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We consider hyper-differential sensitivity analysis (HDSA) of nonlinear Bayesian inverse problems governed by partialdifferential equations (PDEs) with infinite-dimensional parameters. In previous works, HDSA has been used to assessthe sensitivity of the solution of deterministic inverse problems to additional model uncertainties and also different types of measurement data. In the present work, we extend HDSA to the class of Bayesian inverse problems governed by PDEs. The focus is on assessing the sensitivity of certain key quantities derived from the posterior distribution. Specifically, we focus on analyzing the sensitivity of the MAP point and the Bayes risk and make full use of the information embedded in the Bayesian inverse problem. After establishing our mathematical framework for HDSA of Bayesian inverse problems, we present a detailed computational approach for computing the proposed HDSA indices. We examine the effectiveness of the proposed approach on an inverse problem governed by a PDE modeling heat conduction.
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Inverse Problems for Physics-Based Process Models
We describe and compare two formulations of inverse problems for a physics-based process model in the context of uncertainty and random variability: the Bayesian inverse problem and the stochastic inverse problem. We describe the foundations of the two problems in order to create a context for interpreting the applicability and solutions of inverse problems important for scientific and engineering inference. We conclude by comparing them to statistical approaches to related problems, including Bayesian calibration of computer models.
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- PAR ID:
- 10534185
- Publisher / Repository:
- Annual Review of Statistics and Its Application
- Date Published:
- Journal Name:
- Annual Review of Statistics and Its Application
- Volume:
- 11
- Issue:
- 1
- ISSN:
- 2326-8298
- Page Range / eLocation ID:
- 461 to 482
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1146/annurev-statistics-031017-100108
