Hamiltonian Monte Carlo in Inverse Problems; Ill-Conditioning and Multi-Modality.

IF 1.5 4区工程技术 Q2 ENGINEERING, MULTIDISCIPLINARY

International Journal for Uncertainty Quantification Pub Date : 2021-03-12 DOI:10.1615/int.j.uncertaintyquantification.2022038478

I. Langmore, M. Dikovsky, S. Geraedts, P. Norgaard, R. V. Behren

引用次数: 3

Abstract

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 encoun-tered 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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反问题中的哈密顿蒙特卡罗病态和多模态。

哈密顿蒙特卡罗(HMC)方法允许从连续密度中采样。良好的维度缩放使得HMC被统计社区广泛采用。现代自动微分软件应该允许在贝叶斯反问题中得到更广泛的应用。本文分析了用HMC求解反问题所遇到的两个主要困难:条件差和多模态。在光谱学的背景下，提出了预处理和副本交换蒙特卡罗参数选择的新结果。给出了积分步骤数、步长、预调节器类型和拟合、退火形式和程序的建议。这些建议在高斯情况下进行了严格的分析，并在聚变等离子体重建中得到推广。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

International Journal for Uncertainty Quantification ENGINEERING, MULTIDISCIPLINARY-MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

CiteScore

3.60

自引率

5.90%

发文量

期刊介绍： The International Journal for Uncertainty Quantification disseminates information of permanent interest in the areas of analysis, modeling, design and control of complex systems in the presence of uncertainty. The journal seeks to emphasize methods that cross stochastic analysis, statistical modeling and scientific computing. Systems of interest are governed by differential equations possibly with multiscale features. Topics of particular interest include representation of uncertainty, propagation of uncertainty across scales, resolving the curse of dimensionality, long-time integration for stochastic PDEs, data-driven approaches for constructing stochastic models, validation, verification and uncertainty quantification for predictive computational science, and visualization of uncertainty in high-dimensional spaces. Bayesian computation and machine learning techniques are also of interest for example in the context of stochastic multiscale systems, for model selection/classification, and decision making. Reports addressing the dynamic coupling of modern experiments and modeling approaches towards predictive science are particularly encouraged. Applications of uncertainty quantification in all areas of physical and biological sciences are appropriate.