Zydrunas Gimbutas

A wideband fast multipole method for the Helmholtz equation in three dimensions

We describe a wideband version of the Fast Multipole Method for the Helmholtz equation in three dimensions. It unifies previously existing versions of the FMM for high and low frequencies into an algorithm which is accurate and efficient for any frequency, having a CPU time of O(N) if low-frequency computations dominate, or O(NlogN) if high-frequency computations dominate. The performance of the algorithm is illustrated with numerical examples.

On the Compression of Low Rank Matrices

A procedure is reported for the compression of rank-deficient matrices. A matrix A of rank k is represented in the form

A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions

A = U \circ B \circ V

A Nonlinear Optimization Procedure for Generalized Gaussian Quadratures

, where B is a

A Nyström method for weakly singular integral operators on surfaces

k\times k

Overcoming Low-Frequency Breakdown of the Magnetic Field Integral Equation

submatrix of A, and U, V are well-conditioned matrices that each contain a

Fast multi-particle scattering: A hybrid solver for the Maxwell equations in microstructured materials

k\times k

Coulomb Interactions on Planar Structures: Inverting the Square Root of the Laplacian

identity submatrix. This property enables such compression schemes to be used in certain situations where the singular value decomposition (SVD) cannot be used efficiently. Numerical examples are presented.

A fast multipole method for the Rotne-Prager-Yamakawa tensor and its applications

We present a numerical algorithm for the construction of efficient, high-order quadratures in two and higher dimensions. Quadrature rules constructed via this algorithm possess positive weights and interior nodes, resembling the Gaussian quadratures in one dimension. In addition, rules can be generated with varying degrees of symmetry, adaptable to individual domains. We illustrate the performance of our method with numerical examples, and report quadrature rules for polynomials on triangles, squares, and cubes, up to degree 50. These formulae are near optimal in the number of nodes used, and many of them appear to be new.

On the numerical evaluation of the singular integrals of scattering theory

We present a new nonlinear optimization procedure for the computation of generalized Gaussian quadratures for a broad class of square integrable functions on intervals. While some of the components of this algorithm have been previously published, we present a simple and robust scheme for the determination of a sparse solution to an underdetermined nonlinear optimization problem which replaces the continuation scheme of the previously published works. The new algorithm successfully computes generalized Gaussian quadratures in a number of instances in which the previous algorithms fail. Four applications of our scheme to computational physics are presented: the construction of discrete plane wave expansions for the Helmholtz Greens function, the design of linear array antennae, the computation of a quadrature for the discretization of Laplace boundary integral equations on certain domains with corners, and the construction of quadratures for the discretization of Laplace and Helmholtz boundary integral equations on smooth surfaces.