From b9331d53510f7a775f5e418517f33fffc52cc991 Mon Sep 17 00:00:00 2001
From: gergaud <joseph.gergaud@toulouse-inp.fr>
Date: Fri, 21 Oct 2022 16:22:03 +0200
Subject: [PATCH] ajout TP C14

---
 Optimisation/TP/TP-C14/Algo_Gauss_Newton.m |  60 ++++
 Optimisation/TP/TP-C14/Algo_Newton.m       |  59 ++++
 Optimisation/TP/TP-C14/C14_figures.png     | Bin 0 -> 102578 bytes
 Optimisation/TP/TP-C14/C14_results.txt     |  65 +++++
 Optimisation/TP/TP-C14/Modelisation_C14.m  | 316 +++++++++++++++++++++
 Optimisation/TP/TP-C14/TP_Opti1A.pdf       | Bin 0 -> 595741 bytes
 6 files changed, 500 insertions(+)
 create mode 100644 Optimisation/TP/TP-C14/Algo_Gauss_Newton.m
 create mode 100644 Optimisation/TP/TP-C14/Algo_Newton.m
 create mode 100644 Optimisation/TP/TP-C14/C14_figures.png
 create mode 100644 Optimisation/TP/TP-C14/C14_results.txt
 create mode 100644 Optimisation/TP/TP-C14/Modelisation_C14.m
 create mode 100644 Optimisation/TP/TP-C14/TP_Opti1A.pdf

diff --git a/Optimisation/TP/TP-C14/Algo_Gauss_Newton.m b/Optimisation/TP/TP-C14/Algo_Gauss_Newton.m
new file mode 100644
index 0000000..2573a0d
--- /dev/null
+++ b/Optimisation/TP/TP-C14/Algo_Gauss_Newton.m
@@ -0,0 +1,60 @@
+function [beta, norm_grad_f_beta, f_beta, norm_delta, nb_it, exitflag] ...
+          = Algo_Gauss_Newton(residu, J_residu, beta0, option)
+%*****************************************************************
+% Fichier  ~gergaud/ENS/Optim1a/TP-optim-20-21/TP-ref/GN_ref.m   *
+% Novembre 2020                                                  *
+% Université de Toulouse, INP-ENSEEIHT                           *
+%*****************************************************************
+%
+% GN resout par l'algorithme de Gauss-Newton les problemes aux moindres carres
+% Min 0.5||r(beta)||^2
+% beta \in \IR^p
+%
+% Paramètres en entrés
+% --------------------
+% residu : fonction qui code les résidus
+%          r : \IR^p --> \IR^n
+% J_residu : fonction qui code la matrice jacobienne
+%            Jr : \IR^p --> real(n,p)
+% beta0 : point de départ
+%         real(p)
+% option(1) : Tol_abs, tolérance absolue
+%             real
+% option(2) : Tol_rel, tolérance relative
+%             real
+% option(3) : n_itmax, nombre d'itérations maximum
+%             integer
+%
+% Paramètres en sortie
+% --------------------
+% beta      : beta
+%             real(p)
+% norm_gradf_beta : ||gradient f(beta)||
+%                   real
+% f_beta : f(beta)
+%          real
+% r_beta : r(beta)
+%          real(n)
+% norm_delta : ||delta||
+%              real
+% nbit : nombre d'itérations
+%        integer
+% exitflag   : indicateur de sortie
+%              integer entre 1 et 4
+% exitflag = 1 : ||gradient f(beta)|| < max(Tol_rel||gradient f(beta0)||,Tol_abs)
+% exitflag = 2 : |f(beta^{k+1})-f(beta^k)| < max(Tol_rel|f(beta^k)|,Tol_abs)
+% exitflag = 3 : ||delta)|| < max(Tol_rel delta^k),Tol_abs)
+% exitflag = 4 : nombre maximum d'itérations atteint
+%      
+% ---------------------------------------------------------------------------------
+
+% TO DO %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+    beta = [0;0];
+    norm_grad_f_beta = 0;
+    f_beta = 0;
+    norm_delta = 0;
+    nb_it = 0;
+    exitflag = 0;
+
+end
diff --git a/Optimisation/TP/TP-C14/Algo_Newton.m b/Optimisation/TP/TP-C14/Algo_Newton.m
new file mode 100644
index 0000000..e7ab662
--- /dev/null
+++ b/Optimisation/TP/TP-C14/Algo_Newton.m
@@ -0,0 +1,59 @@
+function [beta, norm_grad_f_beta, f_beta, norm_delta, nb_it, exitflag] ...
