Inverse problems: From regularization to Bayesian inference
Inverse problems: From regularization to Bayesian inference
复制标题
DOI：
10.1002/wics.1427
复制
发表时间：
2018-05-01
影响因子：
3.2
通讯作者：
Somersalo, E.
中科院分区：
文献类型：
--
作者：
Calvetti, D.;Somersalo, E.
逆问题涉及基于预测模型（称为正向模型）寻找观察到的后果的未知原因，该模型按因果顺序将前一个量与后一个量关联起来。正向模型通常是适定的，因为原因以独特且稳定的方式决定结果。另一方面，逆问题通常是不适定的：数据可能不足以明确地识别原因，可能不存在精确的解决方案，并且就像在神秘故事中一样，在没有额外信息的情况下发现原因往往对测量噪声和建模误差高度敏感。贝叶斯方法提供了一种通用且自然的方式，通过将未知数据建模为随机变量来突出其值的不确定性，从而合并额外信息来补充噪声数据。以后验分布的形式呈现解决方案为计算有用的估计提供了广泛的可能性。逆问题传统上是从正则化的角度来处理的，这是一个用附近的适定问题代替不适定问题的过程。虽然许多正则化技术可以通过事先设计在贝叶斯框架中重新解释，但贝叶斯形式主义提供了新技术来丰富传统逆问题的范式。特别是，正演模型的不准确性和不足之处自然会在统计框架中得到处理。类似地，关于解决方案的定性信息可以以具有未知参数的先验形式重新表述，这些参数可以在分层贝叶斯环境中成功处理。本文分类为：数据分析的统计和图形方法 > 贝叶斯方法和理论算法和计算方法 > 计算统计的数值方法应用 > 计算数学
Inverse problems deal with the quest for unknown causes of observed consequences, based on predictive models, known as the forward models, that associate the former quantities to the latter in the causal order. Forward models are usually well-posed, as causes determine consequences in a unique and stable way. Inverse problems, on the other hand, are usually ill-posed: the data may be insufficient to identify the cause unambiguously, an exact solution may not exist, and, like in a mystery story, discovering the cause without extra information tends to be highly sensitive to measurement noise and modeling errors. The Bayesian methodology provides a versatile and natural way of incorporating extra information to supplement the noisy data by modeling the unknown as a random variable to highlight the uncertainty about its value. Presenting the solution in the form of a posterior distribution provides a wide range of possibilities to compute useful estimates. Inverse problems are traditionally approached from the point of view of regularization, a process whereby the ill-posed problem is replaced by a nearby well-posed one. While many of the regularization techniques can be reinterpreted in the Bayesian framework through prior design, the Bayesian formalism provides new techniques to enrich the paradigm of traditional inverse problems. In particular, inaccuracies and inadequacies of the forward model are naturally handled in the statistical framework. Similarly, qualitative information about the solution may be reformulated in the form of priors with unknown parameters that can be successfully handled in the hierarchical Bayesian context. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory Algorithms and Computational Methods > Numerical Methods Applications of Computational Statistics > Computational Mathematics