We present a study on the optimization of the updating magnitude for a class of free energy methods based on flat-distribution sampling, including the Wang-Landau (WL) algorithm and metadynamics. These methods rely on adaptive construction of a bias potential that offsets the potential of mean force by histogram-based updates. The convergence of the bias potential can be improved by decreasing the updating magnitude with an optimal schedule. We show that while the asymptotically optimal schedule for the single-bin updating scheme (commonly used in the WL algorithm) is given by the known inverse-time formula, that for the Gaussian updating scheme (commonly used in metadynamics) is often more complex. We further show that the single-bin updating scheme is optimal for very long simulations, and it can be generalized to a class of bandpass updating schemes that are similarly optimal. These bandpass updating schemes target only a few long-range distribution modes and their optimal schedule is also given by the inverse-time formula. Constructed from orthogonal polynomials, the bandpass updating schemes generalize the WL and Langfeld-Lucini-Rago algorithms as an automatic parameter tuning scheme for umbrella sampling.

中文翻译：

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自适应平坦分布采样中的最优更新幅度


我们提出了一项基于平坦分布采样的一类自由能方法更新幅度优化的研究，包括 Wang-Landau (WL) 算法和元动力学。这些方法依赖于偏置势的自适应构造，该偏置势通过基于直方图的更新来抵消平均力的势。可以通过以最佳调度减小更新幅度来改善偏置势的收敛。我们表明，虽然单仓更新方案（常用于 WL 算法）的渐近最优调度是由已知的反时公式给出的，但高斯更新方案（常用于元动力学）的渐近最优调度通常更为复杂。我们进一步表明，单箱更新方案对于很长的模拟来说是最佳的，并且它可以推广到一类类似最佳的带通更新方案。这些带通更新方案仅针对少数远程分布模式，并且它们的最佳调度也由反时公式给出。带通更新方案由正交多项式构成，将 WL 和 Langfeld-Lucini-Rago 算法推广为伞式采样的自动参数调整方案。