+          = Algo_Newton(Hess_f_C14, beta0, option)
+%************************************************************
+% Fichier  ~gergaud/ENS/Optim1a/TP-optim-20-21/Newton_ref.m *
+% Novembre 2020                                             *
+% Université de Toulouse, INP-ENSEEIHT                      *
+%************************************************************
+%
+% Newton r�sout par l'algorithme de Newton les problemes aux moindres carres
+% Min 0.5||r(beta)||^2
+% beta \in R^p
+%
+% Parametres en entrees
+% --------------------
+% Hess_f_C14 : fonction qui code la hessiennne de f
+%              Hess_f_C14 : R^p --> matrice (p,p)
+%              (la fonction retourne aussi le residu et la jacobienne)
+% beta0  : point de départ
+%          real(p)
+% option(1) : Tol_abs, tolérance absolue
+%             real
+% option(2) : Tol_rel, tolérance relative
+%             real
+% option(3) : nitimax, nombre d'itérations maximum
+%             integer
+%
+% Parametres en sortie
+% --------------------
+% beta      : beta
+%             real(p)
+% norm_gradf_beta : ||gradient f(beta)||
+%                   real
+% f_beta    : f(beta)
+%             real
+% res       : r(beta)
+%             real(n)
+% norm_delta : ||delta||
+%              real
+% nbit       : nombre d'itérations
+%              integer
+% exitflag   : indicateur de sortie
+%              integer entre 1 et 4
+% exitflag = 1 : ||gradient f(beta)|| < max(Tol_rel||gradient f(beta0)||,Tol_abs)
+% exitflag = 2 : |f(beta^{k+1})-f(beta^k)| < max(Tol_rel|f(beta^k)|,Tol_abs)
+% exitflag = 3 : ||delta)|| < max(Tol_rel delta^k),Tol_abs)
+% exitflag = 4 : nombre maximum d'itérations atteint
+%      
+% ---------------------------------------------------------------------------------
+
+% TO DO %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+    beta = [0;0];
+    norm_grad_f_beta = 0;
+    f_beta = 0;
+    norm_delta = 0;
+    nb_it = 0;
+    exitflag = 0;
+
+end
diff --git a/Optimisation/TP/TP-C14/C14_figures.png b/Optimisation/TP/TP-C14/C14_figures.png
new file mode 100644
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diff --git a/Optimisation/TP/TP-C14/C14_results.txt b/Optimisation/TP/TP-C14/C14_results.txt
new file mode 100644
index 0000000..0109f4f
--- /dev/null
+++ b/Optimisation/TP/TP-C14/C14_results.txt
@@ -0,0 +1,65 @@
+[Warning: Cannot set Position while WindowStyle is 'docked'] 
+[> In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('Modelisation_C14', '/Users/gergaud/git-ENS/Optimisation-et-EDP-1A/enseignants/Optimisation/TP/TP-C14/src_etudiants/Modelisation_C14.m', 59)" style="font-weight:bold">Modelisation_C14</a> (<a href="matlab: opentoline('/Users/gergaud/git-ENS/Optimisation-et-EDP-1A/enseignants/Optimisation/TP/TP-C14/src_etudiants/Modelisation_C14.m',59,0)">line 59</a>)] 
+[Warning: Contour not rendered for constant ZData] 
+[> In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('contour', '/Applications/MATLAB_R2022b.app/toolbox/matlab/specgraph/contour.m', 51)" style="font-weight:bold">contour</a> (<a href="matlab: opentoline('/Applications/MATLAB_R2022b.app/toolbox/matlab/specgraph/contour.m',51,0)">line 51</a>)
