The project proposes development of efficient, accurate and rapidly-convergent algorithms for the simulation of scattering of acoustic and electromagnetic fields within and around structures that consist of complex combinations of penetrable structures and possess complex geometrical characteristics. These are configurations of fundamental importance in diverse fields, with application in communications, remote sensing and surveillance, geophysical prospecting, materials science and biomedical imaging – amongst many others. The simulation of electromagnetic wave propagation in such complex structures gives rise to a host of significant computational challenges that arise from presence of complex material arrangements, geometric singularities, cavity resonances, and ill-conditioning, amongst other complicating factors.