Latest results: Bayesian inversion on unstructured lattices

Our paper on prior modelling for Bayesian statistical inverse problems has been accepted for publication in Inverse Problems and Imaging. The reference is

[1] L. Roininen, J. M. J. Huttunen and S. Lasanen, Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography, Inverse Problems and Imaging, Accepted March 2014.

The paper considers Gaussian smoothness priors on unstructured lattices. Such priors have been widely used in Bayesian inversion, however there has been a lack of systematic construction of proper integrable, computationally efficient and discretisation-invariant smoothness priors. In this paper, we show how to construct such priors via solutions of stochastic partial differential equations. The paper is a continuation for two earlier papers also published in IPI:

[2] L. Roininen, M. S. Lehtinen, S. Lasanen, M. Orispää and M. Markkanen, Correlation priors, Inverse Problems and Imaging 5 (2011) 167-184.

[3] L. Roininen, P. Piiroinen and M. Lehtinen, Constructing continuous stationary covariances as limits of the second-order stochastic difference equations, 7 (2013) 611-647.

The papers [2] and [3] offer convergence studies of the stochastic difference equations to some continuous objects. The paper [1] concentrates more on unstructured meshes, i.e. finite element approach, and is more application oriented with a strong emphasis on scientific computing. From a point of view of a geophysical observatory, the methods developed can be used especially in ground prospecting as well as in ionospheric tomography.

We will post a link to the full paper, as soon as it is online. Meanwhile below you may find the abstract of the paper. If you are interested in the topic, the authors will gladly respond to any questions or comments.

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

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.