+In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('Modelisation_C14', '/Users/gergaud/git-ENS/Optimisation-et-EDP-1A/enseignants/Optimisation/TP/TP-C14/src_etudiants/Modelisation_C14.m', 89)" style="font-weight:bold">Modelisation_C14</a> (<a href="matlab: opentoline('/Users/gergaud/git-ENS/Optimisation-et-EDP-1A/enseignants/Optimisation/TP/TP-C14/src_etudiants/Modelisation_C14.m',89,0)">line 89</a>)] 
+[Warning: Contour not rendered for constant ZData] 
+[> In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('contour', '/Applications/MATLAB_R2022b.app/toolbox/matlab/specgraph/contour.m', 51)" style="font-weight:bold">contour</a> (<a href="matlab: opentoline('/Applications/MATLAB_R2022b.app/toolbox/matlab/specgraph/contour.m',51,0)">line 51</a>)
+In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('Modelisation_C14', '/Users/gergaud/git-ENS/Optimisation-et-EDP-1A/enseignants/Optimisation/TP/TP-C14/src_etudiants/Modelisation_C14.m', 109)" style="font-weight:bold">Modelisation_C14</a> (<a href="matlab: opentoline('/Users/gergaud/git-ENS/Optimisation-et-EDP-1A/enseignants/Optimisation/TP/TP-C14/src_etudiants/Modelisation_C14.m',109,0)">line 109</a>)] 
+Algorithme de Gauss-Newton
+--------------------------
+residu_C14(beta0, Donnees)
+     0
+
+J_residu_C14(beta0, Donnees)
+     0
+
+--------------------------------------------------------------------------------------------
+     nb_iter         A0        lambda    ||f'(beta)||     f(beta)    ||delta||     exitflag 
+--------------------------------------------------------------------------------------------
+            0           10       0.0001            0            0
+
+     0     0     0     0     0     0     0
+
+     0     0     0     0     0     0     0
+
+     0     0     0     0     0     0     0
+
+     0     0     0     0     0     0     0
+
+     0     0     0     0     0     0     0
+
+     0     0     0     0     0     0     0
+
+     0     0     0     0     0     0     0
+
+     0     0     0     0     0     0     0
+
+--------------------------------------------------------------------------------------------
+Algorithme de Newton
+--------------------
+Hessienne f(beta^{(0)})
+     0
+
+--------------------------------------------------------------------------------------------
+     nb_iter         A0        lambda    ||f'(beta)||     f(beta)    ||delta||     exitflag 
+--------------------------------------------------------------------------------------------
+            0           10       0.0001            0            0
+
+     0     0     0     0     0     0     0
+
+     0     0     0     0     0     0     0
+
+     0     0     0     0     0     0     0
+
+     0     0     0     0     0     0     0
+
+     0     0     0     0     0     0     0
+
+     0     0     0     0     0     0     0
+
+     0     0     0     0     0     0     0
+
+     0     0     0     0     0     0     0
+
+--------------------------------------------------------------------------------------------
diff --git a/Optimisation/TP/TP-C14/Modelisation_C14.m b/Optimisation/TP/TP-C14/Modelisation_C14.m
new file mode 100644
index 0000000..d1844bb
--- /dev/null
+++ b/Optimisation/TP/TP-C14/Modelisation_C14.m
@@ -0,0 +1,316 @@
+%-------------------------------------------------------------------------%
+% 1SN - TP Optimisation                                                   %
+% INP Toulouse - ENSEEIHT                                                 %
+% Novembre 2020                                                           %
