Martingale-based gradient descent algorithm for estimating free energy values of diffusions
Martingale-based gradient descent algorithm for estimating free energy values of diffusions
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用于估计扩散自由能值的基于 Martingale 的梯度下降算法
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
C. Hartmann
中科院分区：
文献类型：
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作者：
H. Lie;C. Schütte;C. Hartmann
热力学自由能或累积生成函数在平衡系统的罕见事件统计估计中发挥着重要作用，因为它们被解释为归一化常数。在本文中，我们讨论最近提出的一种方法[C. Hartmann 和 C. Schutte，J. Stat。机甲。理论。 Exp., (2012), P11004]通过最小化某个控制函数来减少可逆扩散的自由能估计的方差。我们使用 Cameron-Martin-Girsanov 公式推导该方法，在控制泛函中添加了鞅项。使用涉及罕见事件概率计算的数值示例，我们表明基于鞅的函数在次优控制下表现出较小的方差，并且通过梯度下降最小化控制函数产生表现出更高稳定性的自由能函数近似。
Thermodynamic free energies or cumulant generating functions play a significant role in the estimation of rare event statistics of equilibrated systems because of their interpretation as normalisation constants. In this article we discuss a recently proposed method [C. Hartmann and C. Schutte, J. Stat. Mech. Theor. Exp., (2012), P11004] for variance reduction of free energy estimates of reversible diffusions by minimisation of a certain control functional. Our derivation of the method using the Cameron-Martin-Girsanov formula adds a martingale term to the control functional. Using numerical examples involving the calculation of rare event probabilities, we show that the martingale-based functional exhibits smaller variance under suboptimal controls, and that minimisation of the control functional by gradient descent yields free energy function approximations that exhibit more stability.