Semivariogram Methods for Modeling Whittle-Matérn Priors in Bayesian Inverse Problems

Document Type

Presentation Abstract

Presentation Date

5-6-2020

Abstract

We present a new technique, based on semivariogram methodology, for obtaining point estimates for use in prior modeling for solving Bayesian inverse problems. This method requires a connection between Gaussian processes with covariance operators defined by the Matérn covariance function and Gaussian processes with precision (inverse-covariance) operators defined by the Green's functions of a class of elliptic stochastic partial differential equations (SPDEs). We present a detailed mathematical description of this connection. We will show that there is an equivalence between these two Gaussian processes when the domain is infinite which breaks down when the domain is finite due to the effect of boundary conditions on Green's functions of PDEs. We show how this connection can be re-established using extended domains. We then introduce the semivariogram method for estimating the Matérn covariance parameters, which specify the Gaussian prior needed for stabilizing the inverse problem. Results are extended from the isotropic case to the anisotropic case where the correlation length in one direction is larger than another. The situation where the correlation length is spatially dependent rather than constant will also be considered. Finally, we compare and contrast the semivariogram method with a fully-Bayesian approach of finding estimates for and quantifying uncertainty in the hyperparameters. We implement each method in two-dimensional image inpainting test cases to show that it works on practical examples.

Additional Details

Doctoral Dissertation Defense. Link to the presenter's dissertation.

May 6, 2020 at 1:00 p.m. online

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