Wave propagation and imaging in complex environments is an important topic in applied mathematics with a wide range of applications, including not only radar and seismic reconstruction but also many others, such as laser beam propagation through clouds, light propagation through the atmosphere in astronomy, secure communications in scattering media, medical imaging, and nondestructive evaluation of materials. This cluster will involve contemporary topics on waves in random media. Recent progress in this area has been motivated on the one hand by a range of applications that involve partly or fully incoherent waves, such as time reversal, active-array imaging, passive imaging and hybrid imaging, and on the other hand by advances in sensor technology that have brought new and massive amounts of data.
Mathematically, the problem of waves in random media may be placed in a stochastic framework, where the complex environments are realizations of random fields and the scattering regimes are determined by important physical scales, such as the wavelength, the distance of propagation, and the scale of the fluctuations. The theory for waves in random media is concerned in particular with describing the statistics of the wave field in a concise manner. It is fundamental to applications in imaging through complex environments.
This cluster brings together scientists with expertise in stochastic analysis and theoretical, numerical, and experimental wave propagation and imaging, random matrix theory, and compressed sensing with the goal of exchanging ideas and advancing the field. The main topics of interest are:
Radar imaging is a highly developed field that involves a rich variety of mathematical and computational areas, such as electromagnetic theory and partial differential equations, functional analysis, harmonic analysis, coding theory, Lie groups, and statistical signal processing. Still, many challenges remain. For example, more of the physics needs to be incorporated into solutions to the radar inverse problem, including physical scattering mechanisms, multiple scattering, moving objects, and corrections for propagation through random or complex media. Recent hardware developments make it possible to collect an unprecedented amount of data, sampled at extremely high rates and often including polarimetry, and mathematical techniques are needed for fast, accurate image formation and interpretation of this data. Seismology is faced with many similar mathematical problems; this program provides an opportunity for synergistic development of both fields.
This cluster will bring together laboratory scientists and academic researchers with the overarching goal of advancing the field of radar imaging. Particular attention will be paid to exploring mathematical commonalities between radar and seismic imaging and to possibilities for applying seismic imaging techniques to radar imaging. Specific topics of interest include:
Seismic inversion is the process of transforming seismic data generated by active or passive sources into a quantitative description of the subsurface properties of the earth. It addresses important problems related to our energy needs, to hazards such as earthquakes and volcanic eruptions, and to the general study of Earth's interior on a planetary scale. It draws extensively from the mathematical sciences by applying tools from signal processing, elastic and electromagnetic theory, partial differential equations, harmonic analysis, inverse-problem theory, numerical analysis, optimization, and statistics. The sheer volume of seismic data also makes it arguably the oldest area with "big data." Theoretical and engineering developments have advanced this field tremendously in the past several decades; however, there remain many fundamental open questions, ranging from uniqueness and uncertainty through the nonlinear nature of these problems.
This cluster will bring together academic and industrial researchers with the goal of addressing some of the key challenges in the analysis of inverse problems, with emphasis on reconstruction, big data and fast algorithms. Thematic topics include: