Abstract
The theoretical analysis of many problems in physics, astronomy, and applied mathematics requires an efficient numerical exploration of multimodal parameter spaces that exhibit broken ergodicity. Monte Carlo methods are widely used to deal with these classes of problems, but such simulations suffer from a ubiquitous sampling problem: The probability of sampling a particular state is proportional to its entropic weight. Devising an algorithm capable of sampling efficiently the full phase space is a long-standing problem. Here, we report a new hybrid method for the exploration of multimodal parameter spaces exhibiting broken ergodicity. Superposition enhanced nested sampling combines the strengths of global optimization with the unbiased or athermal sampling of nested sampling, greatly enhancing its efficiency with no additional parameters. We report extensive tests of this new approach for atomic clusters that are known to have energy landscapes for which conventional sampling schemes suffer from broken ergodicity. We also introduce a novel parallelization algorithm for nested sampling.
- Received 26 February 2014
DOI:https://doi.org/10.1103/PhysRevX.4.031034
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Published by the American Physical Society
Popular Summary
Nested sampling is a general Bayesian computational method that has applications ranging from numerical computations of Bayesian evidence for Bayesian model comparison to thermodynamic sampling of potential energy landscapes. The approach has found widespread use in many fields, including cosmology, condensed matter physics, and systems biology. We report a new hybrid Monte Carlo method, combining the complementary techniques of nested sampling and global optimization. We show that this superposition enhanced nested sampling scheme extends the applicability of nested sampling to important systems that were previously beyond the reach of conventional simulation techniques.
Nested sampling samples parameter spaces uniformly under some likelihood constraint that decreases iteratively throughout the calculation. This procedure enables the density of states for each point in parameter space to be determined statistically, thus providing direct access to computing the partition function (Bayesian evidence) and thermodynamic properties at arbitrary temperatures for the case of a potential energy landscape. This methodology, however, is not immune to the problem of broken ergodicity, where ergodicity refers to time and spatial averages. Since the probability of sampling a point in parameter space is proportional to its entropic weight, this technique is susceptible to broken ergodicity. For this reason, designing algorithms capable of sampling the full space efficiently has been a long-standing problem, further exacerbated by the complexity of sampling landscapes with kinetic traps. The enhanced nested sampling scheme that we develop can be run in parallel on multiple processors; the optimization aspect of the scheme ensures that no critical minima are missed in phase space.
The results show that both exact and approximate implementations of the new technique can be useful. The corresponding superposition enhanced nested sampling codes are available publicly in Python/C.
