Inverse Problems via Layer Splitting

通过层分割的反演问题

• 批准号: 0099838
• 负责人: John Sylvester
• 金额: $ 11.13万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0099838&HistoricalAwards=false
• 关键词: Inverse; Problems; via; Layer; Splitting

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.

该项目的目标是开发“层剥离”，或更准确地说，“层分裂”技术，以解决一维或多维的逆散射问题。我们正在努力利用因果关系原理并尽可能地表征散射数据来创建稳定的算法。逆散射问题通常出现在“频域”或“时域”中。理论上，两者是等价的，一组数据通过傅里叶变换与另一组数据相关。然而，在一个领域中很容易看到的特征在另一个领域中可能显得复杂得多。例如，频域中的散射算子很容易看出满足某些界限。 时域中的类似界限显得更加复杂。 类似地，反射的时间（即，您先听到附近物体的反射，然后再听到远处物体的反射）转换到频域后，就成为分析函数空间中理想的属性。我们方法的一个主要特点是仔细分析如何在两种情况下表达每个此类特征，并使用它们来帮助表征散射数据并增强稳定性。科学的基本任务是研究世界。大多数情况下，我们通过将波（例如光、X 射线、声音）引导至某个物体并在波与该物体相互作用后观察它们来实现此目标。 在某些情况下，此类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，实验中的数据越来越不可能具有直接意义。越来越需要复杂的数学和统计技术将数据转化为对人类研究人员有意义的东西（例如 CAT 扫描、中子散射实验）。这就是反问题在当今科学中扮演的一般角色。 它是解释实验的数学科学。当我们解决逆问题时，我们将物理学倒推，从结果中推断出原因。虽然物理直觉通常建议最好的成像实验，但成像算法并不是自然过程的模型。特别是，这些问题通常是不适定的，并且物理原理的应用方式通常必须与它们在直接模拟自然的“正向问题”中的运作方式截然不同。因此，它们提供了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来开发稳定的成像技术。这是一个在以前的 NSF 支持下发现的示例。我们观察波速未知的层状无损介质的波反射。如果人们对部分介质中的波速进行猜测，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到达接收器而违反了因果关系原理。我们利用这个原理开发了非常稳定的算法。