantum register based on
tweezer traps for neutral atoms.Comment: 4+ pages, 3 figures, revised version. To appear in Phys. Rev. A
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