Phase Retrieval Problems in Inverse Scattering

萨克斯研究员研究量子力学的一维逆散射问题，其中要找到的解是薛定谔方程中的势，并开发了计算解的有效数值方法。 当以数据形式给出复值函数（称为反射系数）时，已知各种逆散射问题是唯一可解的。 然而，在许多有趣的应用中，反射系数无法测量，而只能通过实验确定其幅度。 逆散射中的相位问题在于解决逆散射问题，尽管缺乏反射系数中包含的相位信息。 一般来说，这些缺失的数据会给问题带来真正的非唯一性，人们必须通过使用其他类型的信息来补偿。 例如，人们可以以某种与情况的物理性质一致的方式限制可接受的解决方案的类别，或者可以使用与通常的散射数据相关的一些其他类型的数据。 除了唯一性问题之外，还需要开发准确可靠的计算技术。 模型问题——一维逆散射问题——出现在表面结构的中性和X射线反射研究中。 还研究了光学中的相关问题。 在此类应用中，自然要求势能在半线上消失，并且这基本上（尽管不是完全）消除了由于丢失相位信息而导致的模糊性。 该项目的主要目标是识别各种类型的进一步补充信息，其规范允许对潜力进行独特的恢复。 此外，还开发了数值方法来计算每种情况下的电势。 这里开发的数值方法的一个重要方面是，最终只需要解决相对较小维参数空间上的优化问题。 在科学和工程的许多领域中，在非常小的长度尺度上表征表面和界面的化学结构是很有意义的。 除了其内在的理论兴趣之外，这种能力还具有技术应用，特别是在材料科学和生物学领域。 近年来受到广泛关注的一项实验技术是使用所谓的反射率数据：将精心准备的 X 射线或中子束瞄准要寻找材料特性的表面，并仔细测量反射束。 由于光束与表面的相互作用，反射光束编码了大量关于表面的信息，因此人们面临着正确解释这些数据的问题。 该项目的主要目标是研究用于做出此类推论的某些分析和数值方法。 反射光束的特征在于振幅和相位分量，但传统的测量装置仅对振幅分量敏感。 众所周知，反射率数据的解释相当模糊与相位分量的缺失有关。 为了弥补这一点，人们希望使用各种可用的补充信息。 在这个项目中，研究人员试图从数学角度阐明如何将这些额外信息纳入自动数据处理技术中。

Sacks The investigator studies the one-dimensional inverse scattering problem of quantum mechanics, in which the solution to be found is a potential in the Schrodinger equation, and develops effective numerical methods for computing the solution. A variety of inverse scattering problems are known to be uniquely solvable when a complex valued function, known as the reflection coefficient, is given as data. In many interesting applications, however, the reflection coefficient cannot be measured, but instead only its amplitude can be determined experimentally. The phase problem in inverse scattering consists then in solving the inverse scattering problem despite the lack of phase information contained in the reflection coefficient. In general these missing data introduce genuine non-uniqueness into the problem, and one must compensate by making use of other kinds of information. For example, one might restrict the class of admissible solutions in some way consistent with the physics of the situation, or alternatively one might have available some other kinds of data related to the usual scattering data. Aside from uniqueness questions, there is also a need to develop accurate and reliable computational techniques. The model problem --- the one-dimensional inverse scattering problem --- arises in neutr on and x-ray reflection studies of surface structure. Related problems in optics are also studied. It is natural in such applications to require the potential to vanish on a half line, and this substantially, although not completely, removes the ambiguity due to the missing phase information. A main goal of this project is to identify various types of further supplementary information whose specification allows for unique recovery of the potential. Also, numerical methods are developed for computation of the potential in each case. An important aspect of the numerical approach developed here is that ultimately it is only necessary to solve an optimization problem over a relatively small dimensional parameter space. In a number of areas of science and engineering it is of interest to characterize the chemical structure of surfaces and interfaces on very small length scales. Aside from its intrinsic theoretical interest, this kind of capability has technological applications, especially in materials science and biology. One experimental technique which has received considerable attention in recent years is the use of so-called reflectivity data: a carefully prepared beam of x-rays or neutrons is aimed at the surface whose material properties are sought, and the reflected beam is carefully measured. Due to the interaction of the beam with the surface, the reflected beam encodes a great deal of information about the surface, and one is thus confronted with the problem of properly interpreting such data. The main goal of this project is to investigate certain analytical and numerical methods for making such inferences. The reflected beam may be characterized by amplitude and phase components, but conventional measuring devices are sensitive only to the amplitude component. It is well understood that considerable ambiguity in the interpretation of reflectivity data is associated with the absence of the phase component. To compensate for this one hopes to use various kinds of supplementary information which may be available. In this project the investigator seeks to elucidate from a mathematical point of view how it may be possible to incorporate such extra information into automatic data processing techniques.