08/14/2023
By Min Hyung Cho

The Department of Mathematics and Statistics and the Department of Computer Science at the Kennedy College of Sciences invite you to attend a doctoral dissertation proposal by Jared Weed on “Accurate Numerical Methods for Evaluating Layer Potentials and Integral Equations Using Quadrature by Two Expansions.”

Date: Tuesday, Sept. 5, 2023
Time: 1:30 to 3 p.m.
Location: Dan 321

Thesis/Dissertation Title: Accurate Numerical Methods for Evaluating Layer Potentials and Integral Equations Using Quadrature by Two Expansions

Committee Members

  • Min Hyung Cho (Co-chair), Department of Mathematics & Statistics, Kennedy College of Sciences, UMass Lowell
  • Jingfang Huang (Co-chair), Department of Mathematics, University of North Carolina at Chapel Hill
  • Reza Ahmadzadeh, Miner School of Computer & Information Sciences, UMass Lowell
  • Bow Wu, Department of Mathematics & Statistics, Kennedy College of Sciences, UMass Lowell
  • Liyi Dai, Raytheon Technologies

Abstract:
One of the main numerical challenges to applying quadrature rules that discretize continuous integral equation formulations found in classical potential theory is the accurate and efficient evaluation of layer potentials with singular or nearly-singular kernels, particularly when the target point is close to—or on—the boundary. We propose a Quadrature by Two Expansions (QB2X) numerical integration technique that is developed for the single- and double-layer potentials of the Helmholtz equation in two dimensions. This method uses both local complex Taylor expansions and plane wave type expansions to achieve a resulting representation which is numerically accurate for all target points inside a leaf box of the Fast Multipole Method (FMM) hierarchical tree structure. QB2X explicitly includes a nonlinear dependency of the boundary geometry in the plane wave expansions, providing for higher-order representations of both the boundary geometry and density functions in the integrand.
 
First, we show that numerical results of the QB2X method for Helmholtz layer potentials using one expansion center in the entire FMM leaf box with various boundaries and densities outperforms simplified solutions of current state-of-the-art methods. Second, we demonstrate preliminary results that the QB2X method extends to Yukawa layer potentials with minimal modification. Finally, we outline a proposal to implement the QB2X method into the chunkie framework, a MATLAB integral equation toolbox that parameterizes curves into chunkers and evaluates layer potentials defined on those chunkers to represent solutions of the Helmholtz equations.

All interested students and faculty members are invited to attend.