Nonlinear Inverse Problems in Imaging

Chapter 3

Basics of Forward Problem

To solve an inverse problem, we need to establish a mathematical model of the underlying physical phenomena as a forward problem. Correct formulation of the forward problem is essential to obtain a meaningful solution of the associated inverse problem. Since the partial differential equation (PDE) is a suitable mathematical tool to describe most physical phenomena, we will study the different kinds of PDEs commonly used in physical science.

When we set up a forward problem associated with an inverse problem, we should take account of the well-posedness (Hadamard 1902) as described in Chapter 1. In constructing a mathematical model that transforms physical phenomena into a collection of mathematical expressions and data, we should consider the following three properties.

• Existence: at least one solution exists. For example, the problem u′′(x) = 1 in [0, 1] has at least one possible solution, u = x²/2, whereas the problem |u′′(x)|² = − 1 has no solution.
• Uniqueness: only one solution exists. For example, the boundary value problem

has the unique solution u(x) = e^x, whereas the boundary value problem

has infinitely many solutions, u(x) = x, x², x³, ….

• Continuity or stability: a solution depends continuously on the data.

A problem without the ...