Simons Foundation Lectures are free public colloquia related to basic science and mathematics. These high-level talks are intended for professors, students, postdocs and business professionals, but interested people from the metropolitan area are welcome as well.
Integral equation methods play an important role in the numerical simulation of electromagnetic scattering. They are easy to employ in complex geometry and impose the desired radiation conditions at infinity without the need for artificial numerical boundaries. Two of the obstacles faced by current forward simulation tools are “low-frequency breakdown” and the lack of easy to use high order quadrature rules for complicated surfaces. In this talk, I will review the relevant background material, discuss a new mathematical formalism for scattering from perfect conductors and briefly describe a new quadrature technique that yields easily implementable high order rules for singular and weakly singular integrals. The scheme, denoted QBX (quadrature by expansion) is compatible with fast hierarchical algorithms such as the fast multipole method. I will also present some preliminary results on inverse obstacle scattering. This is joint work with A. Barnett, C. Borges, C. L. Epstein, M. Ferrando, Z. Gimbutas, A. Kloeckner, M. O’Neil, and F. Vico.