Efficient Methods for Image Reconstruction and Uncertainty Quantification with Application to Photo-acoustic Tomography.

PETROSKE, KATRINA ELISE. Efficient Methods for Image Reconstruction and Uncertainty Quantification with Application to Photo-acoustic Tomography. (Under the direction of Arvind Krishna Saibaba.) Inverse problems arise in various scientific applications such as biomedical and geophysical imaging applications. A significant amount of effort has focused on developing efficient and robust methods to compute solutions to inverse problems by reconstructing parameters of interest. In addition to parameter reconstruction, there is a critical need to be able to obtain valuable uncertainty information (e.g., solution variances, samples, and credible intervals) to assess the reliability of computed solutions and to aid in decisionmaking. To demonstrate the techniques developed in this work we use a model problem, photo-acoustic tomography (PAT), an imaging modality that is used in breast and brain imaging. The goal of PAT is to recover the spatial distribution of the optical properties, such as the absorption coefficient, from ultrasound measurements. The PAT reconstruction process can be mathematically formulated as two coupled inverse problems involving partial differential equations. We take a two-step approach to solving PAT: the first step is a linear inverse problem for which we take a Bayesian approach and the second step is non-linear for which we take a deterministic approach. In the first part of the thesis, we focus on uncertainty quantification for linear inverse problems with Gaussian posterior distributions. We exploit Krylov subspace methods to develop and analyze new techniques for large-scale uncertainty quantification in inverse problems by exploring the posterior distribution. In particular, we use the generalized Golub-Kahan bidiagonalization to derive an approximation of the posterior covariance matrix, and we provide theoretical results that quantify the accuracy of the approximate posterior covariance matrix and of the resulting posterior distribution. Then, we describe efficient methods that use the approximation to compute measures of uncertainty, such as the Kullback-Liebler divergence and optimality criteria, which are important in the context of optimal experimental design. Additionally, we present two methods that use the preconditioned Lanczos algorithm to efficiently generate samples from the posterior distribution. Numerical examples, including a model problem from PAT, demonstrate the effectiveness of the described approaches. In the second part of the thesis, we focus on the non-linear inverse problem, also known as Quantitative photo-acoustic tomography (QPAT). We take a deterministic approach and formulate QPAT a nonlinear PDE-constrained optimization problem. We develop Newton and Gauss-Newton solvers for QPAT in which the search directions are computed inexactly using the preconditioned Conjugate Gradient method. We study various aspects of the solvers such as the type of regularization used, choice of preconditioner, choice of stopping criteria, and the behavior as the number of sources is increased. The performance of the solvers is demonstrated through a synthetic model problem from QPAT.