Phase Retrieval Problems in Inverse Scattering

研究人员研究了量子力学中的一维逆散射问题，其中要找到的解是薛定谔方程中的势，并发展了有效的数值方法来计算该解。当一个被称为反射系数的复值函数被作为数据给出时，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，反射系数不能测量，而只能通过实验确定其幅度。因此，逆散射中的位相问题在于解决逆散射问题，尽管反射系数中缺少位相信息。一般来说，这些缺失的数据会给问题带来真正的非唯一性，人们必须利用其他类型的信息来弥补。例如，可以根据实际情况以某种方式限制可接受解的类别，或者可以获得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，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-产生于神经网络表面结构的ON和x射线反射研究。对光学中的相关问题也进行了研究。在这样的应用中，需要在半条线上消失的可能性是很自然的，这基本上(尽管不是完全)消除了由于丢失的相位信息而引起的模糊性。该项目的一个主要目标是确定各种类型的进一步补充信息，这些补充信息的规格允许以独特的方式恢复潜力。此外，还发展了计算每种情况下的势的数值方法。这里提出的数值方法的一个重要方面是，最终只需要在相对较小的参数空间上求解优化问题。在许多科学和工程领域中，在很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。除了其固有的理论兴趣外，这种能力还具有技术应用，特别是在材料科学和生物学方面。近年来受到广泛关注的一种实验技术是使用所谓的反射率数据：精心准备的X射线或中子光束对准其材料特性所在的表面，并仔细测量反射光束。由于……的相互作用带有表面的光束，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。这个项目的主要目标是研究做出这种推论的某些分析和数值方法。反射波束可以用幅度和相位分量来表征，但传统的测量设备只对幅度分量敏感。可以很好地理解，反射率数据的解释相当模糊，这与没有相位分量有关。为了弥补这一点，人们希望使用可能获得的各种补充信息。在这个项目中，研究人员试图从数学的角度阐明如何可能将这种额外的信息纳入自动数据处理技术。

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.