Fast solvers for frequency-domain wave-scattering problems and applications

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The computation of wave phenomena is widely needed in many application areas, for example models of radar and telecommunications devices require the computation of electromagnetic waves while the implementations of seismic and medical imaging algorithms use acoustic, elastic, and electromagnetic waves to obtain information about the earth's subsurface and the human body respectively. Computer models of the propagation of waves arise naturally in the design and implementation of these technologies. Medical imaging technicians use computer models of how the material composition of the human body scatters incoming electromagnetic waves in order to solve the "inverse problem'' of reconstructing the internal makeup of a human being from an observed scattered wave field. Similarly, seismologists use computer models of how the material properties of the earth's subsurface affects the transmission of elastic waves in order to reconstruct the earth's subsurface properties from observed echoes of elastic wavesThis technology is hugely useful, for example in the medical context it means we can often diagnose health problems without a need for more invasive techniques. In the seismology case it makes something seemingly impossible become possible - since it is never physically possible to explore all of the earth's subsurface properties by simply boring holes. However the fast and accurate computer modelling of such wave phenomena is complicated and costly (in terms of computer time), principally (but not solely) because of the highly oscillatory nature of the waves and the complicated media through which they pass. Thus there is a strong need for new methods that speed up such models and that task is a principal focus of this research. This project will devise and mathematically justify new families of fast methods for implementing these computer wave models, and will make the new methods available through two software platforms which are accessible to a wide range of scientists as well as in an additional specialist high performance computing library. As well as devising new methods for modelling (which work well on today's multiprocessor computers), the project will also involve direct collaboration with two companies - Schlumberger (a Project Partner, interested in seismology) and ABB (interested in electromagnetic computations) - as well as two academic groups, one in geosciences and one in electromagnetics.