Leslie Greengard

TL;DR: Greengard et al. as discussed by the authors proposed a fast algorithm for the evaluation of heat potentials, which requires an amount of work of the order O(N M ) for large-scale problems.

TL;DR: The algorithm is direct, assumes that the source distribution is discretized using an adaptive quad-tree, and allows for Dirichlet, Neumann, periodic, and free-space conditions to be imposed on the boundary of a square.

TL;DR: In this paper, a fast algorithm for the calculation of elastostatic fields in locally isotropic composites is presented, which uses an integral equation approach due to Sherman, combined with the fast multipole method and an adaptive quadrature technique.

TL;DR: In this paper, the authors presented a new numerical method for the solution of linear two-point boundary value problems of ordinary differential equations by discretizing the differential equation to a second kind integral equation via a high order Nystrom scheme.

Abstract: A number of problems in solid state physics and materials science can be resolved by the evaluation of certain lattice sums (sums over all sites of an infinite perfect lattice of some potential energy function). One classical example, the calculation of lattice sums of circular and spherical harmonics, dates back to the last century, to Lord Rayleigh’s work on computing the effective conductivity of a simple composite. While Lord Rayleigh presented an efficient asymptotic method for two‐dimensional problems, he resorted to direct evaluation of the lattice sums in the three‐dimensional case. More recent methods, based on Ewald summation, have been developed by Nijboer and De Wette, Schmidt and Lee, and others. In this article, a fast method for evaluating lattice sums which is based on a new renormalization identity is described.