Image-Driven Two-Point Boundary Value Inverse Problems: A Case Study

Abstract

The collage method for treating boundary value inverse problems, given observational data values across the domain, is well-established in the literature. Here, instead, we formulate an inverse problem where the information about the dependent variable is given in the form of a greyscale image. The image gives no actual values, but does give some comparative information across the domain. In this paper, for the Sturm-Liouville two-point boundary value problem

$$\begin{aligned} (k(x)u'(x))' + q(x)u(x)= & {} f(x) \\ u(0)= & {} u_0 \\ u(L)= & {} u_L, \end{aligned}$$

we consider the inverse problem:

$$\begin{aligned}&Given\, f(x), q(x), \,the\, BCs, \,and\, a \,{\textit{256}}{\text {-}}grayscale\, image\, {\textit{of}}\, the \,level\\&values\, {\textit{of}}\, u(x),\, recover\, an\, estimate\, {\textit{of}}\, k(x). \end{aligned}$$

For context, we can think of the greyscale image as representing the isotherms of the steady-state heat distribution, concentrations in a chemical system, or population densities. After summarizing the mathematical framework, we focus on a particular example, considering several scenarios for which we can solve this inverse problem and exploring the impact of observation noise and image resolution on the recovered approximation.