Inverse Problems and Imaging (IPI)

Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography

Pages: 561 - 586, Volume 8, Issue 2, May 2014      doi:10.3934/ipi.2014.8.561

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Lassi Roininen - University of Oulu, Sodankylä Geophysical Observatory, Tähteläntie 62, FI-99600 Sodankylä, Finland (email)
Janne M. J. Huttunen - University of Eastern Finland, Department of Applied Physics, Yliopistonranta 1 F, FI-70211 Kuopio, Finland (email)
Sari Lasanen - Department of Mathematical Sciences, P.O. Box 3000, 90014 University of Oulu, Finland (email)

Abstract: We study flexible and proper smoothness priors for Bayesian statistical inverse problems by using Whittle-Matérn Gaussian random fields. We review earlier results on finite-difference approximations of certain Whittle-Matérn random field in $\mathbb{R}^2$. Then we derive finite-element method approximations and show that the discrete approximations can be expressed as solutions of sparse stochastic matrix equations. Such equations are known to be computationally efficient and useful in inverse problems with a large number of unknowns.
    The presented construction of Whittle-Matérn correlation functions allows both isotropic or anisotropic priors with adjustable parameters in correlation length and variance. These parameters can be used, for example, to model spatially varying structural information of unknowns.
    As numerical examples, we apply the developed priors to two-dimensional electrical impedance tomography problems.

Keywords:  Electrical impedance tomography, Bayesian statistical inversion, smoothness priors.
Mathematics Subject Classification:  Primary: 65N21; Secondary: 60J22.

Received: April 2013;      Revised: February 2014;      Available Online: May 2014.