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Inverse Problems
Here's a prototype of an inverse problem. Suppose we want to determine the electrical conductivity inside an object by imposing electrical potentials at the boundary and measuring the resulting induced boundary current flux. Mathematically the problem takes the following form.
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^3$, and let $\gamma$ be a positive function on $\Omega$, representing the electrical conductivity. If we impose the electrical potential $u|_{\partial \Omega} = f$ on the boundary, then physics says that inside $\Omega$, the potential $u$ must solve the boundary value problem \begin{equation}\label{CauchyProblem} \begin{split} \nabla \cdot \gamma \nabla u &= 0 \\ u|_{\partial \Omega} & = f. \end{split} \end{equation} Then the resulting boundary current flux is given by $\gamma \partial_{\nu} u$. Since this boundary value problem has unique solutions, each electrical conductivity function $\gamma$ defines a map $\Lambda_{\gamma}$ which takes a boundary potential $f$ to its resulting current flux $\gamma \partial_{\nu} u$. This map is called a voltage-to-current or Dirichlet-to-Neumann map, and our question is now whether knowledge of $\Lambda_{\gamma}$ can be used to determine $\gamma$.
This problem is sometimes referred to as Calderón's problem, after Alberto Calderón, and it is a classic example of an inverse problem: we are given information about the solutions to an equation with unknown coefficients, and we want to know if we can use that information to determine the coefficients.
This sort of problem arises from all sorts of practical situations: CT scans and MRIs, for example, can be understood mathematically as inverse problems. So can optical tomography, in which measurements of scattered light are used to create clear images -- in fact some models for optical tomography are closely related to Calderón's problem. Applications aren't just limited to medical imaging, either: seismic tomography, in which pressure waves from earthquakes are used to image the earth's interior, is another classic example of an inverse problem.
Solving inverse problems typically requires deep ideas from harmonic analysis and PDE. If we consider Calderón's problem, for example, and suppose $u$ is a solution to the boundary value problem above, we can integrate by parts to see that \begin{equation} \int_{\Omega} \gamma |\nabla u |^2 \,dx = \int_{\partial \Omega} \overline{u}\Lambda_{\gamma} u \,dS. \end{equation} If we can always measure $u$ and $\Lambda_{\gamma} u$ at the boundary, it follows that we can recover $\gamma$ as long as we have enough solutions $u$ that the set of all $|\nabla u|^2$ are dense in some appropriate function space.
In general, though, showing that some set of functions is dense in a given function space is hard. To solve the inverse problem this way, you therefore need a very thorough understanding of the fine properties of solutions to the given boundary value problem, and this takes you deep into various rabbit holes of mathematical knowledge.
For a more thorough introduction to Calderón's problem, see these notes by Mikko Salo.
The Finnish Centre of Excellence in Inverse Problems also maintains a semi-comprehensive list of conferences related to inverse problems.
In February 2017 I gave a recorded talk at the IMA on some of my work in acousto-optics.
And for more information on my past work on inverse problems, you can return here and look at some of my papers.