62,283 research outputs found
Statistical mechanics approach to the phase unwrapping problem
The use of Mean-Field theory to unwrap principal phase patterns has been
recently proposed. In this paper we generalize the Mean-Field approach to
process phase patterns with arbitrary degree of undersampling. The phase
unwrapping problem is formulated as that of finding the ground state of a
locally constrained, finite size, spin-L Ising model under a non-uniform
magnetic field. The optimization problem is solved by the Mean-Field Annealing
technique. Synthetic experiments show the effectiveness of the proposed
algorithm
Optimization by linear kinetic equations and mean-field Langevin dynamics
Probably one of the most striking examples of the close connections between
global optimization processes and statistical physics is the simulated
annealing method, inspired by the famous Monte Carlo algorithm devised by
Metropolis et al. in the middle of the last century. In this paper we show how
the tools of linear kinetic theory allow to describe this gradient-free
algorithm from the perspective of statistical physics and how convergence to
the global minimum can be related to classical entropy inequalities. This
analysis highlight the strong link between linear Boltzmann equations and
stochastic optimization methods governed by Markov processes. Thanks to this
formalism we can establish the connections between the simulated annealing
process and the corresponding mean-field Langevin dynamics characterized by a
stochastic gradient descent approach. Generalizations to other selection
strategies in simulated annealing that avoid the acceptance-rejection dynamic
are also provided
Deterministic Modularity Optimization
We study community structure of networks. We have developed a scheme for
maximizing the modularity Q based on mean field methods. Further, we have
defined a simple family of random networks with community structure; we
understand the behavior of these networks analytically. Using these networks,
we show how the mean field methods display better performance than previously
known deterministic methods for optimization of Q.Comment: 7 pages, 4 figures, minor change
Quantum Annealing and Analog Quantum Computation
We review here the recent success in quantum annealing, i.e., optimization of
the cost or energy functions of complex systems utilizing quantum fluctuations.
The concept is introduced in successive steps through the studies of mapping of
such computationally hard problems to the classical spin glass problems. The
quantum spin glass problems arise with the introduction of quantum
fluctuations, and the annealing behavior of the systems as these fluctuations
are reduced slowly to zero. This provides a general framework for realizing
analog quantum computation.Comment: 22 pages, 7 figs (color online); new References Added. Reviews of
Modern Physics (in press
Asymptotic behavior of memristive circuits
The interest in memristors has risen due to their possible application both
as memory units and as computational devices in combination with CMOS. This is
in part due to their nonlinear dynamics, and a strong dependence on the circuit
topology. We provide evidence that also purely memristive circuits can be
employed for computational purposes. In the present paper we show that a
polynomial Lyapunov function in the memory parameters exists for the case of DC
controlled memristors. Such Lyapunov function can be asymptotically
approximated with binary variables, and mapped to quadratic combinatorial
optimization problems. This also shows a direct parallel between memristive
circuits and the Hopfield-Little model. In the case of Erdos-Renyi random
circuits, we show numerically that the distribution of the matrix elements of
the projectors can be roughly approximated with a Gaussian distribution, and
that it scales with the inverse square root of the number of elements. This
provides an approximated but direct connection with the physics of disordered
system and, in particular, of mean field spin glasses. Using this and the fact
that the interaction is controlled by a projector operator on the loop space of
the circuit. We estimate the number of stationary points of the approximate
Lyapunov function and provide a scaling formula as an upper bound in terms of
the circuit topology only.Comment: 20 pages, 8 figures; proofs corrected, figures changed; results
substantially unchanged; to appear in Entrop
Application of Quantum Annealing to Nurse Scheduling Problem
Quantum annealing is a promising heuristic method to solve combinatorial
optimization problems, and efforts to quantify performance on real-world
problems provide insights into how this approach may be best used in practice.
We investigate the empirical performance of quantum annealing to solve the
Nurse Scheduling Problem (NSP) with hard constraints using the D-Wave 2000Q
quantum annealing device. NSP seeks the optimal assignment for a set of nurses
to shifts under an accompanying set of constraints on schedule and personnel.
After reducing NSP to a novel Ising-type Hamiltonian, we evaluate the solution
quality obtained from the D-Wave 2000Q against the constraint requirements as
well as the diversity of solutions. For the test problems explored here, our
results indicate that quantum annealing recovers satisfying solutions for NSP
and suggests the heuristic method is sufficient for practical use. Moreover, we
observe that solution quality can be greatly improved through the use of
reverse annealing, in which it is possible to refine a returned results by
using the annealing process a second time. We compare the performance NSP using
both forward and reverse annealing methods and describe how these approach
might be used in practice.Comment: 20 pages, 13 figure
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