反问题中的哈密顿蒙特卡罗方法;病态性和多模态性
Hamiltonian Monte Carlo in Inverse Problems; Ill-Conditioning and Multi-Modality
摘要：
哈密顿蒙特卡罗（HMC）方法允许从连续密度采样。有利的缩放维度导致统计界广泛采用HMC。现代自动微分软件应该允许更广泛地使用贝叶斯逆问题。本文分析了HMC在求解反问题时遇到的两个主要困难：条件差和多模态。新的结果预处理和副本交换蒙特卡罗参数选择的背景下，光谱。对积分步骤的数目、步长、预处理器的类型和安装、退火形式和时间表提出了建议。这些建议进行了严格的分析，在高斯的情况下，并概括在融合等离子体重建。
英文摘要：
The Hamiltonian Monte Carlo (HMC) method allows sampling from continuous densities. Favorable scaling with dimension has led to wide adoption of HMC by the statistics community. Modern auto-differentiating software should allow more widespread usage in Bayesian inverse problems. This paper analyzes two major difficulties encountered using HMC for inverse problems: poor conditioning and multi-modality. Novel results on preconditioning and replica exchange Monte Carlo parameter selection are presented in the context of spectroscopy. Recommendations are given for the number of integration steps as well as step size, preconditioner type and fitting, annealing form and schedule. These recommendations are analyzed rigorously in the Gaussian case, and shown to generalize in a fusion plasma reconstruction.
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