Inverse Problems via Layer Splitting

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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.