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a/Claudio_Maggioni_2/Claudio_Maggioni_2.tex b/Claudio_Maggioni_2/Claudio_Maggioni_2.tex index 024e451..ed75527 100755 --- a/Claudio_Maggioni_2/Claudio_Maggioni_2.tex +++ b/Claudio_Maggioni_2/Claudio_Maggioni_2.tex @@ -130,9 +130,33 @@ The code to generate the plots below can be found in Section 1.5 of the script \ \subsection{Finally, explain why the Conjugate Gradient method is a Krylov subspace method.} -Because theorem 5.3 holds, which itself holds mainly because of this (5.10, page 106 [127]): +We can say that the Conjugate Gradient method is a Krylov subspace method since we can say that both all residuals $r_i$ and all search directions $p_i$ are contained in the span of $0$ to $k$ repeated applications of matrix transformer $A$ onto the initial search direction $p_0$. Broadly speaking, this property is directly connected with the fact that the CG methods performs up to $n$ steps exactly for a $n$ by $n$ matrix, walking steps that are all $A$-orthogonal with each other (i.e. for all couples of search steps $p_i$, $p_j$ where $i \neq j$, $\langle Ap_i, p_j \rangle = 0$). +To sum up our claim, we can say CG is indeed a Krylov subspace method because: -\[r_{k+1} = r_k + a_k * A * p_k\] +\[\text{span}\{r_0, r_1, \ldots, r_n\} = \text{span}\{r_0, A r_0, \ldots, A^k r_0\}\] +\[\text{span}\{p_0, p_1, \ldots, p_n\} = \text{span}\{r_0, A r_0, \ldots, A^k r_0\}\] + +These statements have been already proven in class and the proof can be found in Theorem 5.3 of Nocedal's book. + +% These statements can be proven by induction over $k$. The base case holds trivially for $k=0$, while we have by the induction hypothesis for any $k$ that: + +% \[r_k \in \text{span}\{r_0, A r_0, \ldots, A^k r_0\}\;\;\; p_k \in \text{span}\{r_0, A r_0, \ldots, A^k r_0\}\] + +% We'd like to prove that the two properties hold from $k+1$ starting from the hypothesis on $k$. We first multiply the first hypothesis by $A$ from the left: + +% \[A r_k \in \text{span}\{A r_0, A^2 r_0, \ldots, A^{k+1} r_0\}\] + +% By the alternative definition of the residual for the CG method (i.e. $r_{k+1} = r_k + \alpha_k A p_k$), we find that: + +% \[r_{k+1} \in \text{span}\{r_0, A r_0, A^2 r_0, \ldots, A^{k+1} r_0\}\] + +% We need to add $r_0$ in the span again since one of the components that defines $r_1$ is indeed $r_0$. We don't need to add other terms in the span since $r_1$ to $r_k$ are in the span already by the induction hypothesis. + +% Combining this expression with the induction hypothesis we prove induction of the first statement by having: + +% \[\text{span}\{r_0, r_1, \ldots, r_n, r_{n+1}\} \subseteq \text{span}\{r_0, A r_0, \ldots, A^k r_0, A^{k+1} r_0\}\] + +% To prove $\supseteq$ as well to achieve equality, we use the induction hypothesis for the second statement to find that $A^kr_0 \in \text{span}\{A$: \section{Exercise 2} diff --git a/Claudio_Maggioni_2/smarthut.md b/Claudio_Maggioni_2/smarthut.md deleted file mode 100644 index 03572f8..0000000 --- a/Claudio_Maggioni_2/smarthut.md +++ /dev/null @@ -1,247 +0,0 @@ -\documentclass{scrartcl} -\usepackage[utf8]{inputenc} -\usepackage{float} -\usepackage{graphicx} -\usepackage[ruled,vlined]{algorithm2e} -\usepackage{subcaption} -\usepackage{amsmath} -\usepackage{pgfplots} -\pgfplotsset{compat=newest} -\usetikzlibrary{plotmarks} -\usetikzlibrary{arrows.meta} -\usepgfplotslibrary{patchplots} -\usepackage{grffile} -\usepackage{amsmath} -\usepackage{subcaption} -\usepgfplotslibrary{external} -\tikzexternalize -\usepackage[margin=2.5cm]{geometry} - -% To compile: -% sed -i 's#title style={font=\\bfseries#title style={yshift=1ex, font=\\tiny\\bfseries#' *.tex -% luatex -enable-write18 -shellescape main.tex - -\pgfplotsset{every x tick label/.append style={font=\tiny, yshift=0.5ex}} -\pgfplotsset{every title/.append style={font=\tiny, align=center}} -\pgfplotsset{every y tick label/.append style={font=\tiny, xshift=0.5ex}} -\pgfplotsset{every z tick label/.append style={font=\tiny, xshift=0.5ex}} - -\setlength{\parindent}{0cm} -\setlength{\parskip}{0.5\baselineskip} - -\title{Optimization methods -- Homework 2} -\author{Claudio Maggioni} - -\begin{document} - -\maketitle - -\section{Exercise 1} - -\subsection{Implement the matrix $A$ and the vector $b$, for the moment, without taking into consideration the -boundary conditions. As you can see, the matrix $A$ is not symmetric. Does an energy function of -the problem exist? Consider $N = 4$ and show your answer, explaining why it can or cannot exist.} - -The implementation of a function that generates $A$ w.r.t. the parameter $N$ can be found in the MATLAB script -\texttt{main.m} under Section 1.1. - -The matrix $A$ and the vector $b$ appear in the following form: - -\[A = \begin{bmatrix} - 1 & 0 & 0 & 0 & \ldots & 0 \\ - -1 & 2 & -1 & 0 & \ldots & 0 \\ - 0 & -1 & 2 & -1 & \ldots & 0 \\ - \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ - 0 & 0 & 0 & 0 & \ldots & 1 \\ - \end{bmatrix}\;\;\;b = \begin{bmatrix}0\\h^2\\h^2\\\vdots\\0\end{bmatrix}\] - -For $N = 4$, we can attempt to build a minimizer to the solve the system $Ax = b$. In order to find an $x$ such that $Ax = b$, we could define a minimizer $\phi(x)$ such that: - -\[\phi(x) = \|b - Ax\|^2\] - -Here $\phi(x) = 0$ would mean that $x$ is an exact solution of the system $Ax = b$. We can then attempt to write such minimizer for $N = 4$: - -\[\phi(x) = \|b - Ax\|^2 = \left|\begin{bmatrix}0 - x_1\\\frac19 -x_1 + 2x_2 -x_3\\ -\frac19 -x_2 +2x_3 -x_4\\0 - x_4\end{bmatrix}\right|^2 =\]\[= x_1^2 + \left(\frac19 - x_1 + 2x_2 - x_3\right)^2 + \left(\frac19 - x_2 + 2x_3 - x_4\right)^2 + x_4^2\] - -\[\Delta \phi(x) = \begin{bmatrix}4x_1 - 4x_2 + 2x_3 -\frac29\\ --4x_1 +10x_2 -8x_3 + 2x_4 +\frac29\\ -2x_1 -8x_2 +10x_3 + -4x_4 +\frac29\\ - 2x_2 - 4x_3 + 4x_4 -\frac29\end{bmatrix}\;\;\;\Delta^2 \phi(x) = \begin{bmatrix}4&-4&2&0\\-4&10&-8&2\\2&-8&10&-4\\0&2&-4&4\end{bmatrix}\] - - As it can be seen from the Hessian calculation, the Hessian is positive definite forall $x$s. This means, by the sufficient condition of minimizers, that we can find a minimizer by solving $\Delta \phi(x) = 0$ (i.e. finding stationary points in the hyperspace defined by $\phi(x)$. Solving that, we find: - -\[x = \begin{bmatrix}0\\\frac19\\\frac19\\0\end{bmatrix}\;\;\] - -which is indeed the minimizer and solution of $Ax = b$. Therefore, $\phi(x)$ is a valid energy function and we can say an energy function for the problem exists. - -\subsection{Once the new matrix has been derived, write the energy function related to the new problem -and the corresponding gradient and Hessian.