The investigator and his colleagues develop numerical methods for scattering
problems, and perform a detailed analysis of these methods. Specifically, they develop boundary element methods for integral formulations of the Helmholtz equation, in which the basis functions are asymptotically derived. Each basis function is the product of a smooth amplitude and a highly oscillatory phase factor, like the asymptotic solution. The phase factor is determined a-priori by geometrical optics and the geometrical theory of diffraction, while the amplitude is a smooth function. This choice of oscillatory phase factor, ensures that the number of unknowns is virtually independent of the frequency. The basis functions being studied are designed in a way that the matrix, resulting from the discretization, has a number of significant entries which is proportional to the number of basis functions, in the large frequency limit. Hence, this matrix can be approximated with a sparse approximate matrix. The investigator and his colleagues explore the use of the sparse matrix instead of the original matrix, in order to obtain an approximate solution to the problem, and they study the solution of the approximate problem with iterative methods, and multi-grid methods. In addition they study the use of the sparse matrix as a preconditioner. This new methodology is being developed by studying problems of increasing complexity, such as convex domains with corners in two dimensions, and convex three dimensional domains. The impact of this research is therefore in the development of discretization schemes for the Helmholtz equation in which the number of unknowns is virtually independent of the frequency, and the design of optimally fast algorithms for solving those discrete problems. This transforms problems which are computationally exceedingly difficult if not intractable, into ones which are readily solved.
The solution of scattering problems has many applications in technology. Applications include remote sensing, ultra-sound imaging of tissue, non-destructive testing, geophysical prospecting, the design of reduced radar signature airplanes, and the determination of the acoustic signature of submersible vehicles. For the current configurations of interest, geometrical complications and/or compositional complexity in the underlying structure make manual solutions practically impossible. Moreover, in the mid to high frequency regime, standard computational approaches require an exceedingly large amount of calculations, leaving some problems computationally intractable. The fast and efficient algorithms for scattering problems developed by the investigator and his colleagues will broadly impact design and discovery tools in the application areas mentioned above.
