Commit cf24164d authored by COULAUD Olivier's avatar COULAUD Olivier
Browse files

Remoce Old Docs and add new FMath.hpp

parent 1ed80141
@article{fong09a,
author = {Fong, W and Darve, E},
doi = {DOI: 10.1016/j.jcp.2009.08.031},
file = {:Users/coulaud/Recherche/Bibliographie/Mendeley/Fong, Darve - 2009 - The black-box fast multipole method.pdf:pdf},
issn = {0021-9991},
journal = {Journal of Computational Physics},
keywords = {Chebyshev polynomials,Fast multipole method,Interpolation,Singular value decomposition},
mendeley-groups = {FMM},
number = {23},
pages = {8712--8725},
title = {{The black-box fast multipole method}},
url = {http://www.sciencedirect.com/science/article/pii/S0021999109004665},
volume = {228},
year = {2009}
}
@inproceedings{Warren1993,
address = {New York, NY, USA},
author = {Warren, M S and Salmon, J K},
booktitle = {Proceedings of the 1993 ACM/IEEE conference on Supercomputing},
doi = {http://doi.acm.org/10.1145/169627.169640},
file = {:Users/coulaud/Recherche/Bibliographie/Mendeley/Warren, Salmon - 1993 - A hashed Oct-Tree N-body algorithm.pdf:pdf},
isbn = {0-8186-4340-4},
mendeley-groups = {FMM},
pages = {12--21},
publisher = {ACM},
series = {Supercomputing '93},
title = {{A hashed Oct-Tree N-body algorithm}},
url = {http://doi.acm.org/10.1145/169627.169640},
year = {1993}
}
@Book{mason2003chebyshev,
author = {J C Mason and D C Handscomb},
title = {Chebyshev polynomials},
publisher = {Chapman \& Hall/CRC},
year = {2003}
}
This diff is collapsed.
\section{Splitting the outer product in the interpolation polynomial}
\label{sec:splitouter}
We introduce the notation of the interpolation polynomial based on the
notation from \cite{mason2003chebyshev}, the interpolation of a function
$p(x)$ reads as
\begin{equation}
\label{eq:masondef}
p(x) \sim \sum_{n=0}^{\ell-1} \gamma_n c_n T_n(x)
\end{equation}
with $\gamma_0=1$ and $\gamma_n=2$ for $n\ge1$ and with
\begin{equation}
\label{eq:chebcoeff}
c_n = \frac{1}{\ell} \sum_{k=1}^\ell p(\bar x_n) T_k(\bar x_n)
\end{equation}
After inserting Eqn.~(\ref{eq:chebcoeff}) into Eqn.~(\ref{eq:masondef}) we end
up with
\begin{equation}
\label{eq:outprod}
p(x) \sim \frac{1}{\ell} \sum_{n=1}^\ell p(\bar x_n) \sum_{k=0}^{\ell-1}
\gamma_k T_k(\bar x_n) T_k(x)
\end{equation}
In the discrete case, i.e., we have $N$ scattered points $\{x_i\}_{i=1}^N
\subset [-1,1]$, we write Eqn.~\eqref{eq:outprod} in matrix notation as
\begin{equation}
\label{eq:interpolmatnot}
\Mat{p} \sim \Mat{T}_x \Mat{\bar T}_x^\top \Mat{\bar p}
\end{equation}
with the matrices $\Mat{T}_x \in \mathbb{R}^{N\times\ell}$ with
$(\Mat{T}_x)_{ik} = T_{k-1}(x_i)$ and $\Mat{\bar T}_x \in
\mathbb{R}^{\ell\times\ell}$ with $(\Mat{\bar T}_x)_{nk} =
\nicefrac{\gamma_{k-1}}{\ell} \, T_{k-1}(\bar x_n)$. Using this notation the
interpolated kernel function $K(x,y)$ reads as
\begin{equation}
\label{eq:kinterpolmatnot}
\Mat{K} \sim \Mat{T}_x \Mat{\bar T}_x^\top \Mat{\bar K} (\Mat{T}_y
\Mat{\bar T}_y^\top)^\top
\end{equation}
With the low-rank representation $\Mat{\bar K} \sim \Mat{UV}^T$ we compute
$\Mat{\bar U} = \Mat{\bar T}_x^\top \Mat{U}$ and $\Mat{\bar V} = \Mat{\bar
T}_y^\top \Mat{V}$ and write Eqn.~\eqref{eq:kinterpolmatnot} as
\begin{equation}
\label{eq:kinterpolmatnot1}
\Mat{K} \sim \Mat{T}_x \Mat{\bar U} \Mat{\bar V}^\top \Mat{T}_y^\top
\end{equation}
\paragraph{Can we still use symmetries?} Recall the approach we proposed in
Sec.~\ref{sec:m2l}. It exploits the symmetries in the arrangement of the
far-field interactions. As we see from Eqn.~\eqref{eq:permut} in that case we
need permutation matrices $\Mat{P}_t$ for the $t$-th far-field interaction and
Eqn.~\eqref{eq:kinterpolmatnot} becomes
\begin{equation}
\label{eq:symsplit}
\Mat{K}_t \sim \Mat{T}_x \Mat{\bar T}_x^\top (\Mat{P}_t \Mat{\bar K}_{p(t)}
\Mat{P}_t^\top) (\Mat{T}_y \Mat{\bar T}_y^\top)^\top.
