Inverse Source Problems, Splitting, and Uncertainty

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This project studies fundamental problems which arise in the development of remote sensing technologies. Remote sensing systems are either active or passive. An underwater array of acoustic sensors that seeks to locate a submarine (the source) based on the noise radiated from the submarine's engine is a passive remote sensing system. A mathematical model for passive remote sensing is the Inverse Source Problem. A sonar array that transmits its own (incident) wave to locate the submarine based on the properties of the echo, is an active remote sensing system. In this case, the array, not the submarine, is the primary source and the submarine is referred to as the scatterer because it scatters the sound wave in new directions. A mathematical model for active remote sensing is the Inverse Scattering Problem. Scatterers are also called secondary sources or induced sources because the primary or incident wave causes them to radiate a secondary or scattered wave. This terminology emphasizes the close relationship between the inverse source problem and the inverse scattering problem. Every scattered wave is a wave radiated by an induced source, the source is different for different incident waves, but the support (size and location) is always the same. The principal investigator seeks to elucidate the fundamental limitations that physics places on these technologies, and to develop algorithms to extract all information within these limits.Rephrasing the uncertainty principle of Donoho and Stark in terms of inequalities for inner products and estimates on an underlying operator has made it possible to look for applications of these principles in new contexts, including the inverse source problem. Although the inverse source problem for the Helmholtz equation does not have a unique solution, the investigator has developed a well-defined notion of support for far fields radiated by unions of well-separated sources. The splitting operator decomposes a far field radiated by such a union of sources into the far fields radiated by each individual source. A major goal is to estimate the condition number of the splitting operator, and to develop algorithms to do the splitting and locate the individual sources. The condition number is estimated in terms of physically meaningful parameters, which are wavelength, diameters, and distance between sources. Another related objective is the development of norms and estimates for partial differential equations related to wave phenomena that, unlike many weighted norms, can be readily interpreted in physical units, and are therefore more easily accessible to the broader scientific community.