Mathematical Sciences: Inverse Nodal Problems and Fluid Mechanics

哈尔德是从事反问题和流体力学研究的研究员。McLaughlin和研究者证明了矩形区域上薛定谔方程中的势可以由节线(即特征函数消失的点的集合)唯一地确定(直到一个附加常数)。目前的证明假定电势在矩形边界附近为常数。这一假设排除了一大类势能，一个目标是消除这一限制性假设。在流体力学领域，巴特克最近提出了一种非常有趣的方法来求解二维和三维流动的欧拉方程。利用对计算速度核的一种新的解释，研究人员旨在建立该方法的收敛。弹性材料(如梁或板)的性质可以用弹性参数来描述。在许多问题中，弹性参数是不可直接观测的。例如，人们在地球表面看不到由于石油的存在而在两英里深处密度发生变化的情况。同样，金属内部深处的裂纹在外部也可能看不到。在这种情况下，人们可以通过发出波来探测材料，并研究波的反射或材料的振动。反问题理论的一个重要目标是证明可以从观测到的数据中推断出弹性材料的性质，并开发有效的算法来估计材料的性质。在工程、气象学和医学中，能够准确地计算流体的运动是很重要的。例如：(1)流经管道的水，(2)天气预报中的空气流动，以及(3)通过人造心脏的血液流动。目前二维流动的计算是常规的，但计算三维水流的运动仍然是一个挑战，特别是在水流湍动的情况下。这是一个拥有足够强大的计算机的问题，也是描述资本流动的一种非常有效的方式。研究人员的工作重点是特别适合于研究非常复杂的流动的数值方法。

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.