\end{equation}
With $\Mat{\bar K}_{p(t)} \sim \Mat{U}_{p(t)} \Mat{V}_{p(t)}^\top$ the
matrices in the outer product in Eqn.~\eqref{eq:kinterpolmatnot1} become
$\Mat{\bar U}_t = \Mat{\bar T}_x^\top \Mat{P}_t \Mat{U}_{p(t)}$ and $\Mat{\bar
V}_t = \Mat{\bar T}_y^\top \Mat{P}_t \Mat{V}_{p(t)}$. Thus, it is not
possible anymore to use the fact that due to symmetries we can express all
$316$ far-field interactions by permutations of $16$ only.
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "main"
%%% End:
This diff is collapsed.
@article{Schmidt1991,
abstract = {The Rokhlin-Greengard fast multipole algorithm for evaluating Coulomb and multipole potentials has been implemented and analyzed in three dimensions. The implementation is presented for bounded charged systems and systems with periodic boundary conditions. The results include timings and error characterizations},
author = {Schmidt, K. E. and Lee, Michael a.},
doi = {10.1007/BF01030008},
file = {:Users/coulaud/Recherche/Bibliographie/Mendeley/Schmidt, Lee - 1991 - Implementing the fast multipole method in three dimensions.pdf:pdf},
issn = {0022-4715},
journal = {Journal of Statistical Physics},
keywords = {Ewald summation,Periodic boundary conditions,fast multipole method,macro summation,many-body problem,n-body,spherical harmonique},
mendeley-groups = {FMM},
month = jun,
number = {5-6},
pages = {1223--1235},
title = {{Implementing the fast multipole method in three dimensions}},
url = {http://link.springer.com/10.1007/BF01030008},
volume = {63},
year = {1991}
}
@article{Heyes1981,
abstract = {The Coulomb potentials and fields in infinite point charge lattices are represented by series expansions in real and reciprocal space using a method similar to that devised by Bertaut. A systematic investigation is made into the relative convergence characteristics of series derived by varying the charge spreading function which is required in the theory. A number of formulas which are already in use are compared with new expressions derived by this method.},
author = {Heyes, D. M.},
doi = {10.1063/1.441285},
file = {:Users/coulaud/Recherche/Bibliographie/Mendeley/Heyes - 1981 - Electrostatic potentials and fields in infinite point charge lattices.pdf:pdf},
issn = {00219606},
journal = {The Journal of Chemical Physics},
keywords = {Electrostatics,Lattice theory},
mendeley-groups = {Chimie},
mendeley-tags = {Electrostatics,Lattice theory},
number = {3},
pages = {1924},
title = {{Electrostatic potentials and fields in infinite point charge lattices}},
url = {http://link.aip.org/link/?JCP/74/1924/1\&Agg=doi},
volume = {74},
year = {1981}
}
@article{DeLeeuw1980,
author = {{De Leeuw}, S W and Perram, J W and Smith, E R},
file = {:Users/coulaud/Recherche/Bibliographie/Mendeley/De Leeuw, Perram, Smith - 1980 - Simulation of electrostatic systems in periodic boundary conditions. I. Lattice sums and dielectric con.pdf:pdf},
journal = {Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences},
keywords = {Periodic Boundary Conditions,dieclectric constant,lattice sums},
mendeley-groups = {Chimie},
mendeley-tags = {Periodic Boundary Conditions,dieclectric constant,lattice sums},
number = {1752},
pages = {27--56},
title = {{Simulation of electrostatic systems in periodic boundary conditions. I. Lattice sums and dielectric constants}},
url = {http://www.jstor.org/stable/2397238 http://rspa.royalsocietypublishing.org/content/373/1752/27.short},
volume = {373},
year = {1980}
}
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment