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Mathematics of Computation

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Variable order smoothness priors for ill-posed inverse problems


Authors: Daniela Calvetti, Erkki Somersalo and Ruben Spies
Journal: Math. Comp. 84 (2015), 1753-1773
MSC (2010): Primary 65F22; Secondary 65C20
Published electronically: November 13, 2014
MathSciNet review: 3335890
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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.


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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

DOI: https://doi.org/10.1090/S0025-5718-2014-02909-8
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)
Article copyright: © Copyright 2014 American Mathematical Society