Turkish Journal of Physics

Synthesis of Wavelet Theory and the Multigrid Approach: Numerical Solution of Problems Involving Long-Range Interactions




Multigrid algorithms represent the state of the art in the numerical solution of physical problems involving long-range interactions, such as finding the potential due to a charge distribution. Wavelets provide an efficient representation of functions which exhibit localized bursts of short length-scale behavior, such as the total electronic charge density of a molecule. Computing the electrostatic field in and around a molecule should benefit from both approaches. In this work, we demonstrate how a novel interpolating wavelet transform may be used as the mathematical bridge to connect the two approaches. The result is a specialized multigrid algorithm which may be applied to problems expressed in wavelet bases. With this approach, optimal interpolation and restriction operators and optimal grids are predetermined by an interpolating multiresolution analysis. Moreover, problems on irregular meshes may be treated efficiently without invoking a dense underlying regular mesh. We will present the new method and contrast its efficiency with standard wavelet and multigrid.

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