Inverse Problems via Layer Splitting

这个项目的目标是开发“层剥离”，或更恰当地说，“层分裂”技术在一个或多个维度的逆散射问题。我们正在努力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。逆散射问题通常在“频域”或“时域”中提出。从理论上讲，两者是等价的，一组数据通过傅立叶变换与另一组数据相关。然而，在一个领域中容易看到的特征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。 时域中的类似边界看起来要复杂得多。 同样，反射的时间（即，在你听到来自更远的物体的反射之前，你听到来自附近物体的反射）在转换到频域时成为解析函数空间中的理想的属性。我们方法的主要特征是仔细分析如何在两种情况下表达每个这样的特征，并使用这些来帮助描述散射数据并加强稳定性。科学的根本任务是研究世界。大多数情况下，我们通过将波（例如光，X射线，声音）指向物体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，这种实验的结果可以很容易地理解（例如照片，单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有直接的意义。越来越多的复杂的数学和统计技术需要将数据转换为对人类研究者有意义的东西（例如CAT扫描，中子散射实验）。这就是反问题在当今科学中的普遍作用。 它是口译实验的数学科学。当我们解决逆问题时，我们将物理学向后运行，从结果中推导出原因。虽然物理直觉通常建议最好的成像实验，但成像算法不是自然过程的模型。特别是，这些问题往往是不适定的，物理原理必须经常以与它们在直接模拟自然的“正向问题”中的功能完全不同的方式应用。因此，它们提供了一个利用数学直觉来补充物理直觉的独特机会。该项目旨在以新颖的方式利用物理原理开发稳定的成像技术。这里有一个例子，在以前的NSF支持下发现的。我们观察波的反射从一个分层的无损介质与未知的波速。如果人们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，那么要么猜测是正确的，要么计算出的反射由于过早地返回到接收器而违反了因果关系原理。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.