add doc about periodic and wip for sse intel

Doc/ScalFmm-PeriodicModel/periodicmodel.tex

+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Simple Sectioned Essay Template
+% LaTeX Template
+%
+% This template has been downloaded from:
+% http://www.latextemplates.com
+%
+% Note:
+% The \lipsum[#] commands throughout this template generate dummy text
+% to fill the template out. These commands should all be removed when 
+% writing essay content.
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%----------------------------------------------------------------------------------------
+%	PACKAGES AND OTHER DOCUMENT CONFIGURATIONS
+%----------------------------------------------------------------------------------------
+
+\documentclass[12pt]{article} % Default font size is 12pt, it can be changed here
+
+\usepackage{geometry} % Required to change the page size to A4
+\geometry{a4paper} % Set the page size to be A4 as opposed to the default US Letter
+
+\usepackage{graphicx} % Required for including pictures
+
+\usepackage{float} % Allows putting an [H] in \begin{figure} to specify the exact location of the figure
+\usepackage{wrapfig} % Allows in-line images such as the example fish picture
+
+\usepackage{lipsum} % Used for inserting dummy 'Lorem ipsum' text into the template
+
+\linespread{1.2} % Line spacing
+
+%\setlength\parindent{0pt} % Uncomment to remove all indentation from paragraphs
+
+\graphicspath{{./Pictures/}} % Specifies the directory where pictures are stored
+
+%%%%%%%%% PUT BY ME %%%%%%%%%%%%%%%%%%
+\usepackage[usenames,dvipsnames]{xcolor}
+
+\usepackage{caption}
+\usepackage{subcaption}
+
+\usepackage{tikz}
+\usetikzlibrary{decorations.markings}
+
+\usepackage[ruled,vlined]{algorithm2e}
+
+\usepackage{amsmath}
+
+\usepackage{pgfplots}
+
+\newcommand{\Olivier}[1]{\footnote{\color{green}Olivier: #1}}
+\newcommand{\Berenger}[1]{\footnote{\color{blue}Berenger: #1}}
+\newcommand{\TODO}[1]{\textbf{\color{red}Todo: #1}}
+%%%%%%%%% END PUT BY ME %%%%%%%%%%%%%%%%%%
+
+\begin{document}
+
+%----------------------------------------------------------------------------------------
+%	TITLE PAGE
+%----------------------------------------------------------------------------------------
+
+\begin{titlepage}
+
+\newcommand{\HRule}{\rule{\linewidth}{0.5mm}} % Defines a new command for the horizontal lines, change thickness here
+
+\center % Center everything on the page
+
+\textsc{\LARGE INRIA}\\[1.5cm] % Name of your university/college
+\textsc{\Large HiePACS Team}\\[0.5cm] % Major heading such as course name
+\textsc{\large Bordeaux}\\[0.5cm] % Minor heading such as course title
+
+\HRule \\[0.4cm]
+{ \huge \bfseries ScalFMM Periodic Model}\\[0.4cm] % Title of your document
+\HRule \\[1.5cm]
+
+\begin{minipage}{0.4\textwidth}
+\begin{flushleft} \large
+\emph{Author:}\\
+Berenger \textsc{Bramas} % Your name
+\end{flushleft}
+\end{minipage}
+~
+\begin{minipage}{0.4\textwidth}
+\begin{flushright} \large
+\emph{Supervisor:} \\
+\textsc{} % Supervisor's Name
+\end{flushright}
+\end{minipage}\\[4cm]
+
+{\large \today}\\[3cm] % Date, change the \today to a set date if you want to be precise
+
+\vfill % Fill the rest of the page with whitespace
+
+\end{titlepage}
+
+%----------------------------------------------------------------------------------------
+%	TABLE OF CONTENTS
+%----------------------------------------------------------------------------------------
+
+\tableofcontents % Include a table of contents
+
+\newpage % Begins the essay on a new page instead of on the same page as the table of contents 
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Abstract}
+
+This study proposes a new approach to compute a boundary/periodic Fast Multipole Method (FMM) in any direction.
+The proposed model uses the standard FMM operators ($M2M$, $M2L$, $L2L$, $P2P$) which make it kernel independent.
+The algorithm can simulate a large repetition of the simulation box for a low computational cost.
+
+Keywords: FMM, octree, boundary.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Introduction}
+The FMM is widely use to run physical simulations where pairs interactions, also known as nbody problem,
+can have a complexity reduced from O(n2) to O(n).
+But many physical problem could also have the need to compute periodic simulations.
+Many analytical methods have been developed but none of them is using the power of the FMM to compute the periodicity.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{State of the art}
+
+In APPLICATION OF FAST MULTIPOLE METHOD TO MICROMAGNETIC SIMULATION OF PERIODIC SYSTEMS, an approach is proposed but
