From 66c3bbfa94d8a41e58adf154be25e6d86fee8e30 Mon Sep 17 00:00:00 2001 From: "martial.simon" Date: Sun, 13 Apr 2025 19:54:19 +0200 Subject: init: initial commit --- ...u gradiant pour syst\303\250me matriciel.ipynb" | 31807 +++++++++++++++++++ 1 file changed, 31807 insertions(+) create mode 100644 "PVCM/cama/fr/ma52 M\303\251thode du gradiant pour syst\303\250me matriciel.ipynb" (limited to 'PVCM/cama/fr/ma52 Méthode du gradiant pour système matriciel.ipynb') diff --git "a/PVCM/cama/fr/ma52 M\303\251thode du gradiant pour syst\303\250me matriciel.ipynb" "b/PVCM/cama/fr/ma52 M\303\251thode du gradiant pour syst\303\250me matriciel.ipynb" new file mode 100644 index 0000000..ecfffc6 --- /dev/null +++ "b/PVCM/cama/fr/ma52 M\303\251thode du gradiant pour syst\303\250me matriciel.ipynb" @@ -0,0 +1,31807 @@ +{ + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.10.12" + } + }, + "nbformat": 4, + "nbformat_minor": 4, + "cells": [ + { + "cell_type": "markdown", + "metadata": { + "lang": "fr" + }, + "source": [ + "# A x = b vu comme un problème d'optimisation\n", + "\n", + "Pour résoudre $A {\\bf x} = {\\bf b}$ on va chercher le minimum de la fonctionnelle\n", + "\n", + "$$\n", + "J({\\bf x}) = \\frac{1}{2} \\, {\\bf x}^T \\, A \\, {\\bf x} - {\\bf b}\\, . {\\bf x}\n", + "$$\n", + "\n", + "En ce point la dérivée s'annule et dans un cas précis la dérivée est justement $A {\\bf x} - {\\bf b}$.\n", + "\n", + "## Calcul de dérivée\n", + "\n", + "La notion de dérivée en dimension supérieure à 1 doit être manipulée avec précaution car il y a différents\n", + "types de dérivées, les dérivées partielles et la dérivée totale.\n", + "\n", + "Ce qui nous intéresse est la dérivée dans une direction ce qui correpond à la dérivée partielle en $y$ si on\n", + "va dans la direction de l'axe $y$.\n", + "\n", + "### Définition\n", + "\n", + "On dit que $f : \\Omega \\subset X \\rightarrow Y$ ($\\Omega$ ouvert) est dérivable en ${\\bf p} \\in \\Omega$ si\n", + "\n", + "$$\n", + "\\exists \\; f'({\\bf p}) \\in L(X,Y) \\textrm{ tel que} \\\\\n", + "f ({\\bf p} + {\\bf h}) = f({\\bf p}) + f'({\\bf p}) ( {\\bf h} ) + {\\bf h} \\; \\varepsilon({\\bf h}) \\quad\n", + "\\textrm{ avec } \\lim_{\\bf h \\rightarrow 0} \\varepsilon({\\bf h}) = {\\bf 0}\n", + "$$\n", + "\n", + "où $L(X,Y)$ représente l'espace des applications linéraires continues de $X$ dans $Y$. Attention c'est\n", + "$f'(p) \\in L(X,Y)$ et non $f'$.\n", + "\n", + "\n", + "On note que si $f$ est dérivable en ${\\bf p}$ alors $\\forall \\; {\\bf h} \\in X$\n", + "\n", + "$$\n", + "f'({\\bf p}) ({\\bf h})= \\lim_{\\theta \\rightarrow 0} \\frac{f({\\bf p} + \\theta \\, {\\bf h}) - f({\\bf p})}{\\theta}\n", + "$$\n", + "\n", + "Attention à bien regarder le type de chaque terme. Pour bien visualiser j'ai considéré que $f$ est une \n", + "fonction scalaire donc $Y = ℝ$. Par contre $X = ℝ^n$ avec $n > 1$. \n", + "\n", + "#### Autre notation\n", + "\n", + "Je trouve la notation avec le gradient plus claire. On voit mieux où sont les vecteurs et les scalaires :\n", + "\n", + "$$\n", + "f ({\\bf p} + {\\bf h}) = f({\\bf p}) + \\nabla \\! f\\,({\\bf p})^T \\; {\\bf h} + {\\bf h}^T \\; \\varepsilon({\\bf h}) \\quad\n", + "$$\n", + "\n", + "Comme j'ai pris $Y = ℝ$ l'application $f$ est scalaire. $\\nabla \\! f \\, ({\\bf p})$ est un vecteur dont le\n", + "produit scalaire avec ${\\bf h}$ donne un réel.