We propose a method to reduce the relaxation time towards equilibrium in stochastic sampling of complex energy landscapes in statistical systems with discrete degrees of freedom by generalizing the platform previously developed for continuous systems. The method starts from a master equation, in contrast to the Fokker-Planck equation for the continuous case. The master equation is transformed into an imaginary-time Schr\"odinger equation. The Hamiltonian of the Schr\"odinger equation is modified by adding a projector to its known ground state. We show how this transformation decreases the relaxation time and propose a way to use it to accelerate simulated annealing for optimization problems. We implement our method in a simplified kinetic Monte Carlo scheme and show an acceleration by an order of magnitude in simulated annealing of the symmetric traveling salesman problem. Comparisons of simulated annealing are made with the exchange Monte Carlo algorithm for the three-dimensional Ising spin glass. Our implementation can be seen as a step toward accelerating the stochastic sampling of generic systems with complex landscapes and long equilibration times.
Numerical sampling of rugged energy landscapes is notoriously difficult [1]. Transition rates between two states are exponential functions of the energy barrier between them divided by the temperature. The sampling of complex systems is a daunting task because there are many states of comparable energies separated by large barriers. One of the most widespread sampling methods at finite temperatures is the Monte Carlo method, where one follows one or more trajectories of a virtual Brownian particle as it moves through the state space. To sample a system at a given temperature, there exists a plethora of approaches, and amongst many others particularly notable are the standard (Metropolis) [2] and the kinetic Monte Carlo [3] methods. At low temperatures these stochastic schemes tend to take a long time before a satisfactory result is reached. If one is interested in the behavior of a specific problem in the low-temperature limit, the common method is simulated annealing [4]. If, however, one samples systems where below a certain temperature the state space splits into regions which are separated by huge barriers, like for example in spin glass systems, simulated annealing does not always lead to satisfactory results. Examples of accepted solutions to this problem include the exchange Monte Carlo method [5] and the population annealing [6]. Although good results are obtainable for most systems using these techniques, relaxation time still stays a crucial factor. For the protein folding problem [7] for example, the longest simulations available on present day computers are still far from the equilibrium distributions. A possible step in resolving the problem of over-long relaxation times was proposed in Ref. [8] where a method to accelerate the sampling of continuous systems was introduced.
The basic idea is to rewrite the Fokker-Planck equation, which describes the time evolution of the probability distribution for continuous systems, into an imaginary-time Schrödinger equation, for which one artificially introduces an energy gap between the ground state and the first excited state. Then the relaxation time, which is proportional to the inverse of the energy gap, is reduced.
In the present paper, we extend the idea of Ref. [8] in Sec. II to discrete systems and implement it into a stochastic sampling scheme. Next, in Sec. III, we analyze how our method can be used to improve the performance of simulated annealing of the traveling salesman problem [9] and the three-dimensional Ising spin glass. In the case of the traveling salesman problem we find that the simulated annealing is significantly accelerated and the modified sampling finds the approximately optimal solution much faster than unmodified simulations. Our method also leads to improvements for the three-dimensional Gaussian Ising spin glass. Sec. IV concludes this paper.
In this section we extend idea of Ref. [8] to discrete systems and discuss its implementation into the kinetic Monte Carlo algorithm. We furthermore introduce a simplified version of the kinetic Monte Carlo scheme to speed up the calculations and save computational resources.
The master equation whose transition rates w ab fulfill the detailed balance condition reads,
If we set P a = f a Q a , with f a = e -βEa/2 / √ Z, where Z = b exp(-βE b ) is the partition function of the system and E a is the energy of state a, we get
Since H ab , as defined in Eq. ( 2), is a real, symmetric matrix, we call it a Hamiltonian and thus, Eq. ( 2) may be regarded as an imaginary-time Schrödinger equation. It has a zero eigenvalue with eigenvector f b , b H ab f b = 0. This follows directly from the definitions of f b and H ab and the detailed balance condition in Eq. ( 1). The lowest eigenvalue of H ab is therefore zero as guaranteed by the Perron-Frobenius theorem.
Following the idea in Ref. [8], we make the transformation H ab → H ab + λ(δ ab -P 0 ab ), where P 0 ab is the projector to the state of zero eigenvalue, which is expected to shorten the relaxation time toward equilibrium. The spectrum of H ab is then shifted by λ for all states except the ground state. It is easy to see that the matrix elements of the projector are
The solutions of this equation are,
where the {Q (n) } and { n } are the eigenvectors and corresponding eigenvalues of H ab , respectively.
Making use of the relation
and deduce its solution from Eq. ( 4) as
For large times this decays to
i.e. the Boltzmann weight, as expected. However, as can be seen from Eq. ( 6), the relaxation is faster than the case where λ is absent.
In Ref. [8], a similar idea was tested for continuous systems using a diffusion Monte Carlo calculation. In the present discrete case, a straightforward implementation of our method is through the kinetic Monte Carlo algorithm [3]. Let us first describe this Monte Carlo method on the original master equation (1) in order to make is clear what parts need modifications to accommodate the λ-term in
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