+is not using the power of FMM and the usual shape of its operators.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Usual Operators}
+In this section we remind the common FMM operators in a algorithm view.
+
+\subsection{Octree}
+The FMM operators relies on the spacial decomposition of the problem which is done using an octree.
+The system, which can be particles for example, is include in a cube which is the root of the tree at level $0$.
+Then we divide this root-cell by $2$ in each dimension which create $2^{DIM}$ sub-cells.
+We repeat the previous step until a level $L$. We have a maximum of $2^{DIM*l}$ cells at level $l$.
+Some implementation subdivides a cell until a condition is respected, like for example having less than $N$ particles in each leaves.
+Such approaches leads to a tree where all leaves may not be at the same level, but our method can even be applied them.
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=0.6]{../Images/Octree}
+\caption{$2D$ Octree}
+\end{figure}
+
+\subsection{Top Down Operators}
+These operators are performing computation between a cell and its children (the cells it contains one lower level in the tree) called $M2M$ and $L2L$,
+and inside leaves between cells and particles called $P2M$ and $L2P$.
+Some implementation defines optimizations which make them having operators that skips level during $M2M$ or $L2L$.
+But in most FMM it exists a $M2M$ (and a $L2L$) which is able to compute the interaction between a cells and its $2^{DIM}$ children.
+The children are positioned at equal distance from their common parent $r$ and can be positioned using spherical coordinates:
+$(r,\theta,\phi)$, with $(\theta,\phi) \in \{\pi/4,3\pi/4\} \times \{\pi/4, 3\pi/4, 5\pi/4, 7\pi/4\}$ in $3D$ and
+$(r,\phi)$, with $(\phi) \in \{\pi/4, 3\pi/4, 5\pi/4, 7\pi/4\}$ in $2D$.
+Depending on the level where the operator is applied, $r$ is equal to half of a cube diagonal:
+$r = Diagonal/2 = \sqrt{ 3 * Edge^2 }/2 = \sqrt{ 3 * (BoxWidth/2^{l})^2 }/2$.
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=0.6]{../Images/TopDownOperators}
+\caption{Top-Down Operators}
+\end{figure}
+
+\subsection{Interactions operators}
+The operators that operate on one level and do not rely on the parent or sub-cells are the $P2P$ and the $M2L$.
+The $P2P$ operates between neighbors leaves. If the tree is full (each cell exists), each leaf has a number of neighbors $ng$ with
+$2^{DIM} - 1 \leq ng \leq 3^{DIM}-1$.
+The minimum is for leaves that are positioned in the corners.
+If the octree is not full then the minimum can be $0$ since leaf may not have any neighbors.
+
+The $M2L$ operates between a cell and what is called its interaction list.
+The interaction list is composed of cells from the same level and which are not neighbors but are closed.
+The closed properties is true for the cells which have their parents as neighbors respectively.
+So each cell has a unique interactions list.
+Depending on the position of a cell relatively to its parent, the interactions list is taken cells from $[-3;+2]$ or $[-2;+3]$ in each dimension.
+In $3D$ and with a full octree, each cell has a maximum of $6^3-3^3 = 189$ interactions.
+The $6^3$ is coming from the cube of size $5$ which is encapsulate all the potential interactions and the $3^3$ represents the cell itself and its neighbors.
+Any FMM implementation has the capacity to compute the $M2L$ between two cell of the same level that are separated by at least one and a maximum of two in each direction.
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=0.6]{../Images/InteractionsOperators}
+\caption{Interactions Operators, P2P (Blue), M2L (Green)}
+\end{figure}
+
+\subsection{Global view of the algorithm}
+The FMM algorithm can be split in three majors parts, an upward pass, a transfer pass and a downward pass.
+The upward pass is composed by the $P2M$ and the $M2M$, it centralized the information contains by each cell from leaf level to the top of the tree.
+The transfer pass, composed by the $M2L$, is translating the values at each level using the interactions lists.
+Then the downward pass, using $L2L$ and $L2P$, merge the results of the transfer pass and impact the leaves by starting from the root.
+In the non periodic FMM, there is no computation above level $2$ since no $M2L$ happens at level $1$ because the interactions lists are empty.
+
+In addition of these steps called the far field, we compute the near field which is the $P2P$ operators.
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Virtual Repetition}
+
+When using periodicity, a cell or a leaf can have its interactions or neighbors lists extended.
+In fact, when working with periodicity the simulation box should be repeated.
