Chapter 9 - Applications of vector spaces

Abstract

This chapter uses geometrical concepts drawn from the theory of vector spaces to solve inverse problems. The model parameters are considered one point in a space of all possible model parameters, the data are considered one point in a space of all possible data, and the theory is considered a way of mapping a point in model space into a point in data space. The coordinate axes of these spaces can be rotated or transformed, without changing any of the fundamental relationships. This process suggests methods of transforming a complicated inverse problem into one that is in some sense simpler, either with a rotation that preserved length (e.g., a Householder rotation) or with one that changes the measure of length in some specifically tailored fashion. This process is shown to clarify ideas of minimum error and uniqueness and leads to the definition of the natural solution of an inverse problem (and a corresponding natural generalized inverse). Singular-value decomposition is developed as a way to identify the null subspaces associated with solutions that do not affect the prediction error and to construct the natural solution. Vector space ideas are also useful in understanding problems with inequality constraints and can be used to derive general properties of its solution, as embodied in the Kuhn-Tucker theorem. These considerations lead to several new types of inverse problem solutions, including nonnegative least squares.