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Numerical determination of potentials

机译：电位的数值确定

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Abstract: In this paper we will describe some computational methods for the recovery of potentials under a variety of types of data measurements. We will look at both inverse Sturm-Liouville problems on a finite interval and inverse scattering problems on the line. The unifying approach to all of this is the fact that many of these types of problems can be solved by converting the given spectral or scattering data into boundary data for a certain hyperbolic partial differential equation. In all cases the problem is an overdetermined one and it is precisely this fact that allows us to recover the potential. Further, we will show that this translated problem can be solved in a numerically stable way, and indeed this approach leads to an excellent, as well as a unifying, scheme for the reconstruction of the potential. There is a classical method of solving many of the above types of problems and the definitive formulation is due to Gel'fand, Levitan and Marchenko some forty years ago. Our approach has many similarities, and indeed the same starting point, but the crucial difference is while the original scheme reduced the problem to a Fredholm integral equation, we will exploit instead an equivalent hyperbolic partial differential equation.!19
机译：摘要：在本文中，我们将描述一些用于在各种类型的数据测量下恢复电势的计算方法。我们将研究有限区间上的Sturm-Liouville逆问题和在线上的逆散射问题。所有这些的统一方法是，可以通过将给定的光谱或散射数据转换为某个双曲型偏微分方程的边界数据来解决许多这类问题。在所有情况下，问题都是一个无法解决的问题，正是这一事实使我们得以恢复潜力。此外，我们将证明可以以数值稳定的方式解决此转换后的问题，并且实际上，这种方法可导致极好的以及统一的电势重构方案。有一种经典的方法可以解决许多上述类型的问题，而最终的公式归因于40年前的Gel'fand，Levitan和Marchenko。我们的方法有许多相似之处，并且实际上是相同的起点，但是关键的区别是，当原始方案将问题简化为Fredholm积分方程时，我们将改用等价的双曲型偏微分方程！19

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