Figure 1: The measurement situation where we seek the temperature on a surface and measurements are available inside the material.
Figure 2: Temperature distribution below the earth's surface in Tibet computed from surface measurements. We also see the structure of the surface and the model of the thermal properties of the earth.
An application is that sometimes one wants to estimate the time dependent temperature on the surface of a body. Is the surface not accessible for measurements it is sometimes possible to instead measure inside the body, close to the surface, and compute the sought surface temperature by solving an ill-posed problem.
The measurement situation can be described as a Cauchy problem for the heat equation where measurements are available at x=1, and the temperature history is sought at x=0, see Figure 1. Applications exist within several scientific fields. In geology one wants to estimate the temperature several kilometers below the surface by using measurements at, or at least close to, the surface. Here one solves the stationary heat equation, with a non-linear thermal model, using an efficient numerical technique that adds stability to the calculations. Similar applications are found in the steel industry where, for instance, one wants to study in detail what happens to the surface of metal alloys during heat treatment.
Also the Helmholtz equation, which describes wave propagation, is of interest in applications. The problem of locating a sound source from measurements at a distance can be formulated as a Cauchy problem for the Helmholtz equation. The problem is ill-posed and special regularization techniques are required. Here I study, in collaboration with other researchers, a class of methods that are based on the fact that a certain quadratic form can be made positive definite by introducing an artificial boundar, see Figure 3. Doing this we can introduce a scalar product which is natural for the problem, and that can be used to implement efficient iterative regularisation methods.
Figure 3: We seek the solution to the Helmholtz on a part of the boundary given Cauchy data on a different part of the boundary. The interior boundaries are selected so that a certain quadratic form is positive definite.