I started with mathematics, moved into computer science, writing a dissertation in multipole algorithms. Fast Multipole Method is one of the top 10 algorithms of the 20th century introduced by Prof. Greengard of the Courant institute at NYU. My work focused on generalizing the algorithm for higher order potentials/forces and when the particles are in motion. This has applications in molecular dynamics simulations.
Kasthuri Kannan, Hemant Mahawar and Vivek Sarin, A Multipole Based Treecode using Spherical Harmonics for the Potentials of the Form \(r^{-\lambda}\). Lecture Notes in Computer Science. 3514, pp. 107-114, May 2005.
Description: Spherical harmonics are eigenfunctions of the Laplace-Beltrami operator in spherical coordinates. And hence they serve as orthogonal basis functions, which can be used to represent Coulomb-like potentials. We make use of this property to efficiently compute these potentials using ultraspherical (Gegenbauer) polynomials.
Kasthuri Kannan and Vivek Sarin, A Treecode for Accurate Force Calculations. Lecture Notes in Computer Science. 3991, pp. 92-99, May 2006.
Description: Computing the forces in N-body simulations is of the order \(O(N^2)\). Treecodes present a fast approximation to such computations. However, accuracy in such computations are limited when using cartesian tensors. Spherical tensors provide greater accuracy for these simulations.
Kasthuri Kannan and Vivek Sarin, A Treecode for Potentials of the Form \(r^{-\lambda}\), International Journal of Computer Mathematics. 84, 1249-1260, Jan. 2007.
Description: This work presents a fast algorithm to compute potentials of the form \(r^{-\lambda}\) that are used in molecular dynamics simulations. This is the exension of the first work above, where I describe the complete algorithm along with complexity analysis.