DEPARTMENT OF MATHEMATICS AND STATISTICS

03/13 Alex Barnett, Mathematics, Dartmouth College, Efficient and robust integral equation methods for 3D acoustic scattering from doubly-periodic media , .

A growing number of our technologies (telecommunications, radar, solar energy, etc) rely on the manipulation of linear waves at the wavelength scale. Advances in numerical modeling continue to be key to such progess. I will focus on scattering of single-frequency scalar waves from periodic media, either a lattice of isolated obstacles, or a grating with a connected periodically-modulated surface. We extend recently-developed QBX quadratures to 3D, and combine this with the fast multipole method, and with a new quasi-periodizing scheme based on matching on unit cell walls. No quasi-periodic Green's function nor lattice sums are needed, and the solver is robust for all scattering parameters including Wood's anomalies. (Joint work with Leslie Greengard and Zydrunas Gimbutas)

04/17 Andreas Kloeckner, Mathematics, NYU, Quadrature by Expansion: A New Method for the Evaluation of Layer Potentials , .

The changing computing landscape surrounding numerical methods for partial differential equations (PDEs) provides motivation for a reconsideration of the benefits of various types of numerical schemes. As an example, high-order methods and methods based on integral equations, previously viewed as costly, are seeing much renewed interest. Integral equation methods for the solution of partial differential equations, when coupled with suitable fast algorithms, yield geometrically flexible, asymptotically optimal and well-conditioned schemes. The practical application of these methods, however, requires the accurate evaluation of boundary integrals with singular, weakly singular or nearly singular kernels. We will examine a new systematic, high-order approach that works for any singularity. The scheme, denoted QBX (quadrature by expansion), is easy to implement and compatible with fast hierarchical algorithms such as the fast multipole method. We will see how the analysis-based mathematical properties of QBX position it ideally to tackle very general engineering problems while making good use of today's high-performance computational resources. (with L. Greengard, A. Barnett, M. O'Neil)

04/24 Catalin Turc, Mathematics, NJIT, Regularized Combined Field Integral Equations for solution of scattering problems in the frequency domain, .

Whenever applicable, numerical methods based on integral equation formulations of frequency domain scattering problems have been successfully used for large scale simulations. We will discuss in this talk a general framework capable of producing integral equation formulations for scattering problems that require small numbers of Krylov subspace iterations for all frequencies and for all types of material properties of the scatterers. We refer to this new class of integral equations as Regularized Combined Field Integral Equations. These integral formulations are amenable to both Galerkin and Nystrom discretizations, and consistenly outperform the classical formulations. In addition, these formulations enjoy certain surprising coercivity properties that allow for an error analysis of the Galerkin and Nystrom discretizations that exhibits explicit frequency dependent bounds. (with Y. Boubendir, O. Bruno, and D. Levadoux)

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