+A cell on the border or separated by one cell from the border should have interactions from the other side of the box.
+In the next paragraph we supposed when speaking about interactions or neighbors lists that they are a periodic lists.
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=0.6]{../Images/Repetitions}
+\caption{Repetitions of the simulation box create new neighbors}
+\end{figure}
+
+\subsection{Periodicity during normal pass}
+During the upward pass the periodicity does not change the operators.
+In fact, they are operate between cells and sub-cells.
+
+Whereas, the transfer pass ($M2L$ and $P2P$) should not work with periodic lists.
+The normal FMM stops at level $2$ because interactions lists are empty at level $1$.
+By doing this, it makes a repetition of original box $rep(2) = 1/2$.
+If we use the upward pass, transfer pass and downward pass at level $1$ then $rep(1) = 1$.
+In fact, at level $1$ the periodic interactions lists are not empty and the $M2L$ at level $1$ is covering the neighbors of our root.
+It is important to have in mind that without any special operation the box is already repeated once it we go at level $1$ and use periodic lists.
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=0.35]{../Images/PeriodicL2}
+\caption{Extension of the box with a periodicity until level $2$}
+\end{figure}
+
+\subsection{Periodicity at root level}
+If the $M2M$ goes until the root at level $0$, the root cell is containing all the information of the entire tree.
+Using this cell we can perform a $M2L$ from $-3$ to $3$ by using it same cell inside the interactions lists.
+This $M2L$ will not compute the interaction with the neighbors cells, but this has been already done.
+We then have a repetition of $2$ cells in each direction so it like having a grid of $(2*2)+1$ and working on the cell in the middle.
+Finally we can compute the downward pass from level $0$ to the leaves.
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=0.45]{../Images/PeriodicL0}
+\caption{Extension of the box with a periodicity until level $0$}
+\end{figure}
+
+\subsection{Periodicity above the root}
+In the previous paragraph we stopped the operation at level $0$.
+But the key idea of the periodic algorithm is to continue with the usual operators above the level $0$ of the tree.
+
+If we compute the $M2L$ between $[-3;+3]$ at level $0$ it is then impossible to continue with the usual operators.
+In fact, the $M2L$ is covering what sub-cells do not cover and should let too far cells to upper $M2L$.
+The $M2L$ at level $0$ has to be limited to $[-3;+2]$ or $[-2;+3]$ in each direction.
+
+This choice is made by choosing where does our current root is situated relatively to its parent.
+Any choice can be made, but we consider that the root cell it at position $(1,1,1)$ relatively to its parent.
+So when performing the $M2L$ at level $0$ the repetition of the box is not odd and so it is not equilibrate.
+It is similar as having a grid of width $6$ and working on the cell of coordinate $(4,4,4)$.
+We can continue to perform similar operations, $M2M$ followed by $M2L$ and considering that our working cell is in $(1,1,1)$ relatively to its parent.
+
+Then, at the last level above the root, the final $M2L$ should be perform by considering that the working cell is in position $(0,0,0)$ relatively to its parent.
+This put our working cell in the middle of the grid.
+
+It remains that the repetition is not symmetric.
+In fact, the grid has a width of $(6*2^d)$ which is odd.
+There are $(6*2^d)/2$ repetition in one side and $((6*2^d)/2)-1$ in the other side for all the dimensions.
+The have a perfect symmetric periodicity we need to add a border.
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=0.45]{../Images/PeriodicL-1}
+\caption{Extension of the box with a periodicity until level $-1$ without border}
+\end{figure}
+
+We can group the root cell to perform a $M2M$ from level $0$ to level $-1$.
+The results contains the information of a box which is nothing more than our original box repeated $2$ times in each direction.
+Again, we can do a $M2L$ at level $-1$ and we choose that our cell is at position $(0,0,0)$ relatively to its parent.
+These operations can be repeated until we have enough repetition.
+
+Every time we perform a $M2M$ $d$ levels above $0$ we create a new cell which contain $2^d$ times our original box.
+Then when we perform a $M2M$ at a level we repeat the cell from $-2$ to $3$ in any direction.
+So at $d$ levels above $0$ performing a $M2L$ leads to create a repetition of $6*2^d$ original box in all the directions.
+So we create a grid of size $(6*2^d)^3$ and we are working on the center cell at position $((6*2^d)/2,(6*2^d)/2,(6*2^d)/2)$.
