11/03/2026
By Min Hyung Cho

The Kennedy College of Sciences, Department of Mathematics and Statistics and the Miner School of Computer and Information Sciences, invite you to attend a doctoral dissertation defense by Jared Weed titled “Accurate Numerical Methods for Wave Scattering Problems.”

Details

  • Date: Nov. 18, 2026
  • Time: 2 – 3:30 p.m.
  • Location: Southwick 350W

Committee members

  • Min Hyung Cho (advisor), Ph.D., Mathematics and Statistics, UMass Lowell
  • Bobbie Wu, Ph.D., Mathematics and Statistics, UMass Lowell
  • Reza Azadeh, Ph.D., Miner School of Computer and Information Science, UMass Lowell
  • Jingfang Huang, Ph.D., Department of Mathematics, UNC Chapel Hill

Abstract

This dissertation develops accurate numerical methods for wave scattering problems governed by the Helmholtz and time-harmonic Maxwell equations. Green’s function representations provide a common approach to these problems, but their numerical accuracy depends on the treatment of boundary geometry, singular kernels, and the conditions imposed on the scattered field. The present work considers these difficulties in the evaluation of Helmholtz layer potentials and in electromagnetic scattering by periodic multilayered structures.

For the Helmholtz equation in two dimensions, Quadrature by Two Expansions (QB2X) combines local complex Taylor expansions with plane wave type expansions to capture the nonlinear contributions of boundary geometry. The resulting representation is accurate for target points throughout a fast multipole method leaf box using a single expansion center. For Maxwell’s equations in three dimensions, a periodization scheme is developed using the method of fundamental solutions. The electric and magnetic fields are expressed as a sum of near and distant interactions using only the free-space Green’s function, with the unknown coefficients determined by enforcing interface continuity, quasi-periodicity, and radiation conditions.

Numerical results demonstrate the accuracy and convergence of the methods for flat and curved boundaries and for media containing many layers. The QB2X representation achieves accuracy approaching machine precision for a range of boundary geometries and density functions. The Maxwell solver exhibits exponential convergence in benchmark problems, remains robust at Wood anomalies, and provides reflection and transmission spectra for multilayered structures. These results establish accurate computational tools for layer-potential evaluation and periodic electromagnetic scattering, and provide a foundation for further development of boundary-based wave scattering methods.