An Introduction to Data Analysis and Uncertainty Quantification for Inverse Problems
An Introduction to Data Analysis and Uncertainty Quantification for Inverse Problems
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DOI：
10.1137/1.9781611974928
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发表时间：
2017-07
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影响因子：
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通讯作者：
L. Tenorio
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作者：
L. Tenorio
粗略地说，解决逆问题就是从噪声（通常是间接）观察中恢复对象（例如参数或函数）。在大多数情况下，这种恢复无法精确完成，因为将数据链接到对象的数学模型是近似值，数据有噪声，观测值的数量是有限的，并且获得解决方案可能需要进一步的近似以进行有效的数值计算。考虑到这种潜在的误差源，评估反问题解的可靠性的重要性是显而易见的。此评估步骤是现在所谓的不确定性量化 (UQ) 的一部分。反问题和其他工程问题的不确定性量化需要熟悉数学、概率和统计学的一些基本方法。但我在与科学家和应用数学家研究逆问题的多年合作中观察到，他们对概率或统计知识的感觉往往不如对应用数学背景的感觉那么舒服。反之亦然：我遇到过一些统计学家，他们有兴趣为反演问题做出贡献，但他们还没有接触过反演问题的基本理论及其应用中出现的问题。因此，本书的目标是成为应用数学和统计学界之间的桥梁。我尝试利用读者的数学背景，主要在反问题的背景下，为昆士兰大学的概率和统计提供基本介绍，反演问题是一个具有许多重要实际应用的领域。此外，本书还为具有统计学背景的人员提供了逆问题统计正则化的基本介绍。由于假定读者熟悉数学、工程和物理科学领域高年级本科生和初级研究生的数学方法，因此可以涵盖很多内容：从本科生统计学和概率到无限维空间上的概率分布。对于统计学家来说，本书使用经典的线性回归和统计推断来介绍不适定反问题的框架，并解释其应用中出现的统计问题。附录中还回顾了反问题所需的数学分析工具。由于所涵盖的统计和概率方法的应用超出了反演问题，因此从事数据科学或昆士兰大学其他应用领域工作的人员也可能会对本书感兴趣。
Roughly speaking, to solve an inverse problem is to recover an object (e.g., parameter or function) from noisy (typically indirect) observations. In most cases such recovery cannot be done exactly because the mathematical models that link data to the object are approximations, data are noisy, the number of observations is finite, and obtaining a solution may require further approximations for efficient numerical computations. The importance of assessing the reliability of solutions to inverse problems is evident given such potential sources of errors. This assessment step is part of what is now called uncertainty quantification (UQ). Uncertainty quantification for inverse problems and other problems in engineering requires familiarity with some basic methods from mathematics, probability, and statistics. But what I have observed during years of collaborations with scientists and applied mathematicians working on inverse problems is that they often do not feel as comfortable with their knowledge of probability or statistics as they do with their background in applied mathematics. The converse is also true: I have encountered statisticians interested in making contributions to inverse problems but who have not been exposed to the basic theory of inverse problems and the questions that arise in their applications. The objective of this book is therefore to serve as a bridge between the applied mathematics and statistics communities. I try to take advantage of the reader's mathematical background to provide a basic introduction to probability and statistics for UQ mainly in the context of inverse problems, a field with many important practical applications. In addition, the book provides a basic introduction to statistical regularization of inverse problems for those with a background in statistics. Since the reader is assumed to be comfortable with mathematical methods at the level of senior undergraduates and beginning graduate students in mathematics, engineering, and physical sciences, much ground can be covered: from undergraduate statistics and probability to probability distributions on infinite-dimensional spaces. For statisticians, the book uses classic linear regression and statistical inference to introduce the framework of ill-posed inverse problems and explain statistical questions that arise in their applications. A review of the mathematical analysis tools required for inverse problems is also included in the appendix. Since the statistical and probability methods covered have applications beyond inverse problems, the book may also be of interest to people working in data science or in other applications of UQ.