Hald The investigator undertakes studies il inverse problems and fluid mechanics. McLaughlin and the investigator have shown that the potential in a Schroedinger equation on a rectangular domain can be uniquely determined (up to an additive constant) by the nodal lines, i.e. the set of points where the eigenfunctions vanish. The present proof assumes that the potential is constant near the boundaries of the rectangle. This assumption excludes a large class of potentials and one goal is to remove this restrictive assumption. In the field of Fluid Mechanics Buttke has recently proposed a very interesting method for solving Euler's equations for two and three dimensional flows. Using a new interpretation of the computed velocity kernels, the investigator aims to establish the convergence of the method. The properties of elastic materials (such as beams or plates) can be described by elastic parameters. In many problems the elastic parameters are not available for direct observation. For example, one cannot see on the surface of the earth that the density changes at the depth of two miles due to the presence of oil. Similarly, a crack deep inside a piece of metal may not be observable on the outside. In such cases one may probe the material by sending waves through it, and study either the reflections of the waves or the vibrations of the material. One important goal of the theory of inverse problems is to show that the property of the elastic material can be inferred from the data thus observed, and to develop efficient algorithms for estimating the material properties. In engineering, meteorology, and medicine it is important to be able to compute the movement of a fluid accurately. Examples are: (1) water running through pipes, (2) the movement of air in weather prediction, and (3) the flow of blood through an artificial heart. At present calculations of two-dimensional flows are done routinely, but it is still a challenge to calculate the mo vement of water in three dimensions, especially when the water is turbulent. It is a question of having sufficiently powerful computers, and a very efficient way of describing the flows. The work of the investigator is focused on numerical methods that are specially suited for studying the movements of very complicated flows.