+%                                                                         %
+% Ce fichier contient le programme principal permettant l'estimation      %
+% des parametres de la fonction de desintegration radioactive du          %
+% carbone 14 par une approche des moindres carres.                        %                                      %
+% Modele : A(t)= A0*exp(-lambda*t)                                        %
+% Les algorithmes utilises pour la minimisation sont :                    %
+% - l'algorithme de Gauss-Newton                                          %
+% - l'algorithme de Newton.                                               %
+%-------------------------------------------------------------------------%
+
+clear
+close all
+clc
+format shortG
+taille_ecran = get(0,'ScreenSize');
+L = taille_ecran(3);
+H = taille_ecran(4);
+
+% Initialisation
+
+% Donnees
+if exist('C14_results.txt','file')
+    delete('C14_results.txt');
+end
+diary C14_results.txt
+Ti = [ 500; 1000; 2000; 3000; 4000; 5000; 6300];
+Ai = [14.5; 13.5; 12.0; 10.8;  9.9;  8.9;  8.0];
+Donnees = [Ti, Ai];
+
+% Estimation a priori des parametres du modele : beta0 = [A0, lambda]
+beta0 = [10; 0.0001];   % Newton, Gauus-Newton, fminunc et leastsq converge
+% beta0 = [15; 0.001];    % Newton, Gauss-Newton, fminunc et leastsq divergent
+% beta0 = [15; 0.0005];   % Newton diverge, Gauss-Newton, fminunc et leastsq convergent
+% beta0 = [10; 0.0005];   % Gauss-Newton converge
+
+%% Calcul et affichage du modele initial ----------------------------------
+xmin = 9; xmax = 20;
+x = linspace(xmin, xmax, 100);
+
+ymin = -0.0001; ymax = 0.0005;
+y = linspace(ymin, ymax, 100);
+
+[A0_plot, lambda_plot] = meshgrid(x, y);
+[m , n] = size(A0_plot);
+f_plot = zeros(m, n);
+
+for i = 1:m
+   for j = 1:n
+      res_plot = residu_C14([A0_plot(i,j); lambda_plot(i,j)], Donnees);
+      % TO DO %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+      f_plot(i,j) = 0.5*(res_plot.')*res_plot;
+   end
+end
+
+figure('Position',[0.1*L,0.1*H,0.8*L,0.8*H]); 
+subplot(1,3,1) % Fonction f en 3D
+    mesh(A0_plot, lambda_plot, f_plot)
+    axis('square')
+    title('Representation de la fonction f des moindres carres')
+    xlabel('A_0');
+    ylabel('\lambda')
+    zlabel('f(A_0,\lambda)')
+
+T = linspace(0,6500,100);
+A = beta0(1)*exp(-beta0(2)*T);
+txt_legend{1} = 'donnees';
+txt_legend{2} = 'depart';
+
+subplot(2, 3, 2) % Donnees + modele initial pour Gauss-Newton
+    title('Desintegration radioactive du carbone 14 (Gauss-Newton)')
+    axis([0 max(T) 0 18])
+    xlabel('dur�e T')
+    ylabel('radioactivit� A')
+    hold on
+    grid on
+    plot(Ti, Ai, 'ok')
+    plot(T, A)
+    legend(txt_legend{1:2}, 'Location', 'SouthWest')
+
+subplot(2, 3, 5) % Courbes de niveaux de f pour Gauss-Newton
+    title('Recherche des parametres (Gauss-Newton)')
+    xlabel('A_0')
+    ylabel('\lambda')
+    hold on
+    contour(A0_plot, lambda_plot, f_plot, 100);
+    plot(beta0(1),beta0(2),'ok')
+    text(beta0(1),beta0(2),'  depart \beta^{(0)}')
+    
+subplot(2, 3, 3) % Donnees + modele initial pour Newton
+    title('Desintegration radioactive du carbone 14 (Newton)')
+    axis([0 max(T) 0 18])
+    xlabel('dur�e T')
+    ylabel('radioactivit� A')
+    hold on
+    grid on
+    plot(Ti, Ai, 'ok')
+    plot(T, A)
+    legend(txt_legend{1:2}, 'Location', 'SouthWest')
+