} - -As by the definitions of $A$ and $b$ given in the assignment, we already enforce $x_1 = x_n = 0$, since the first and the last term of the matrix-vector product $Ax$ are $x_1$ and $x_n$ respectively and the system of equations represented by $Ax = b$ would indeed include the equalities $x_1 = b_1 = 0$ and $x_n = b_n = 0$. Therefore, we can simply alter the matrix $A$ without any need to perform any other transformation in the system. - - We therefore define $\overline{A}$ as a copy of $A$ where $\overline{A}_{2,1} = \overline{A}_{n-1, n} = 0$. - -The objective function then becomes $\phi(x) = \frac12 x^T \overline{A} x - b^T x$. -And since the objective is a standard quadratic form, the gradient is $\overline{A}x - b$ while the Hessian is $\overline{A}$. - -\subsection{Write the Conjugate Gradient algorithm in the pdf and implement it Matlab code in a function -called \texttt{CGSolve}.} - -The Conjugate Gradient algorithm is the following: - -\begin{algorithm}[H] -\SetAlgoLined -\KwResult{Conjugate Gradient Algorithm} -Set $r_0 \gets Ax_0 - b$, $p_0 \gets r_0$, $k \gets 0$\; -\While{$r_k \neq 0$}{% - $\alpha_k \gets \frac{r^T_kr_k}{p^T_kAp_k}$\; - $x_{k+1} \gets x_k + \alpha_k p_k$\; - $r_{k+1} \gets r_k + \alpha_k A p_k$\; - $\beta_{k+1} \gets \frac{r^T_{k+1}r_{k+1}}{r^T_kr_k}$\; - $p_{k+1} \gets -r_{k+1} + \beta_{k+1}p_k$\; - $k \gets k + 1$\; -} -\end{algorithm} - -The MATLAB solution of this task can be found in Section 1.3 of the script \texttt{main.m} under the function \texttt{CGSolve}. - -\subsection{Solve the Poisson problem.} - -The solution of this task can be found in Section 1.4 of the script \texttt{main.m}. -Due to space constraints, the $R^{1000}$ column vector for the solution of $x$ is not shown here but can be easily derived by running the script and checking the variable \texttt{x} after the script execution. - -\subsection{Plot the value of energy function and the norm of the gradient (here, -use semilogy) as functions of the iterations.} - -The code to generate the plots below can be found in Section 1.5 of the script \texttt{main.m}. - -\begin{figure}[H] - \begin{subfigure}{0.5\textwidth} - \resizebox{\textwidth}{\textwidth}{\input{obvalues}} - \caption{Objective function values w.r.t. iteration number} - \end{subfigure} - \begin{subfigure}{0.5\textwidth} - \resizebox{\textwidth}{\textwidth}{\input{gnorms}} - \caption{Norm of the gradient w.r.t. iteration number \\ (y-axis is log scaled)} - \end{subfigure} - \caption{Plots for Exercise 1.4.} -\end{figure} - -\subsection{Finally, explain why the Conjugate Gradient method is a Krylov subspace method.} - -Because theorem 5.3 holds, which itself holds mainly because of this (5.10, page 106 [127]): - -\[r_{k+1} = r_k + a_k * A * p_k\] - -\section{Exercise 2} - -Consider the linear system $Ax = b$, where the matrix $A$ is constructed in three different ways: - -\begin{itemize} - \item $A_1 =$ diag([1:10]) - \item $A_2 =$ diag(ones(1,10)) - \item $A_3 =$ diag([1, 1, 1, 3, 4, 5, 5, 5, 10, 10]) - \item $A_4 =$ diag([1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0]) -\end{itemize} - -\subsection{How many distinct eigenvalues has each matrix?} - -Each matrix has a distinct number of eigenvalues equal to the number of distinct -elements on its diagonal. So, in order, each A has respectively 10, 1, 5, and 10 distinct eigenvalues. - -\subsection{Construct a right-hand side $b=$rand(10,1) and apply the -Conjugate Gradient method to solve the system for each $A$.