12 research outputs found
Mathematical aspects of the Digital Annealer's simulated annealing algorithm
The Digital Annealer is a CMOS hardware designed by Fujitsu Laboratories for
high-speed solving of Quadratic Unconstrained Binary Optimization (QUBO)
problems that could be difficult to solve by means of existing general-purpose
computers. In this paper, we present a mathematical description of the
first-generation Digital Annealer's Algorithm from the Markov chain theory
perspective, establish a relationship between its stationary distribution with
the Gibbs-Boltzmann distribution, and provide a necessary and sufficient
condition on its cooling schedule that ensures asymptotic convergence to the
ground states
Gibbs Measures on Multidimensional Spaces. Equivalences and a Groupoid Approach
We consider some of the main notions of Gibbs measures on subshifts
introduced by different communities, such as dynamical systems, probability,
operator algebras, and mathematical physics. For potentials with -summable
variation, we prove that several of the definitions considered by these
communities are equivalent. In particular, when the subshift is of finite type
(SFT), we show that all definitions coincide. In addition, we introduced a
groupoid approach to describe some Gibbs measures, allowing us to show the
equivalence between Gibbs measures and KMS states (the quantum analogous to the
Gibbs measures).Comment: arXiv admin note: text overlap with arXiv:2008.1372
Mixing time and simulated annealing for the stochastic cellular automata
Finding a ground state of a given Hamiltonian on a graph is an
important but hard problem. One of the potential approaches is to use a Markov
chain Monte Carlo to sample the Gibbs distribution whose highest peaks
correspond to the ground states. In this paper, we investigate a particular
kind of stochastic cellular automata, in which all spins are updated
independently and simultaneously. We prove that (i) if the temperature is fixed
sufficiently high, then the mixing time is at most of order , and that
(ii) if the temperature drops in time as , then the limiting
measure is uniformly distributed over the ground states.Comment: 16 pages, 8 figure
Possible interpretations of the joint observations of UHECR arrival directions using data recorded at the Telescope Array and the Pierre Auger Observatory
Medidas de Gibbs em subshifts
We study the properties of Gibbs measures for functions with d-summable variation defined on a subshift X. Based on Meyerovitch\'s work from 2013, we prove that if X is a subshift of finite type (SFT), then any equilibrium measure is also a Gibbs measure. Although the definition provided by Meyerovitch does not make any mention to conditional expectations, we show that in the case where X is a SFT it is possible to characterize these measures in terms of more familiar notions presented in the literature.Nós estudamos as propriedades de medidas de Gibbs para funções com variação d-somável definidas em um subshift X. Baseado no trabalho de Meyerovitch de 2013, provamos que se X é um subshift de tipo finito (STF), então qualquer medida de equilíbrio é também uma medida de Gibbs. Embora a definição fornecida por Meyerovitch não faz qualquer menção à esperanças condicionais, mostramos que no caso em que X é um STF, é possível caracterizar estas medidas em termos de noções mais familiares apresentadas na literatura
Medidas de Gibbs em subshifts
We study the properties of Gibbs measures for functions with d-summable variation defined on a subshift X. Based on Meyerovitch\'s work from 2013, we prove that if X is a subshift of finite type (SFT), then any equilibrium measure is also a Gibbs measure. Although the definition provided by Meyerovitch does not make any mention to conditional expectations, we show that in the case where X is a SFT it is possible to characterize these measures in terms of more familiar notions presented in the literature.Nós estudamos as propriedades de medidas de Gibbs para funções com variação d-somável definidas em um subshift X. Baseado no trabalho de Meyerovitch de 2013, provamos que se X é um subshift de tipo finito (STF), então qualquer medida de equilíbrio é também uma medida de Gibbs. Embora a definição fornecida por Meyerovitch não faz qualquer menção à esperanças condicionais, mostramos que no caso em que X é um STF, é possível caracterizar estas medidas em termos de noções mais familiares apresentadas na literatura
Stability of energy landscape for Ising models
In this paper, we explore the stability of the energy landscape of an Ising Hamiltonian when subjected to two kinds of perturbations: a perturbation on the coupling coe cients and external fields, and a perturbation on the underlying graph structure. We give su cient conditions so that the ground states of a given Hamiltonian are stable under perturbations of the first kind in terms of order preservation. Here by order preservation we mean that the ordering of energy corresponding to two spin configurations in a perturbed Hamiltonian will be preserved in the original Hamiltonian up to a given error margin. We also estimate the probability that the energy gap between ground states for the original Hamiltonian and the perturbed Hamiltonian is bounded by a given error margin when the coupling coe cients and local external magnetic fields of the original Hamiltonian are i.i.d. Gaussian random variables. In the end we show a concrete example of a system which is stable under perturbations of the second kind
Stochastic Optimization : Glauber Dynamics Versus Stochastic Cellular Automata
The topic we address in this paper concerns the minimization of a Hamiltonian function for an Ising model through the application of simulated annealing algorithms based on (single-site) Glauber dynamics and stochastic cellular automata (SCA). Some rigorous results are presented in order to justify the application of simulated annealing for a particular kind of SCA. After that, we compare the SCA algorithm and its variation, namely the ε-SCA algorithm, studied in this paper with the Glauber dynamics by analyzing their accuracy in obtaining optimal solutions for the max-cut problem on Erdös-Rényi random graphs, the traveling salesman problem (TSP), and the minimization of Gaussian and Bernoulli spin glass Hamiltonians. We observed that the SCA performed better than the Glauber dynamics in some special cases, while the ε-SCA showed the highest performance in all scenarios
Mathematical Aspects of the Digital Annealer’s Simulated Annealing Algorithm
The Digital Annealer is a CMOS hardware designed by Fujitsu Laboratories for high-speed solving of Quadratic Unconstrained Binary Optimization (QUBO) problems that could be difficult to solve by means of existing general-purpose computers. In this paper, we present a mathematical description of the first-generation Digital Annealer’s Algorithm from the Markov chain theory perspective, establish a relationship between its stationary distribution with the Gibbs-Boltzmann distribution, and provide a necessary and sufficient condition on its cooling schedule that ensures asymptotic convergence to the ground states
Stochastic Optimization : Glauber Dynamics Versus Stochastic Cellular Automata
The topic we address in this paper concerns the minimization of a Hamiltonian function for an Ising model through the application of simulated annealing algorithms based on (single-site) Glauber dynamics and stochastic cellular automata (SCA). Some rigorous results are presented in order to justify the application of simulated annealing for a particular kind of SCA. After that, we compare the SCA algorithm and its variation, namely the ε-SCA algorithm, studied in this paper with the Glauber dynamics by analyzing their accuracy in obtaining optimal solutions for the max-cut problem on Erdös-Rényi random graphs, the traveling salesman problem (TSP), and the minimization of Gaussian and Bernoulli spin glass Hamiltonians. We observed that the SCA performed better than the Glauber dynamics in some special cases, while the ε-SCA showed the highest performance in all scenarios