Inverse problems in industry

The term “inverse problems ” is itself a curious one as it does not have a tight mathematical definition. Nevertheless a very large class of what are generally agreed to be inverse problems involves going the opposite way to nature. We think of a “forward problem ” as being a mathematical model for a problem solved naturally in the physical world. Given a body with a specified spatially varying conductivity one can apply a current density at the boundary and the resulting potential can be found by solving a Neumann problem for an elliptic partial differential equation. This problem is “solved ” in the physical world and the voltage is well defined (up to a constant) and depends in a stable way on the conductivity and current. As our Neumann problem for an elliptic partial differential equation is a good model for the physical situation it shares these properties of existence, uniqueness and stability of solution. An example of an inverse problem would be to take measurements of the voltage at the boundary arising from the application of a number of patterns of current density on the boundary and attempt to deduce the conductivity in the interior. This problem often called Electrical Impedance Tomography (EIT: see Box) has application in medical diagnosis, industrial process monitoring and geophysical exploration. Another archetypal inverse problem concerns solving the heat equation back-wards in time. There the temperature u(x, t) satisfies the heat equatio