With this in mind, there are numerous ways to evaluate the action of K approximately, using discrete methods for approximating the solution of Poisson's equation. For example, the multigrid method can be used to do this in a very efficient way. The analogy with quadrature suggests using a grid with q points and solving Poisson's equation, with a multigrid method in O(q) work. Then the complete evaluation could be done in O(qmn) work. This potentially would be faster than direct evaluation of the integrals.


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Ding, Hong-Qiang, Naoki Karasawa, and William A. Goddard III, 1992, Atomic level simulations on a million particles: The cell multipole method for Coulomb and London nonbond interactions, J. Chem. Phys. 97:4309–4315.

Draghicescu, C.I., 1994, An efficient implementation of particle methods for the incompressible Euler equations, SIAM J. Numer. Anal. 31: 1090–1108.

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