# SEMIVARIOGRAM METHODS FOR MODELING WHITTLE-MATERN PRIORS IN BAYESIAN INVERSE PROBLEMS

2020

Dissertation

## Degree Type

Doctor of Philosophy (PhD)

Mathematics

## Department or School/College

Department of Mathematical Sciences

## Committee Chair

Johnathan Bardsley

## Commitee Members

Jonathan Graham, Javier Perez Alvaro, Emily Stone, Jesse Johnson

## Keywords

Bayesian methods, boundary conditions, Gaussian field, inverse problems, variogram, Whittle-Matérn

## Publisher

University of Montana

## 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 Mat ́ern 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 depen- dent 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 imple- ment each method in two-dimensional image inpainting test cases to show that it works on practical examples. The MATLAB code for all of these methods can be found here: https://github.com/rbrown53/DissertationCodes.

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