@article {13990,
	title = {A scalar potential formulation and translation theory for the time-harmonic Maxwell equations},
	journal = {Journal of Computational Physics},
	volume = {225},
	year = {2007},
	month = {2007/07/01/},
	pages = {206 - 236},
	abstract = {We develop a computational method based on the Debye scalar potential representation, which efficiently reduces the solution of Maxwell{\textquoteright}s equations to the solution of two scalar Helmholtz equations. One of the key contributions of this paper is a theory for the translation of Maxwell solutions using such a representation, since the scalar potential form is not invariant with respect to translations. The translation theory is developed by introducing {\textquotedblleft}conversion{\textquotedblright} operators, which enable the representation of the electric and magnetic vector fields via scalar potentials in an arbitrary reference frame. Advantages of this representation include the fact that only two Helmholtz equations need be solved, and moreover, the divergence free constraints are satisfied automatically by construction. Truncation error bounds are also presented. The availability of a translation theory and error bounds for this representation can find application in methods such as the Fast Multipole Method.For illustration of the use of the representation and translation theory we implemented an algorithm for the simulation of Mie scattering off a system of spherical objects of different sizes and dielectric properties using a variant of the T-matrix method. The resulting system was solved using an iterative method based on GMRES. The results of the computations agree well with previous computational and experimental results.
},
	keywords = {Debye potentials, Electromagnetic scattering, Fast Multipole Method, Helmholtz equation, Maxwell equations, Mie scattering, T-matrix method, Translation operators},
	isbn = {0021-9991},
	doi = {10.1016/j.jcp.2006.11.025},
	url = {http://www.sciencedirect.com/science/article/pii/S0021999106005845},
	author = {Gumerov, Nail A. and Duraiswami, Ramani}
}
@article {14017,
	title = {Comparison of the Efficiency of Translation Operators Used in the Fast Multipole Method for the 3D Laplace Equation},
	journal = {Technical Reports from UMIACS UMIACS-TR{\textendash}2005-09},
	year = {2005},
	month = {2005/11/30/T16:2},
	abstract = {We examine the practical implementation of a fast multipole method algorithm for the rapid summation of Laplace multipoles. Several translation operators with different asymptotic computational and memory complexities have been proposed for this problem. These algorithms include: Method 0 {\textemdash} the originally proposed matrix based translations due to Greengard andRokhlin (1987), Method 1 {\textemdash} the rotation, axial translation and rotation algorithm due to White and Martin Head-Gordon (1993), and Method 3 {\textemdash} the plane-wave version of the multipole-to local translation operator due to Greengard and Rokhlin (1997). We compare the algorithms on data sets of varying size and with varying imposed accuracy requirements. While from the
literature it would have been expected that method 2 would always be the method of choice, at least as far as computational speed is concerned, we find that this is not always the case. We find that as far as speed is concerned the choice between methods 1 and 2 depends on problem
size and error requirements. Method 2 is the algorithm of choice for large problems where high accuracy is required, though the advantage is not clear cut, especially if memory requirements are an issue. If memory is an issue, Method 1 is the method of choice for most problems. A new analysis of the computational complexities of the algorithms is provided, which explains
the observed results. We provide guidelines for choosing parameters for FMM algorithms.
},
	keywords = {Fast Algorithms, Fast Multipole Method, Harmonic analysis, Laplace Equation in 3D, Translation operators},
	url = {http://drum.lib.umd.edu/handle/1903/3023},
	author = {Gumerov, Nail A. and Duraiswami, Ramani}
}
