Bayesian Inverse Problems with L[subscript 1] Priors: A Randomize-Then-Optimize Approach
Prior distributions for Bayesian inference that rely on the L[subscript 1]-norm of the parameters are of considerable interest, in part because they promote parameter fields with less regularity than Gaussian priors (e.g., discontinuities and blockiness). These L[subscript 1]-type priors include the...
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Society for Industrial & Applied Mathematics (SIAM)
2018
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| Online Access: | http://hdl.handle.net/1721.1/114625 https://orcid.org/0000-0002-4478-2468 https://orcid.org/0000-0001-8242-3290 |
| _version_ | 1826212189971677184 |
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| author | Bardsley, Johnathan M. Solonen, Antti Cui, Tiangang Wang, Zheng Marzouk, Youssef M |
| author2 | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics |
| author_facet | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Bardsley, Johnathan M. Solonen, Antti Cui, Tiangang Wang, Zheng Marzouk, Youssef M |
| author_sort | Bardsley, Johnathan M. |
| collection | MIT |
| description | Prior distributions for Bayesian inference that rely on the L[subscript 1]-norm of the parameters are of considerable interest, in part because they promote parameter fields with less regularity than Gaussian priors (e.g., discontinuities and blockiness). These L[subscript 1]-type priors include the total variation (TV) prior and the Besov space B[subscript 1,1][superscript s] prior, and in general yield non-Gaussian posterior distributions. Sampling from these posteriors is challenging, particularly in the inverse problem setting where the parameter space is high-dimensional and the forward problem may be nonlinear. This paper extends the randomize-then-optimize (RTO) method, an optimization-based sampling algorithm developed for Bayesian inverse problems with Gaussian priors, to inverse problems with L[subscript 1]-type priors. We use a variable transformation to convert an L[subscript 1]-type prior to a standard Gaussian prior, such that the posterior distribution of the transformed parameters is amenable to Metropolized sampling via RTO. We demonstrate this approach on several deconvolution problems and an elliptic PDE inverse problem, using TV or Besov space B[subscript 1,1][superscript s] priors. Our results show that the transformed RTO algorithm characterizes the correct posterior distribution and can be more efficient than other sampling algorithms. The variable transformation can also be extended to other non-Gaussian priors. (An erratum is attached.) |
| first_indexed | 2024-09-23T15:17:46Z |
| format | Article |
| id | mit-1721.1/114625 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T15:17:46Z |
| publishDate | 2018 |
| publisher | Society for Industrial & Applied Mathematics (SIAM) |
| record_format | dspace |
| spelling | mit-1721.1/1146252022-09-29T14:00:12Z Bayesian Inverse Problems with L[subscript 1] Priors: A Randomize-Then-Optimize Approach Bardsley, Johnathan M. Solonen, Antti Cui, Tiangang Wang, Zheng Marzouk, Youssef M Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Wang, Zheng Marzouk, Youssef M Prior distributions for Bayesian inference that rely on the L[subscript 1]-norm of the parameters are of considerable interest, in part because they promote parameter fields with less regularity than Gaussian priors (e.g., discontinuities and blockiness). These L[subscript 1]-type priors include the total variation (TV) prior and the Besov space B[subscript 1,1][superscript s] prior, and in general yield non-Gaussian posterior distributions. Sampling from these posteriors is challenging, particularly in the inverse problem setting where the parameter space is high-dimensional and the forward problem may be nonlinear. This paper extends the randomize-then-optimize (RTO) method, an optimization-based sampling algorithm developed for Bayesian inverse problems with Gaussian priors, to inverse problems with L[subscript 1]-type priors. We use a variable transformation to convert an L[subscript 1]-type prior to a standard Gaussian prior, such that the posterior distribution of the transformed parameters is amenable to Metropolized sampling via RTO. We demonstrate this approach on several deconvolution problems and an elliptic PDE inverse problem, using TV or Besov space B[subscript 1,1][superscript s] priors. Our results show that the transformed RTO algorithm characterizes the correct posterior distribution and can be more efficient than other sampling algorithms. The variable transformation can also be extended to other non-Gaussian priors. (An erratum is attached.) 2018-04-09T16:05:45Z 2018-04-09T16:05:45Z 2017-01 2018-04-04T15:58:38Z Article http://purl.org/eprint/type/ConferencePaper 1064-8275 1095-7197 http://hdl.handle.net/1721.1/114625 Wang, Zheng, Johnathan M. Bardsley, Antti Solonen, Tiangang Cui, and Youssef M. Marzouk. “Bayesian Inverse Problems with L[subscript 1] Priors: A Randomize-Then-Optimize Approach.” SIAM Journal on Scientific Computing 39, 5 (January 2017): S140–S166 © 2017 Society for Industrial and Applied Mathematics https://orcid.org/0000-0002-4478-2468 https://orcid.org/0000-0001-8242-3290 http://dx.doi.org/10.1137/16M1080938 SIAM Journal on Scientific Computing Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Society for Industrial & Applied Mathematics (SIAM) arXiv |
| spellingShingle | Bardsley, Johnathan M. Solonen, Antti Cui, Tiangang Wang, Zheng Marzouk, Youssef M Bayesian Inverse Problems with L[subscript 1] Priors: A Randomize-Then-Optimize Approach |
| title | Bayesian Inverse Problems with L[subscript 1] Priors: A Randomize-Then-Optimize Approach |
| title_full | Bayesian Inverse Problems with L[subscript 1] Priors: A Randomize-Then-Optimize Approach |
| title_fullStr | Bayesian Inverse Problems with L[subscript 1] Priors: A Randomize-Then-Optimize Approach |
| title_full_unstemmed | Bayesian Inverse Problems with L[subscript 1] Priors: A Randomize-Then-Optimize Approach |
| title_short | Bayesian Inverse Problems with L[subscript 1] Priors: A Randomize-Then-Optimize Approach |
| title_sort | bayesian inverse problems with l subscript 1 priors a randomize then optimize approach |
| url | http://hdl.handle.net/1721.1/114625 https://orcid.org/0000-0002-4478-2468 https://orcid.org/0000-0001-8242-3290 |
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