Variable order smoothness priors for ill-posed inverse problems
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- by Daniela Calvetti, Erkki Somersalo and Ruben Spies PDF
- Math. Comp. 84 (2015), 1753-1773 Request permission
Abstract:
In this article we discuss ill-posed inverse problems, with an emphasis on hierarchical variable order regularization. Traditionally, smoothness penalties in Tikhonov regularization assume a fixed degree of regularity of the unknown over the whole domain. Using a Bayesian framework with hierarchical priors, we derive a prior model, formally represented as a convex combination of autoregressive (AR) models, in which the parameter controlling the mixture of the AR models can dynamically change over the domain of the signal. Moreover, the mixture parameter itself is an unknown and is to be estimated using the data. Also, the variance of the innovation processes in the AR model is a free parameter, which leads to conditionally Gaussian priors that have been previously shown to be much more flexible than the traditional Gaussian priors, capable, e.g., to deal with sparsity type prior information. The suggested method, the Weighted Variable Order Autoregressive model (WVO-AR) is tested with a computed example.References
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Additional Information
- Daniela Calvetti
- Affiliation: Case Western Reserve University, Department of Mathematics, Applied Mathematics and Statistics, 10900 Euclid Avenue, Cleveland Ohio 44106
- Email: daniela.calvetti@case.edu
- Erkki Somersalo
- Affiliation: Case Western Reserve University, Department of Mathematics, Applied Mathematics and Statistics, 10900 Euclid Avenue, Cleveland Ohio 44106
- Email: erkki.somersalo@case.edu
- Ruben Spies
- Affiliation: Instituto de Matemática Aplicada del Litoral, IMAL, CONICET-UNL, Güemes 3450, S3000GLN, Santa Fe, Argentina
- Email: rspies@santafe-conicet.gov.ar
- Received by editor(s): January 27, 2013
- Received by editor(s) in revised form: October 4, 2013
- Published electronically: November 13, 2014
- Additional Notes: The second author wishes to thank NSF for partial support of this work (DMS 1016183)
- © Copyright 2014 American Mathematical Society
- Journal: Math. Comp. 84 (2015), 1753-1773
- MSC (2010): Primary 65F22; Secondary 65C20
- DOI: https://doi.org/10.1090/S0025-5718-2014-02909-8
- MathSciNet review: 3335890