\n", + "\n", + "### Calculons la dérivée de J suivant une direction\n", + "\n", + "On calcule la dérivée de $J({\\bf x})$ au point ${\\bf p}$ suivant la direction donnée par le vecteur $\\bf h$ :\n", + "\n", + "$$\n", + "\\begin{align}\n", + "J'({\\bf p}) ({\\bf h}) &= \\lim_{\\theta \\rightarrow 0} \\; \\frac{J({\\bf p} + \\theta \\, {\\bf h}) - J({\\bf p})}{\\theta} \\\\\n", + " &= \\lim_{\\theta \\rightarrow 0} \\; \\frac{1}{\\theta} \\; \n", + " \\left( \\frac{1}{2} \\; ({\\bf p} + \\theta {\\bf h})^T \\, A \\; ({\\bf p} + \\theta {\\bf h})\n", + " - {\\bf b}^T \\, ({\\bf p} + \\theta {\\bf h})\n", + " - \\frac{1}{2} \\; {\\bf p}^T \\, A \\; {\\bf p} + {\\bf b}^T \\, {\\bf p}\n", + " \\right) \\\\\n", + " &= \\lim_{\\theta \\rightarrow 0} \\; \\frac{1}{\\theta} \\; \n", + " \\left( \\frac{1}{2} \\left( \\theta \\, {\\bf p}^T \\, A \\; {\\bf h} \n", + " + \\theta \\, {\\bf h}^T \\, A \\; {\\bf p} \n", + " + \\theta^2 \\, {\\bf h}^T \\, A \\; {\\bf h} \\right) \n", + " - \\theta \\, {\\bf b}^T \\, {\\bf h}\n", + " \\right) \\\\\n", + " &= \\frac{1}{2} \\left( {\\bf p}^T \\, A \\; {\\bf h} +{\\bf h}^T \\, A \\; {\\bf p} \\right) - {\\bf b}^T \\, {\\bf h} \\\\\n", + " &= \\frac{1}{2} \\left( {\\bf p}^T \\, A \\; {\\bf h} +{\\bf p}^T \\, A^T \\; {\\bf h} \\right) - {\\bf b}^T \\, {\\bf h} \\\\\n", + " &= \\frac{1}{2} \\left( {\\bf p}^T \\, ( A + A^T ) - {\\bf b}^T \\right) \\, {\\bf h} \\\\\n", + " &= \\frac{1}{2} \\left( ( A + A^T ) \\, {\\bf p} - {\\bf b} \\right) \\, . \\, {\\bf h}\n", + "\\end{align}\n", + "$$\n", + "\n", + "donc\n", + "\n", + "$$\n", + "\\begin{align}\n", + "J' : \\; & {\\bf x} \\in \\Omega \\subset ℝ^n \\longrightarrow \\;L(ℝ^n, ℝ) \\\\\n", + " & {\\bf x} \\; \\longmapsto \\; \\frac{1}{2} ( A + A^T ) \\, {\\bf x}^T - {\\bf b}\n", + "\\end{align}\n", + "$$\n", + "\n", + "### A symétrique\n", + "\n", + "Dans le cas où A est symétrique on a\n", + "\n", + "$$\n", + "J'({\\bf x}) = A \\; {\\bf x} - {\\bf b}\n", + "$$\n", + "\n", + "On note que si la dérivée s'annule alors on a résolu notre système matriciel !\n", + "\n", + "Il faut donc que\n", + "\n", + "* A soit symétrique\n", + "* J ait un minimum (par certain, on a vu un cas de point selle)\n", + "\n", + "et dans ce cas on peut utiliser une méthode de gradient pour trouver le minimum qui sera la solution de \n", + "notre système matriciel.\n", + "\n", + "#### Propriété\n", + "\n", + "Si A est symétrique et définie positive alors $J$ est convexe strictement et coervice (c.a.d. \n", + "$\\lim_{||{\\bf p}|| \\rightarrow \\infty} J({\\bf p}) = + \\infty$) ce qui garantit qu'elle a un minimum." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Gradient et dérivée\n", + "\n", + "Le gradient est composé des dérivées partielles suivant chaque direction de la base. On peut rérouler les calculs et calculer $\\nabla J$ facilement (mais c'est long). Faisons le pour une matrice sur la partie compliquée :\n", + "\n", + "$$ \n", + "\\begin{align}\n", + "\\frac{1}{2} {\\bf x}^T A {\\bf x} &= \\frac{1}{2} \\begin{pmatrix} x_1 & x_2 \\end{pmatrix} \\begin{pmatrix} a_{11} & a_{12} \\\\ a_{21} & a_{22} \\end{pmatrix} \\begin{pmatrix} x_1 \\\\ x_2 \\end{pmatrix} \\\\\n", + "&= \\frac{1}{2} \\left( x_1 a_{11} x_1 + x_1 a_{12} x_2 + x_2 a_{21} x_1 + x_2 a_{22} x_2 \\right) \\\\\n", + "& = \\frac{1}{2} \\left( a_{11} x_1^2 + (a_{12} + a_{21}) x_1 x_2 + a_{22} x_2^2 \\right)\n", + "\\end{align}\n", + "$$\n", + "\n", + "Calcul des dérivées partielles de $ \\frac{1}{2} x^T A x $ :\n", + "\n", + "$$\n", + "\\frac{\\partial}{\\partial x_1} \\left( \\frac{1}{2} (a_{11} x_1^2 + (a_{12} + a_{21}) x_1 x_2 + a_{22} x_2^2) \\right)\n", + " = \\frac{1}{2} (2 a_{11} x_1 + (a_{12} + a_{21}) x_2) = a_{11} x_1 + \\frac{a_{12} + a_{21}}{2} x_2\n", + "$$\n", + "\n", + "$$\n", + " \\frac{\\partial}{\\partial x_2} \\left( \\frac{1}{2} (a_{11} x_1^2 + (a_{12} + a_{21}) x_1 x_2 + a_{22} x_2^2) \\right)\n", + " = \\frac{1}{2} ((a_{12} + a_{21}) x_1 + 2 a_{22} x_2) = \\frac{a_{12} + a_{21}}{2} x_1 + a_{22} x_2\n", + "$$\n", + "\n", + "\n", + "Donc le gradient de $\\frac{1}{2} {\\bf x}^T A {\\bf x}$ est :\n", + "\n", + "$$\n", + "\\nabla \\left( \\frac{1}{2} x^T A x \\right) = \\begin{pmatrix} a_{11} x_1 + \\frac{a_{12} + a_{21}}{2} x_2 \\\\ \\frac{a_{12} + a_{21}}{2} x_1 + a_{22} x_2 \\end{pmatrix} = \\frac{1}{2} (A + A^T) \\, {\\bf x}\n", + "$$\n", + "\n", + "Il faut faire le calcul pour une matrice $n \\times n$ mais on arrive au même résultat\n", + "et donc\n", + "\n", + "$$\n", + "J'({\\bf x}) = \\nabla J({\\bf x})\n", + "$$\n", + "\n", + "Ainsi on peut aussi calculer la dérivée suivant la direction $\\bf h$ avec le gradient :\n", + "\n", + "$$\n", + "J'({\\bf x})({\\bf h}) = \\nabla J({\\bf x}) \\, . \\, {\\bf h}\n", + "$$" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import plotly.offline as py\n", + "import plotly.graph_objects as go\n", + "\n", + "%matplotlib inline\n", + "%config InlineBackend.figure_format = 'retina'" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "A = np.array([[2,1], [1,2]])\n", + "b = np.array([-1, 0.5])\n", + "\n", + "# lets make the mesh\n", + "x = np.linspace(-20,20,100)\n", + "y = np.linspace(-20,20,100)\n", + "mx, my = np.meshgrid(x,y) \n", + "M = np.array(np.meshgrid(x,y)).swapaxes(0,2)\n", + "\n", + "# and compute 0.5 x.T A x - b x on each node of the mesh\n", + "MA = np.einsum(\"ijk, ka -> ija\", M, A)\n", + "mz = 0.5 * np.einsum(\"ijk, ijk -> ij\", MA, M) - M @ b" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + " \n", + " " + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "application/vnd.plotly.v1+json": { + "config": { + 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" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "fig = go.Figure(data=[go.Surface(x=mx, y=my, z=mz)])\n", + "py.iplot(fig, show_link=False)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "lang": "fr" + }, + "source": [ + "\n", + "La contrainte de la propriété peut sembler être une limite trop dure mais il y a de nombreux problèmes qui\n", + "génèrent des matrices A qui respectent ces contraintes. Par exemples les problèmes de simuation numérique\n", + "comme celui de l'avion qu'on a vu génère une telle matrice, qui de plus sera très creuse avec une dizaine\n", + "de valeur non nulle par ligne même pour une matrice de 1 milliard par 1 milliard.\n", + "\n", + "C'est pour ce type de matrice qu'un méthode itérative comme le gradient offre un gain de performance incroyable\n", + "puisqu'on a la solution en quasiment $O(n^2)$ au lieu de $O(n^3)$.\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ] +} \ No newline at end of file -- cgit v1.2.3