Duan Chen

Bo Wanga, Duan Chen, Bo Zhang et al.

In this paper, we develop fast multipole methods for 3D Helmholtz kernel in layered media. Two algorithms based on different forms of Taylor expansion of layered media Green's function are developed. A key component of the first algorithm is an efficient algorithm based on discrete complex image approximation and recurrence formula for the calculation of the layered media Green's function and its derivatives, which are given in terms of Sommerfeld integrals. The second algorithm uses symmetric derivatives in the Taylor expansion to reduce the size of precomputed tables for the derivatives of layered media Green's function. Numerical tests in layered media have validated the accuracy and O(N) complexity of the proposed algorithms.

Duan Chen, Wei Cai

In this paper, we propose a hierarchical random compression method (HRCM) for kernel matrices in fast kernel summations. The HRCM combines the hierarchical framework of the H-matrix and a randomized sampling technique of the column and row spaces for far-field interaction kernel matrices. We show that a uniform column/row sampling (with a given sample size) of a far-field kernel matrix, with- out the need and associated cost to pre-compute a costly sampling distribution, will give a low-rank compression of such low-rank matrices, independent of the matrix sizes and only dependent on the separation of the source and target locations. This far-field random compression technique is then implemented at each level of the hierarchical decomposition for general kernel matrices, resulting in an O(N logN) random compression method. Error and complexity analysis for the HRCM are included. Numerical results for electrostatic and Helmholtz wave kernels have vali- dated the efficiency and accuracy of the proposed method with a cross-over matrix size, in comparison of direct O(N^2) summations, in the order of thousands for a 3-4 digits relative accuracy.

Duan Chen, Wei Cai, Brian Zinser et al.

In this paper, we develop an accurate and efficient Nyström volume integral equation (VIE) method for the Maxwell equations for large number of 3-D scatterers. The Cauchy Principal Values that arise from the VIE are computed accurately using a finite size exclusion volume together with explicit correction integrals consisting of removable singularities. Also, the hyper-singular integrals are computed using interpolated quadrature formulae with tensor-product quadrature nodes for several objects, such as cubes and spheres, that are frequently encountered in the design of meta-materials . The resulting Nyström VIE method is shown to have high accuracy with a minimum number of collocation points and demonstrate $p$-convergence for computing the electromagnetic scattering of these objects. Numerical calculations of multiple scatterers of cubic and spherical shapes validate the efficiency and accuracy of the proposed method.

Duan Chen, Wei Cai, Brian Zinser

In this paper, we develop highly accurate Nyström methods for the volume integral equation (VIE) of the Maxwell equation for 3-D scatters. The method is based on a formulation of the VIE equation where the Cauchy principal value of the dyadic Green's function can be computed accurately for a finite size exclusion volume with some explicit corrective integrals of removable singularities. Then, an effective interpolated quadrature formula for tensor product Gauss quadrature nodes in a cube is proposed to handle the hyper-singularity of integrals of the dyadic Green's function. The proposed high order Nyström VIE method is shown to have high accuracy and demonstrates $p$-convergence for computing the electromagnetic scattering of cubes in $R^3$.