Inverse problems: From regularization to Bayesian inference

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 s...

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Published in:	Wiley interdisciplinary reviews. Computational statistics Vol. 10; no. 3; pp. e1427 - n/a
Main Authors:	Calvetti, D., Somersalo, E.
Format:	Journal Article
Language:	English
Published:	Hoboken, USA John Wiley & Sons, Inc 01.05.2018 Wiley Subscription Services, Inc
Subjects:	Bayesian analysis Computation Computational mathematics Computer applications Data Data analysis Data processing Frameworks Graphical methods hierarchical Bayesian models Ill‐posedness Inverse problems likelihood Mathematical models Mathematics measurement and modeling errors Modelling Noise measurement Noise sensitivity Numerical methods posterior estimates Prediction models Probability theory Random variables Regularization sparsity‐promoting and sample‐based priors Statistical analysis Statistical inference Statistical methods Statistics structural Tikhonov regularization
ISSN:	1939-5108, 1939-0068
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Abstract	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 In inverse problems, the ill‐posedness manifests itself in the form of a likelihood density whose support is wide in some directions, or more generally, along some manifolds, where no clear preference to parameter values is given. An informative prior efficiently restricts the support of the posterior density where the likelihood is non‐informative.
AbstractList	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 In inverse problems, the ill‐posedness manifests itself in the form of a likelihood density whose support is wide in some directions, or more generally, along some manifolds, where no clear preference to parameter values is given. An informative prior efficiently restricts the support of the posterior density where the likelihood is non‐informative. 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 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 TheoryAlgorithms and Computational Methods > Numerical MethodsApplications of Computational Statistics > Computational Mathematics
Author	Somersalo, E. Calvetti, D.
Author_xml	– sequence: 1 givenname: D. surname: Calvetti fullname: Calvetti, D. organization: Case Western Reserve University – sequence: 2 givenname: E. surname: Somersalo fullname: Somersalo, E. email: erkki.somersalo@case.edu organization: Case Western Reserve University
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Snippet	Inverse problems deal with the quest for unknown causes of observed consequences, based on predictive models, known as the forward models, that associate the...
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SubjectTerms	Bayesian analysis Computation Computational mathematics Computer applications Data Data analysis Data processing Frameworks Graphical methods hierarchical Bayesian models Ill‐posedness Inverse problems likelihood Mathematical models Mathematics measurement and modeling errors Modelling Noise measurement Noise sensitivity Numerical methods posterior estimates Prediction models Probability theory Random variables Regularization sparsity‐promoting and sample‐based priors Statistical analysis Statistical inference Statistical methods Statistics structural Tikhonov regularization
Title	Inverse problems: From regularization to Bayesian inference
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