+
+
+\subsection{Adding a border}
+After having proceed $d$ levels above $0$, a border of the simulation box is missing.
+To take into consideration this border, we need to use the root cell and create all the configuration in order to have create
+cells one levels above $d$ which contains all the information.
+Then we can apply a final $M2L$ between those cells and our working cell.
+
+In $2D$ there are three kinds of border cells: the horizontal border, the vertical border and the angle.
+In $3D$ there are seven kinds of border cells: the top, the side, the back and the four different links between them.
+More generally, there are $DIM$ different sides and $2^{DIM-1}$ different connexion between sides.
+
+Computing the cell which represent a side is done in $d-1$ $M2M$.
+Each of this $M2M$ is computing the relation between one or two sub-cells and their parent.
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=0.45]{../Images/Border}
+\caption{Extension of the box with a periodicity until level $-1$ with border}
+\end{figure}
+
+\subsection{Limiting the periodicity direction}
+
+If the simulation needs that the periodicity is limited in some direction, for example $(+X,+-Y,No \, Z)$, the only changes is in the construction to the periodic lists including those made above the root.
+Then, during the addition of the border only the periodic directions have to be applied.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Theoretical repetition}
+
+The periodic-algorithm does not allow to choose a specific grid size.
+The only parameter $D$ is the number of level above the root of the box where the algorithm is applied.
+As it has been illustrated the final grid size has a width of $dim = 3*2^{Abs(D)+1} + 1$
+
+\begin{center}
+\begin{tabular}{ | c | c | }
+ \hline                 
+   D & Grid Dim \\ \hline
+   1 & 3 \\
+   0 & 7 \\
+   1 & 13 \\
+   5 & 193 \\
+   20 & 6 291 457 \\
+ \hline  
+ \end{tabular}
+\end{center}
+
+\subsection{Cost above the root}
+We determine the cost of the periodicity in term of FMM operators.
+The following cost is based on a full octree.
+
+There are $2^{DIM}$ $M2M$, $1$ $L2L$, and $6^{DIM}-3^{DIM}$ $M2L$ per level above the root.
+For the border the cost is $\sum_{i=1}^{DIM}{i^2}$ $M2M$ per level above root.
+
+For example, in $3D$, the cost for $D=5$ is $5*(2^3 + 1^2 + 2^2 + 3^2) = 110$ $M2M$ and $5*189 = 945$ $M2L$.
+For example, in $3D$, the cost for $D=20$ is $20*(2^3 + 1^2 + 2^2 + 3^2) = 440$ $M2M$ and $20*189 = 3780$ $M2L$.
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Implementation details}
+
+\subsection{The algorithm}
+
+The algorithm above the root can be spited in 4 steps:
+\begin{itemize}
+\item Upward pass, for level $0$ to $-d$ we compute the $2^{DIM}$ $M2M$ ($8$ per level in $3D$).
+\item Transfer pass, for level $0$ to $-d$ we compute the $6^{DIM}-3^{DIM}$ $M2L$ ($189$ per level in $3D$).
+For level $0$ to $-d+1$ we put the cell at position $(1,1,1)$ relatively to its parent.
+For the last $M2L$ at level $-d$ we put the cell at position $(0,0,0)$ relatively to its parent. 
+\item Border pass, at level $-d-1$ we compute the border and apply it using $M2L$.
+\item Downward pass, from level $-d-1$ to $0$ we perform one $L2L$.
+There is only one sub-cell per $L2L$ and the position has to be the same as the one used during the Transfer pass.
+\end{itemize}
+
+\subsection{Low level details}
+
+In our library, all the main part of the FMM are dissociated.
+To ensure that our kernel is able to compute all the periodic operation, we pass to our kernel initialization method an offset for the box with and
+for the tree height.
+Our kernel compute the requested operation without any difference between the real FMM and the operation above the root for the periodic FMM.
+In fact, let the original FMM box width be $w$ and a tree height be $h$.
+Then, if is required to apply periodicity until level $d$ above the root.
+The kernel should be able to proceed usual FMM operator in a tree of height of size $h+d+5$ with a initial box width of $w*2^{d+3}$.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Conclusion}
+A new approach has been proposed to compute a periodic FMM.
+This approach can be used with any kernel and without the need of developing new operators.
+Moreover, it is possible to compute an important repetition size for a small cost.
+
+\end{document}

Src/Kernels/P2P/FP2P.hpp

@@ -5,8 +5,7 @@
 #include "../../Utils/FMath.hpp"
 
 #ifdef ScalFMM_USE_SSE
-#include <emmintrin.h>
-#include <mmintrin.h>
+#include "../../Utils/FSse.hpp"
 #endif
 
 /**

Src/Utils/FSse.hpp

+#ifndef FSSE_HPP
+#define FSSE_HPP
+
+#endif // FSSE_HPP