Inverse and Ill-posed Problems

Abstract

Nonlinear ill-posed problems arise in a variety of important applications, ranging from medical imaging to geophysics to the nondestructive testing of materials. This chapter provides an overview of the various numerical methods for nonlinear ill-posed problems. For each method, a sequence of subproblems is solved. For each subproblem, the solution depends on a parameter. The method should have the following characteristics: (1) each subproblem must be well-posed, (2) each subproblem must be solved efficiently, and (3) the method must allow the inclusion of a priori information about desired solutions. The chapter also discusses the Levenberg–Marquardt method, which may be viewed as an iterated linearize-and-then-regularize approach to solving the nonlinear problem. As an alternative to the Penalized Least Squares Method, it proposes a constrained least squares approach to solving nonlinear ill-posed problems, in which regularization of the solution comes from imposing explicit bounds on the norm of the approximate solution.