Bayes risk A-optimal experimental design methods for ill-posed inverse problems

Abstract

Optimal experimental design methods are well established and have been applied in a variety of different fields. Most of the classical methods in optimal experimental design however neglect the subject of ill-posedness. Ill-posedness is an issue that is prevalent when solving inverse problems. In order to solve most real-world inverse problems of interest, we must use methods known as regularization to help stabilize the final solution. The use of regularization introduces a bias into the obtained estimates that classical optimal experimental design techniques do not take into account. The primary goal of this thesis is to consider some advancements in optimal experimental design methods that can be applied to ill-posed inverse problems. We present a general method known as Bayes risk A-optimal design that uses the prior moment conditions of the inverse problem. The Bayes risk A-optimal design has a natural connection to the Tikhonov regularization estimator. We demonstrate how this connection arises and how this method can be used in practice. We also address some of the important theoretical considerations behind the use of A-optimal, and more generally all linear-optimal, design criteria. One of the most important results in optimal design theory is the so-called equivalence theorem. We present an updated version, the generalized equivalence theorem, that can be used with Bayes risk A-optimal designs for ill-posed inverse problems. This updated form of the equivalence theorem is paramount as it justifies the use of techniques that address ill-posedness for all linear-optimal design criteria. Optimal designs are typically found by using either a sequential search algorithm or a convex optimization routine. To address the actual computation of an A-optimal design for ill-posed inverse problems We discuss the strengths and weakness behind both general approaches and ultimately present a hybrid method that uses both techniques for determining a design. The result is a method that is able to use the strengths of both approaches, specifically a method that is highly customizable and can be used to find A-optimal designs with specific attributes that are desirable to the user.