Selected research topics

Scattering problems in unbounded interfaces

I have developed a method for solving a class of decomposable scattering problems in complicated unbounded domains. This method starts by splitting the computational domain into simple subdomains (such as straight waveguides or periodic gratings). The domain Green’s function for each piece can then be used to reduce the scattering problem to an integral equation on the domain boundaries. The resulting integral equation can be analytically continued to the complex plane, where it can be safely truncated and discretized using standard integral equation techniques.

A fast solver for surface PDEs

I showed that a broad class of elliptic surface PDEs can be reformulated as well-conditioned surface integral equations using their 2D fundamental solutions. I then developed a singular quadrature scheme and fast direct solver which allow the resulting integral equations to be accurately discretized and efficiently solved. I have used this method to solve a variety of problems such as eigenvalue problems and oscillatory problems on unbounded surfaces. I have also used it as an implicit time stepper for hyperbollic problems on surfaces.

A domain decomposition method for waveguide circuits

I have demonstrated how a complicated photonic circuit can be efficiently be simulated by recursively merging impedance to impedance maps for individual circuit elements. When circuit elements are connected by straight pipes, these maps can be greatly compressed by projecting the impedance data onto the modes for each pipe, resulting in an efficient numerical solver.