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A particle method for solving Fredholm equations of the first kind
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Crucinio, Francesca, Doucet, Arnaud and Johansen, Adam M. (2021) A particle method for solving Fredholm equations of the first kind. Journal of the American Statistical Association . doi:10.1080/01621459.2021.1962328 ISSN 0162-1459. (In Press)
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Official URL: https://doi.org/10.1080/01621459.2021.1962328
Abstract
Fredholm integral equations of the first kind are the prototypical example of ill-posed linear inverse problems. They model, among other things, reconstruction of distorted noisy observations and indirect density estimation and also appear in instrumental variable regression. However, their numerical solution remains a challenging problem. Many techniques currently available require a preliminary discretization of the domain of the solution and make strong assumptions about its regularity. For example, the popular expectation maximization smoothing (EMS) scheme requires the assumption of piecewise constant solutions which is inappropriate for most applications. We propose here a novel particle method that circumvents these two issues. This algorithm can be thought of as a Monte Carlo approximation of the EMS scheme which not only performs an adaptive stochastic discretization of the domain but also results in smooth approximate solutions. We analyze the theoretical properties of the EMS iteration and of the corresponding particle algorithm. Compared to standard EMS, we show experimentally that our novel particle method provides state-of-the-art performance for realistic systems, including motion deblurring and reconstruction of cross-section images of the brain from positron emission tomography.
| Item Type: | Journal Article | |||||||||||||||
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| Subjects: | Q Science > QA Mathematics R Medicine > RC Internal medicine |
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| Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | |||||||||||||||
| Library of Congress Subject Headings (LCSH): | Fredholm equations , Fredholm equations -- Numerical solutions , Expectation-maximization algorithms , Distribution (Probability theory) , Inverse problems (Differential equations), Monte Carlo method , Tomography, Emission -- Mathematical models | |||||||||||||||
| Journal or Publication Title: | Journal of the American Statistical Association | |||||||||||||||
| Publisher: | American Statistical Association | |||||||||||||||
| ISSN: | 0162-1459 | |||||||||||||||
| Official Date: | 9 September 2021 | |||||||||||||||
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| DOI: | 10.1080/01621459.2021.1962328 | |||||||||||||||
| Status: | Peer Reviewed | |||||||||||||||
| Publication Status: | In Press | |||||||||||||||
| Access rights to Published version: | Open Access (Creative Commons) | |||||||||||||||
| Date of first compliant deposit: | 19 July 2021 | |||||||||||||||
| RIOXX Funder/Project Grant: |
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