+subplot(2, 3, 6) % Courbes de niveaux de f pour Newton
+    title('Recherche des parametres (Newton)')
+    xlabel('A_0')
+    ylabel('\lambda')
+    hold on
+    contour(A0_plot, lambda_plot, f_plot, 100);
+    plot(beta0(1),beta0(2),'ok')
+    text(beta0(1),beta0(2),'  depart \beta^{(0)}')
+
+pause(0.5)
+
+%% Algorithmes
+% Choix du nombre d'iteration maximal
+
+nb_iterations_max = 8;
+txt_legend = cell(1,nb_iterations_max+2);
+txt_legend{1} = 'donnees';
+txt_legend{2} = 'depart';
+
+
+%% Gauss-Newton
+% -------------
+% Initialisation de l'affichage
+disp('Algorithme de Gauss-Newton')
+disp('--------------------------')
+disp('residu_C14(beta0, Donnees)')
+disp(residu_C14(beta0, Donnees))
+disp('J_residu_C14(beta0, Donnees)')
+disp(J_residu_C14(beta0, Donnees))
+disp('--------------------------------------------------------------------------------------------')
+disp('     nb_iter         A0        lambda    ||f''(beta)||     f(beta)    ||delta||     exitflag ')
+disp('--------------------------------------------------------------------------------------------')
+
+% Calcul et affichage des valeurs initiales
+res_beta = residu_C14(beta0, Donnees);
+f_beta = 0.5*(res_beta.')*res_beta;
+J_res_beta = J_residu_C14(beta0, Donnees);
+norm_grad_f_beta = norm((J_res_beta.')*res_beta);
+disp([0 beta0(1) beta0(2) norm_grad_f_beta f_beta]);
+
+options = [sqrt(eps) 1.e-12 0];
+
+for i = 1:nb_iterations_max
+    txt_legend{i+2} = ['iterat� ' num2str(i)];
+	options(3) = i;
+	[beta, norm_grad_f_beta, f_beta, norm_delta, k, exitflag] = ...
+	Algo_Gauss_Newton(@(beta) residu_C14(beta, Donnees), ...
+                      @(beta) J_residu_C14(beta, Donnees), ...
+                      beta0, options);
+	disp([k beta(1) beta(2) norm_grad_f_beta f_beta norm_delta exitflag])
+    % Visualisation
+    A = beta(1)*exp(-beta(2)*T);
+	subplot(2, 3, 2)
+	plot(T,A)
+    legend(txt_legend{1:i+2})
+    %  eval(['print -depsc fig_GN_courbe' int2str(i) '_C14'])
+	subplot(2, 3, 5)
+    plot(beta(1),beta(2),'ok')
+	text(beta(1),beta(2),['  \beta^{(' num2str(i) ')}'])
+  
+	pause(0.5)
+end
+
+% Affichage des itérés de beta et sauvegarde des graphique
+disp('--------------------------------------------------------------------------------------------')
+
+%% Newton
+% -------
+% Initialisation de l'affichage
+disp('Algorithme de Newton')
+disp('--------------------')
+% Initialisation de l'affichage
+[H, ~, ~] = Hess_f_C14(beta0, Donnees, ...
+                            @(beta) residu_C14(beta, Donnees), ...
+                            @(beta) J_residu_C14(beta, Donnees));
+disp('Hessienne f(beta^{(0)})')
+disp(H)
+
+disp('--------------------------------------------------------------------------------------------')
+disp('     nb_iter         A0        lambda    ||f''(beta)||     f(beta)    ||delta||     exitflag ')
+disp('--------------------------------------------------------------------------------------------')
+
+% Calcul et affichage des valeurs initiales
+res_beta = residu_C14(beta0, Donnees);
+f_beta = 0.5*(res_beta.')*res_beta;
+J_res_beta = J_residu_C14(beta0, Donnees);
+norm_grad_f_beta = norm((J_res_beta.')*res_beta);
+disp([0 beta0(1) beta0(2) norm_grad_f_beta f_beta]);
+
+options = [sqrt(eps) 1.e-12 0];
+
+for i = 1:nb_iterations_max
+    options(3) = i;
+    % Algorithme de Newton
+    [beta, norm_grad_f_beta, f_beta, norm_delta, k, exitflag] = ...
