Inverse Problems via Layer Splitting

该项目的目标是开发用于一维或多维反散射问题的“层剥离”技术，或者更确切地说，“层分裂”技术。我们正致力于通过利用因果关系原理和尽可能多地描述散射数据来创建稳定的算法。逆散射问题通常是在“频域”或“时间域”中提出的。从理论上讲，这两者是等价的，一组数据通过傅里叶变换相互关联。然而，在一个领域中很容易看到的功能在另一个领域中可能会显得复杂得多。例如，频域中的散射算子很容易满足某些界限。时间域中类似的界限似乎要复杂得多。类似地，反射的时间(即你先听到附近物体的反射，然后再听到远处物体的反射)在转换到频域时成为解析函数空间中的一种理想属性。我们方法的一个主要特征是仔细分析如何在两种背景下表达每个这样的特征，并利用这些特征来帮助表征散射数据并增强稳定性。科学的根本任务是探索世界。大多数情况下，我们通过将波(例如光、X射线、声音)定向到物体上，并在波与该物体相互作用后观察波来实现这一目标。在某些情况下，这种实验的结果很容易理解(例如，一张照片、一张X射线)。然而，随着我们的技术变得越来越复杂，来自实验的数据越来越不可能直接有意义。为了将数据转换成对人类研究人员有意义的东西(例如CAT扫描、中子散射实验)，越来越多的复杂的数学和统计技术是必要的。这就是反问题在当今科学中发挥的普遍作用。它是口译实验的数学科学。当我们解逆问题时，我们倒着运行物理，从结果中推断原因。虽然物理直觉通常会暗示最佳的成像实验，但成像算法并不是自然过程的模型。具体地说，这些问题往往是不适定的，物理原理的应用方式必须经常与它们在直接模拟自然的“正问题”中的作用方式截然不同。因此，它们为使用数学直觉来补充物理直觉提供了一个独特的机会。该项目寻求以新的方式应用物理原理来开发稳定的成像技术。这里有一个例子，是在以前的NSF支持下发现的。我们观察到波在未知波速的分层无损介质中的反射。如果有人猜测部分介质中的波速，并使用该猜测来计算从介质的其余部分看到的反射，那么要么是猜测是正确的，要么是计算的反射过早地到达接收器，违反了因果关系原则。我们利用这一原则开发了一个非常稳定的算法。

The goal of this project is to develop "Layer Stripping", or more properly, "Layer Splitting" techniques for inverse scattering problems in one or more dimensions. We are working to create stable algorithms by utilizing the principle of causality and by characterizing the scattering data, as much as we possibly can. Inverse scattering problems are often posed in either the "frequencydomain" or in the "time domain". Theoretically, the two are equivalent, one set of data being related to the other by the Fourier transform. However, features that are easily seen in one domain can appear much more complicated in the other. For example, the scattering operator, in the frequency domain is easily seen to satisfy certain bounds. The analogous bounds in the time domain appear much more complicated. Similarly, the timing of reflections (i.e. you hear reflections from nearby objects before you hear those from objects further away) becomes a property of ideals in spaces of analytic functions when translated to the frequency domain.A main feature of our approach is to carefully analyze how to express each such feature in both contexts, and use these to help characterize the scattering data and enforce stability.The fundamental task of science is to investigate the world. Most often, we accomplish this goal by directing waves (e.g. light, X-rays, sound) at an object and observing the waves after they have interacted with that object. In some cases, the results of such an experiment can be readily understood (e.g. a photograph, a single X-ray). However, as our technology becomes more and more complex, the data from an experiment are less and less likely to be directly meaningful. More and more, sophisticated mathematical and statistical techniques are necessary to translate data into something which is meaningful to the human investigator (e.g. a CAT scan, a neutron scattering experiment). This is the general role that Inverse Problems plays in science today. It is the mathematical science of interpreting experiment. When we solve inverse problems we run physics backwards, deducing the cause from the effect. While physical intuition often suggests the best imaging experiments, an imaging algorithm is not a model of a natural process. In particular, these problems are often ill-posed, and physical principles must often be applied in ways that are radically different than how they would function in a "forward problem" which directly models nature. Thus they offer a unique opportunity for using mathematical intuition to supplement physical intuition. This project seeks to employ physical principles in novel ways to develop stable imaging techniques. Here is an example, discovered under previous NSF support. We observe reflections of waves from a layered lossless medium with unknown wavespeed. If one makes a guess at the wave speed in part of the medium and uses that guess to compute the reflection one would have seen from the rest of the medium, then either the guess is correct or the computed reflections violate the principle of causality by arriving back at the receiver too soon. We used this principle to develop a very stable algorithm.