Particle-Mesh Wavelet Method (PMW)
The main idea in this method is to formulate the calculation of the
potential energy as matrix-vector operation, where the
entries of the matrix
consist of fixed distances between mesh points in a discretized space and
the vector Q consits of time-dependent charge weights on these mesh points. A
thresholded Wavelet-transform is applied to the matrix in order to
make it sparse. Compression of more than 90% is achieved in that
way. During simulation similar fast Wavelet transforms are
applied to the charge-vector and the product
is evaluated, where A(w) = W A WT, Q(w) = W Q and W is a Wavelet-transform matrix (in the present case Daubechies type). Back-transformation of U(w) then gives the potential energy on the grid points. In order to take into account short range interactions, self energies and contributions from near grid points are subtracted and particle-particle interactions are taken into account explicitly. Forces onto individual particles are calculated by high-order finite difference schemes. Since the Wavelet-transform, acting on the charge vector Q and the back-transform acting on the potential U(w) has complexity O(N), the overall complexity of the method is also of optimal order.
Figure: Wavelet transformed matrix (level 5) of distances between equidistant grid-points in three dimensions. Shown is the thresholded version, where only matrix elements non-equal to zero are shown as symbol.
References:
A Fast Wavelet Based Evaluation of Coulomb Potentials in Molecular Systems, Proc. of NIC Workshop From Computational Biophysics to Systems Biology (submitted).
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Godehard Sutmannlast change 22.06.2006 |
Godehard Sutmann | Print