+    Algo_Newton(@(beta) Hess_f_C14(beta, Donnees, ...
+                                   @(beta) residu_C14(beta, Donnees), ...
+                                   @(beta) J_residu_C14(beta, Donnees)), ...
+                beta0, options);
+    % Affichage des valeurs
+    disp([k beta(1) beta(2) norm_grad_f_beta f_beta norm_delta exitflag])
+    % Visualisation
+    A = beta(1)*exp(-beta(2)*T);
+    subplot(2, 3, 3)
+    plot(T,A)
+    legend(txt_legend{1:i+2})
+    subplot(2, 3, 6)
+    plot(beta(1),beta(2),'ok')
+	text(beta(1),beta(2),['  \beta^{(' num2str(i) ')}'])
+  
+	pause(0.5)
+end
+
+% Affichage des iteres de beta et sauvegarde des courbes
+disp('--------------------------------------------------------------------------------------------')
+print('C14_figures','-dpng')
+diary
+
+%% Fonctions a coder : residu, Jacobienne, Hessienne
+
+%%------------------------------------------------------------------------%                                                                     %
+% Fonction de calcul du residu de la fonction de desintegration           %
+% radioactive du carbone 14                                               %
+%-------------------------------------------------------------------------%
+
+function res = residu_C14(beta, donnees)
+%
+% Paramètres en entrés
+% --------------------
+% beta : vecteur des paramètres
+%         real(p)
+% donnees : Données
+%           real(n,2)
+%
+% Paramètres en sortie
+% --------------------
+% res    : vecteur des résidus
+%          real(n)
+%
+    % TO DO %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+    res = 0;
+
+end
+
+%-------------------------------------------------------------------------%                                                                        %
+% Fonction de calcul de la Jacobienne du residu de la fonction de         %
+% desintegration radioactive du carbone 14                                %
+%-------------------------------------------------------------------------%
+
+function J_res = J_residu_C14(beta, donnees)
+%
+% Paramètres en entrés
+% --------------------
+% beta : vecteur des paramètres
+%         real(p)
+% donnees : Données
+%           real(n,2)
+%
+% Paramètres en sortie
+% --------------------
+% J_res    : Matrice jacobienne des résidus
+%            real(n,p)
+%
+    % TO DO %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+    J_res = 0;
+    
+end
+
+%-------------------------------------------------------------------------%
+% Fonction de calcul de la Hessienne des moindres carres de la fonction   %
+% de desintegration radioactive du carbone 14                             %
+% (residu et J_residu sont des fonctions en entree)                       %
+%-------------------------------------------------------------------------%
+
+function [H_f, res, J_res] = Hess_f_C14(beta, donnees, residu, J_residu)
+%
+% Paramètres en entrés
+% --------------------
+% beta     : vecteur des paramètres
+%            real(p)
+% donnees  : Données
+%            real(n,2)
+% residu   : fonction qui code les résidus
+%            res_beta = residus(beta)
+% J_residu : fonction qui code la matrice jacobienne
+%            J_res_beta = J_residu(beta);
+%
+% Paramètres en sortie
+% --------------------
+% H_f      : Matrice hessienne
+%            real(p,p)
+% res      : vecteur des résidus
+%            real(n)
+% J_res    : Matrice jacobienne des résiduis
+%            real(n,p)
+%
+    % TO DO %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+    res = 0;
+    J_res = 0;
+    H_f = 0;
+
+end
+
+
+
+
+
+
+
+
+
+
+
diff --git a/Optimisation/TP/TP-C14/TP_Opti1A.pdf b/Optimisation/TP/TP-C14/TP_Opti1A.pdf
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-- 
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