随机元胞自动机的混合时间和模拟退火算法
Mixing time and simulated annealing for the stochastic cellular automata
摘要：
在图$G=(V,E)$上寻找伊辛模型的哈密顿量的基态是一个重要而困难的问题。这类问题的标准方法是应用依赖于单自旋翻转马尔可夫链蒙特卡罗方法的算法，例如基于Glauber或Metropolis动力学的模拟退火。在本文中，我们研究了一种特殊的随机细胞自动机，其中所有的自旋是独立和同时更新。我们证明：（i）如果温度足够高，那么混合时间最多为$\log|V|$阶，（ii）如果温度随时间$n$下降为$1/\log n$，那么极限测度均匀分布在基态上。我们还提供了一些模拟本文研究的算法在GPU上实现，并显示其优越的性能相比，传统的模拟退火。
英文摘要：
Finding a ground state of a given Hamiltonian of an Ising model on a graph $G=(V,E)$ is an important but hard problem. The standard approach for this kind of problem is the application of algorithms that rely on single-spin-flip Markov chain Monte Carlo methods, such as the simulated annealing based on Glauber or Metropolis dynamics. 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 $\log|V|$, and that (ii) if the temperature drops in time $n$ as $1/\log n$, then the limiting measure is uniformly distributed over the ground states. We also provide some simulations of the algorithms studied in this paper implemented on a GPU and show their superior performance compared to the conventional simulated annealing.
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