MCMC模拟中位形间距离及回火算法的几何优化
Distance between configurations in MCMC simulations and the geometrical optimization of the tempering algorithms
摘要：
对于一个给定的马尔可夫链蒙特卡罗（MCMC）算法，我们定义的配置之间的距离，量化的过渡的难度。这个距离使我们能够调查MCMC算法的几何方式，我们调查的几何形状的模拟回火算法实现了一个非常多模态系统高度退化真空。我们表明，扩展的配置空间的大规模几何是由一个渐近反德西特度量，并认为在一个简单的，几何的方式，回火参数应该最好地放置指数，以获得高的接受率过渡的额外的维度。我们还讨论了一种求解数值符号问题的算法--回火Lefschetz套环法的几何优化问题。
英文摘要：
For a given Markov chain Monte Carlo (MCMC) algorithm, we define the distance between configurations that quantifies the difficulty of transitions. This distance enables us to investigate MCMC algorithms in a geometrical way, and we investigate the geometry of the simulated tempering algorithm implemented for an extremely multimodal system with highly degenerate vacua. We show that the large scale geometry of the extended configuration space is given by an asymptotically anti-de Sitter metric, and argue in a simple, geometrical way that the tempering parameter should be best placed exponentially to acquire high acceptance rates for transitions in the extra dimension. We also discuss the geometrical optimization of the tempered Lefschetz thimble method, which is an algorithm towards solving the numerical sign problem.
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