} - -The solution of this task can be found in section 2.2 of the \texttt{main.m} MATLAB script. -Below are the chosen $b$ vector (which is unchanged between executions due to fixing the random generator seed in the script) and the solutions of $x$ for each matrix respectively. - -\[b = \begin{bmatrix}0.814723686393179\\ -0.905791937075619\\ -0.126986816293506\\ -0.913375856139019\\ -0.632359246225410\\ -0.0975404049994095\\ -0.278498218867048\\ -0.546881519204984\\ -0.957506835434298\\ -0.964888535199277\end{bmatrix}\; -x_1 = \begin{bmatrix}0.814723686393179\\ -0.452895968537810\\ -0.0423289387645020\\ -0.228343964034755\\ -0.126471849245082\\ -0.0162567341665680\\ -0.0397854598381500\\ -0.0683601899006230\\ -0.106389648381589\\ -0.0964888535199280\end{bmatrix}\;x_2 = \begin{bmatrix}0.814723686393179\\ -0.905791937075619\\ -0.126986816293506\\ -0.913375856139019\\ -0.632359246225410\\ -0.0975404049994095\\ -0.278498218867048\\ -0.546881519204984\\ -0.957506835434298\\ -0.964888535199277\end{bmatrix}\]\[x_3 = \begin{bmatrix}0.814723686393179\\ -0.905791937075619\\ -0.126986816293506\\ -0.304458618713007\\ -0.158089811556352\\ -0.0195080809998819\\ -0.0556996437734097\\ -0.109376303840997\\ -0.0957506835434296\\ -0.0964888535199280\end{bmatrix}\;x_4 = \begin{bmatrix}0.740657896716011\\ -0.754826614267239\\ -0.0976821653904972\\ -0.652411326111541\\ -0.421572830214428\\ -0.0609627567866090\\ -0.163822480881755\\ -0.303823066390868\\ -0.503950965995613\\ -0.482444267601989\end{bmatrix}\] - -\subsection{Compute the logarithm energy norm of the error for each matrix -and plot it with respect to the number of iteration.} - -The code to generate the plots below and to compute the logarithm energy norm of the error can be found in section 2.3 of the \texttt{main.m} MATLAB script. - -\begin{figure}[H] - \begin{subfigure}{0.5\textwidth} - \resizebox{\textwidth}{!}{\input{a1}} - \caption{First matrix} - \end{subfigure} - \begin{subfigure}{0.5\textwidth} - \resizebox{\textwidth}{!}{\input{a2}} - \caption{Second matrix} - \end{subfigure} - \begin{subfigure}{0.5\textwidth} - \resizebox{\textwidth}{!}{\input{a3}} - \caption{Third matrix} - \end{subfigure} - \begin{subfigure}{0.5\textwidth} - \resizebox{\textwidth}{!}{\input{a4}} - \caption{Fourth matrix} - \end{subfigure} - \caption{Plots of logarithm energy norm of the error per iteration. Minus infinity logarithms not shown in the plot.} -\end{figure} - -\subsection{Comment on the convergence of the method for the different matrices. What can you say observing -the number of iterations obtained and the number of clusters of the eigenvalues of the related -matrix?} - -The method converges quickly for each matrix. The fastest convergence surely happens for $A2$, which is -the identity matrix and therefore makes the $Ax = b$ problem trivial. - -For all the other matrices, we observe the energy norm of the error decreasing exponentially as the iterations -increase, eventually reaching $0$ for the cases where the method converges exactly (namely on matrices $A1$ and $A3$). - -Other than for the fourth matrix, the number of iterations is exactly equal -to the number of distinct eigenvalues for the matrix. That exception on the fourth matrix is simply due to the -tolerance termination condition holding true for an earlier iteration, i.e. we terminate early since we find an -approximation of $x$ with residual norm below $10^{-8}$. - -\end{document}