From 3433a3f2801bda7b130fcfde4cc421121d99cc8e Mon Sep 17 00:00:00 2001 From: Orestis <orestis.malaspinas@pm.me> Date: Mon, 4 Apr 2022 01:10:29 +0200 Subject: [PATCH] added beginning of BH. must revamp figs+ --- slides/cours_21.md | 399 +++++++++++++++++++++++++++++- slides/figs/arbre1.png | Bin 0 -> 20740 bytes slides/figs/arbre2.png | Bin 0 -> 21369 bytes slides/figs/arbre3_1.png | Bin 0 -> 21665 bytes slides/figs/arbre3_2.png | Bin 0 -> 60334 bytes slides/figs/arbre3_3.png | Bin 0 -> 74779 bytes slides/figs/corps1.png | Bin 0 -> 15943 bytes slides/figs/corps2.png | Bin 0 -> 8646 bytes slides/figs/corps3_1.png | Bin 0 -> 9965 bytes slides/figs/corps3_2.png | Bin 0 -> 11462 bytes slides/figs/corps3_3.png | Bin 0 -> 12020 bytes slides/figs/nbody_bare.png | Bin 0 -> 19271 bytes slides/figs/nbody_group.png | Bin 0 -> 35163 bytes slides/figs/nbody_n2.png | Bin 0 -> 52545 bytes slides/figs/nbody_qt.png | Bin 0 -> 13205 bytes slides/figs/nbody_qt_withtree.png | Bin 0 -> 55754 bytes 16 files changed, 393 insertions(+), 6 deletions(-) create mode 100644 slides/figs/arbre1.png create mode 100644 slides/figs/arbre2.png create mode 100644 slides/figs/arbre3_1.png create mode 100644 slides/figs/arbre3_2.png create mode 100644 slides/figs/arbre3_3.png create mode 100644 slides/figs/corps1.png create mode 100644 slides/figs/corps2.png create mode 100644 slides/figs/corps3_1.png create mode 100644 slides/figs/corps3_2.png create mode 100644 slides/figs/corps3_3.png create mode 100644 slides/figs/nbody_bare.png create mode 100644 slides/figs/nbody_group.png create mode 100644 slides/figs/nbody_n2.png create mode 100644 slides/figs/nbody_qt.png create mode 100644 slides/figs/nbody_qt_withtree.png diff --git a/slides/cours_21.md b/slides/cours_21.md index 4c8f400..e30a97e 100644 --- a/slides/cours_21.md +++ b/slides/cours_21.md @@ -56,8 +56,8 @@ graph TD; ```C arbre symétrie(arbre) si !est_feuille(arbre) - échanger(arbre.enfant[0], arbre.enfant[1]) - échanger(arbre.enfant[2], arbre.enfant[3]) + échanger(arbre.enfant[0], arbre.enfant[2]) + échanger(arbre.enfant[1], arbre.enfant[4]) pour i de 0 à 3 symétrie(arbre.enfant[i]) retourne arbre @@ -103,10 +103,10 @@ arbre symétrie(arbre) ## Écrire le pseudo-code (5min, matrix) ```C -rien compression_sans_perte(arbre) +rien compression_sans_pertes(arbre) si !est_feuille(arbre) pour i de 0 à 3 - compression_sans_perte(arbre.enfant[i]) + compression_sans_pertes(arbre.enfant[i]) si derniere_branche(arbre) valeur, toutes_égales = valeur_enfants(arbre) si toutes_egales @@ -207,8 +207,8 @@ bool last_value(node *qt, int *val) { * Avec l'écart-type \begin{equation*} -\mu = \frac{1}{4}\sum_{i=0}^{3} p[i],\quad \sigma = \sqrt{\frac{1}{N}\sum_{i=0}^3 (\mu-p[i]) -^2} +\mu = \frac{1}{4}\sum_{i=0}^{3} p[i],\quad \sigma = \sqrt{\frac{1}{4}\sum_{i=0}^3 (\mu-p[i]) +^2} = \sqrt{\frac{1}{4}\left(\sum_{i=0}^3p[i]^2\right)-\mu^2} \end{equation*} ## Que devient l'algorithme? @@ -234,3 +234,390 @@ bool last_value(node *qt, int *val) { . . .  + +# Compression avec perte (6/N) + +## Modifications sur la structure de données? + +. . . + +* On stocke la moyenne, et le carré de la moyenne. + +```C +struct noeud + flottant moyenne, moyenne_carre + node enfants[4] +``` + +* Comment on calcule `moyenne` et `moyenne_carre` sur chaque noeud (pseudo-code)? + +# Calcul de la moyenne + +## Pseudo-code (5min, matrix) + +. . . + +```C +rien moyenne(arbre) { + si !est_feuille(arbre) + pour enfant dans arbre.enfants + moyenne(enfant) + pour enfant dans arbre.enfants + arbre.moyenne += enfant.moyenne + arbre.moyenne_carre += enfant.moyenne_carre + arbre.moyenne /= 4 + arbre.moyenne_carre /= 4 +``` + +# La compression avec pertes + +## Pseudo-code (5min, matrix) + +. . . + +```C +rien compression_avec_pertes(arbre, theta) + si !est_feuille(arbre) + pour i de 0 à 3 + compression_avec_pertes(arbre.enfant[i]) + si derniere_branche(arbre) + si racine(arbre.moyenne*arbre.moyenne - arbre.moyenne_carre) < theta + detruire_enfants(arbre) +``` + +## Le code en entier + +```C +arbre = matrice_à_arbre(matrice) +moyenne(arbre) +compression_sans_pertes(arbre) +``` + +# La dynamique des corps célestes + +## Slides très fortement inspirés du cours de J. Latt, Unige + +## Simulation du problème à $N$-corps + +* Prédiction du mouvement d'un grand nombre de corps célestes. +* Modélisation: + * On se limite aux étoiles; + * Chaque étoile est caractérisée par un point (coordonnées) et une masse; + * On simule en deux dimensions. + * Interactions uniquement par les lois de la gravitation Newtonienne (oui-oui c'est de la **physique**!). + + +# Les équations du mouvement + +## Mouvement de la $i$-ème étoile + +* Algorithme de Verlet ($t_{n+1}=t_n+\delta t$) + + $$ + \vec x_i(t_{n+1})= 2\vec x_i(t_n)-\vec x_i(t_{n-1})+\vec a_i(t_n)\delta t^2. + $$ + +## Force de gravitation + +* $\vec a_i(t_n)=\vec F_i/m_i$. +* Sur l'étoile $i$, la force résultante est donnée par + + $$ + \vec F_i=\sum_{j=1,j\neq i}^N \vec F_{ij}. + $$ + avec + $$ + \vec F_{ij}=\frac{G m_i m_j(\vec x_j-\vec x_i)}{||\vec x_j-\vec x_i||^3}. + $$ + +# Algorithme du problème à $n$-corps + +## Pseudo-code: structure de données + +```C +struct étoile + flottant m + vec x, x_precedent, f +``` + +## Pseudo-code: itération temporelle + +```C +rien iteration_temporelle(étoiles, dt) + pour étoile_une dans étoiles + étoile_une.a = 0 + pour étoile_deux dans étoiles + si (étoile_un != étoile_deux) + étoile_une.f += + force(étoile_une, étoile_deux) + pour étoile dans étoiles + étoile.x, étoile.x_precedent = + verlet(étoile.x, étoile.x_precedent, + étoile.f / étoile.m, dt) +``` + +# Algorithme du problème à $n$-corps + +## Complexité + +* Complexité de chacune des parties? + +. . . + +* $\mathcal{O}(N^2)$, $\mathcal{O}(N)$. + +## En temps CPU pour **une itération** + +* Si temps pour $N=1$ on calcule en $1\mu s$: + ++--------+-------+-------+-----------+ +| N | N^2 | t [s] | t [réel] | ++--------+-------+-------+-----------+ +| 10 | 10^2 | 1e-4 | | ++--------+-------+-------+-----------+ +| 10^2 | 10^4 | 1e-2 | | ++--------+-------+-------+-----------+ +| 10^4 | 10^8 | 1e+2 | ~1min | ++--------+-------+-------+-----------+ +| 10^6 | 10^12 | 1e+6 | ~11j | ++--------+-------+-------+-----------+ +| 10^9 | 10^18 | 1e+12 | ~30k ans | ++--------+-------+-------+-----------+ +| 10^11 | 10^22 | 1e+16 | ~300M ans | ++--------+-------+-------+ ----------+ + +* Typiquement il y a des milliers-millions d'itérations. +* Il y a $10^{11}$ étoiles dans la galaxie. +* Houston we have a problem. + +# Question + +## Comment faire mieux, des idées? + +. . . + +* Si un groupe d'étoiles est suffisamment loin, on le modélise comme un corps unique situé en son centre de masse. +* Exemple: Si on simule plusieurs galaxies, on considère chaque galaxie comme un corps unique! +* Un arbre quaternaire est une structure parfaite pour regrouper les étoiles. + +# Le cas à 10 corps + +::: columns + +:::: {.column width=50%} + +## Illustration: le cas à 10 corps + +{width=60%} + +:::: + +:::: {.column width=50%} + +## Problématique + +* On veut calculer la force sur $1$. + +:::: + +::: + +. . . + + +::: columns + +:::: {.column width=50%} + +## Illustration: le cas à 10 corps + +{width=60%} + + +:::: + +:::: {.column width=50%} + +## Résultat + +* Calcul et somme des forces venant des $9$ autre corps. + +:::: + +::: + +# Le cas à 10 corps + +::: columns + +:::: {.column width=50%} + +## Réduction d'un groupe à un seul corps + +{width=100%} + +:::: + +:::: {.column width=50%} + +## Idée + +* On accélère le calcul en traitant un groupe comme un seul corps. +* Fonctionne uniquement si le groupe est assez loin. +* Autrement l'approximation est trop grossière. + +:::: + +::: + +# Solution: l'arbre quaternaire + +## Corps célestes - arbre + + + +* On omet les noeuds vides pour éviter la surcharge. +* La numérotation est: + * 0: ID + * 1: SD + * 2: ID + * 3: SG + +# Algorithme d'insertion + +::: columns + +:::: {.column width=50%} + +## Insertion corps 1 + +{width=100%} + +:::: + +:::: {.column width=50%} + +## Arbre + +{width=100%} + +* Quadrant ID. +* La feuille est vide, on insère. + +:::: + +::: + +# Algorithme d'insertion + +::: columns + +:::: {.column width=50%} + +## Insertion corps 2 + +{width=100%} + +:::: + +:::: {.column width=50%} + +## Arbre + +{width=100%} + +* Quadrant SD. +* La feuille est vide, on insère. + +:::: + +::: + +# Algorithme d'insertion + +::: columns + +:::: {.column width=50%} + +## Insertion corps 3 (1/N) + +{width=100%} + +:::: + +:::: {.column width=50%} + +## Arbre + +{width=100%} + +* Quadrant SD. +* La feuille est prise par 2. + +:::: + +::: + +# Algorithme d'insertion + +::: columns + +:::: {.column width=50%} + +## Insertion corps 3 (2/N) + +{width=100%} + +:::: + +:::: {.column width=50%} + +## Arbre + +{width=100%} + +* On crée un nouveau noeud. +* Deux corps dans le noeud ID. +* On crée un nouveau noeud. + +:::: + +::: + +# Algorithme d'insertion + +::: columns + +:::: {.column width=50%} + +## Insertion corps 3 (3/N) + +{width=100%} + +:::: + +:::: {.column width=50%} + +## Arbre + +{width=100%} + +* 2 va dans ID. +* 3 va dans SG. +* C'est des feuilles vides, tout va bien. + +:::: + +::: + +# 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