From 71ae37534e652a1f76007b7f4d368eb54c51afe6 Mon Sep 17 00:00:00 2001 From: Claudio Maggioni Date: Wed, 25 Nov 2020 15:20:13 +0100 Subject: [PATCH] mp5: done 1-3 --- mp4/Project_4_Maggioni_Claudio.pdf | Bin 2828422 -> 2828442 bytes mp5/A_eig.tex | 126 +++++++++++ mp5/Makefile | 15 ++ mp5/Project_5_Maggioni_Claudio.pdf | Bin 0 -> 202282 bytes mp5/Project_5_Maggioni_Claudio.tex | 126 +++++++++++ .../blur_data/A.mat | Bin 0 -> 1961992 bytes .../blur_data/B.mat | Bin 0 -> 463814 bytes .../code_template.m | 29 +++ mp5/Project_5_Maggioni_Claudio/myCG.m | 26 +++ mp5/Project_5_Maggioni_Claudio/test.m | 18 ++ .../test_data/A_test.mat | Bin 0 -> 146024 bytes .../test_data/b_test.mat | Bin 0 -> 147756 bytes .../test_data/x_test_exact.mat | Bin 0 -> 146901 bytes mp5/assignment.sty | 95 ++++++++ mp5/test_semilogy.tex | 205 ++++++++++++++++++ mp5/usi_inf.pdf | Bin 0 -> 34678 bytes 16 files changed, 640 insertions(+) create mode 100644 mp5/A_eig.tex create mode 100644 mp5/Makefile create mode 100644 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+\usetikzlibrary{arrows.meta} +\usepgfplotslibrary{patchplots} +\usepackage{grffile} +\usepackage{amsmath} +\usepackage{subcaption} +\usepgfplotslibrary{external} +\tikzexternalize + +\begin{document} + +\setassignment +\setduedate{Wednesday, December 02, 2020, 11:59 PM} + +\serieheader{Numerical Computing}{2020}{Student: Claudio Maggioni}{Discussed with: --}{Solution for Project 5}{} +\newline + +\assignmentpolicy + +The purpose of this assignment is to gain insight on the theoretical and +numerical properties of the Conjugate Gradient method. Here we use this method +in an image processing application with the goal of deblur an image given the +exact (noise-free) blurred image and the original transformation matrix. Note +that the ``noise-free" simplification is essential for us to solve this problem +in the scope of this assignment. + +\section{General Questions [10 points]} +\subsection{What is the size of the matrix $A$?} +$A$ is an $n^2$ by $n^2$ matrix, where $n$ is the width and height in pixels +of the image to transform. + +\subsection{How many diagonals bands does $A$ have?} +$A$ as $d^2$ diagonal bands, where $d$ is strictly an order of magnitude below +$n$. + +\subsection{What is the length of the vectorized blur image $b$?} +$b$ is the row-vectorized form of the image pixel matrix, and thus has +dimensions 1 by $n^2$. + +\section{Properties of A [10 points]} +\subsection{ If $A$ is not symmetric, how would this affect $\tilde{A}$?} +If $A$ were not symmetric, then $\tilde{A}$ would not be positive definite since +by definition $\tilde{A} = A A^T$, thus not satifying the assumptions taken when +solving the system. + +\subsection{ Explain why solving $Ax = b$ for $x$ is equivalent to +minimizing $x^T A x - b^T x$ over $x$.} + +First, we can say that: + +\[f(x) = \frac12 x^T A x - b^T x = \frac12\langle Ax,x \rangle - +\langle b,x \rangle\] + +Then by taking the derivative of $f(x)$ w.r.t. $x$ we have (assuming $A$ is +\textit{spd}, which it is): + +\[f'(x) = \frac12 A^T x + \frac12Ax - b = Ax - b\] + +which for $f'(x) = 0$ will be equivalent to solving $Ax = b$. By taking the +second derivative we have: + +\[f''(x) = A > 0\] + +since $A$ is positive definite. Therefore, we can say that the absolute minima +of $f(x)$ is the solution for $Ax = b$. + + +\section{Conjugate Gradient [40 points]} +\subsection{ Write a function for the conjugate gradient solver +\texttt{[x,rvec]=myCG(A,b,x0,max\_itr,tol)}, where \texttt{x} +and \texttt{rvec} are, respectively, the solution value and a +vector containing the residual at every iteration.} + +The implementation can be found in file \texttt{myCG.m} in the source directory. +The test code for the function \texttt{myCG} can be found in the \texttt{test.m} file. + +\subsection{ In order to validate your implementation, solve the system +defined by \texttt{A\_test.mat} and \texttt{b\_test.mat}. Plot +the convergence (residual vs iteration).} + +The plot of the squared residual 2-norms over all iterations can be found in Figure +\ref{fig:plot1}. + +\begin{figure}[h] + \input{test_semilogy} + \caption{Semilog plot of the plot of the squared residual 2-norms over all iterations}\label{fig:plot1} +\end{figure} + +\subsection{ Plot the eigenvalues of \texttt{A\_test.mat} and comment on the +condition number and convergence rate.} + +The eigenvalues of A can be found in figure +\ref{fig:plot2}. The condition number for matrix $A$ according to \texttt{rcond(...)} is $\approx 3.2720 \cdot 10^7$, which is very low without sitting in the denormalized range (i.e. $< \text{eps}$) and thus very good for the Conjugate Gradient algorithm. +This well conditioning is also reflected in the eigenvalue plot, which shows a not so +drastic increase of the first eigenvalues ordered in increasing order. + +\begin{figure}[h] + \input{A_eig} + \caption{Semilog plot of the eigenvalues of A}\label{fig:plot2} +\end{figure} + +\section{Debluring problem [40 points]} +\subsection{ Solve the debluring problem for the blurred image matrix +\texttt{B.mat} and transformation matrix \texttt{A.mat} using +your routine \texttt{myCG} and Matlab's preconditioned conjugate +gradient \texttt{pcg}. As a preconditioner, use \texttt{ichol} +to get the incomplete Cholesky factors and set routine type to +\texttt{nofill} with $\alpha=0.01$ for the diagonal shift (see +Matlab documentation). Solve the system with both solvers using +$max\_iter=200$ $tol= 10^{-6}$. Plot the convergence (residual +vs iteration) of each solver and display the original and final +deblurred image.} +\subsection{ When would \texttt{pcg} be worth the added computational cost? +What about if you are debluring lots of images with the same +blur operator?} + + +\end{document} diff --git a/mp5/Project_5_Maggioni_Claudio/blur_data/A.mat b/mp5/Project_5_Maggioni_Claudio/blur_data/A.mat new file mode 100644 index 0000000000000000000000000000000000000000..283268a8127ef13a14896089e296c0532fdd373f GIT binary patch literal 1961992 zcma%?dt6f4-v6J|J*{lA)I_4uO4H0tMU#S=>0)K-Sa}(#Q1Q|P>QEzEk(@GXG=b67 z^8Pe7-HZSo(di;;#x_zso>DhysckQ3l2Y(wo7}Q~8>e~B^ZfVoFJGJOTAQ`j`}6sH zzhJ=E2lsvb&1$cWn^p&Web6=Tdv@%G)!(wCevC^xp5TjL+rEeFxncG8T8p zO2n@vC9V!k;;!E8v)bLu*Msc4iM)FACU^4c|KFeg`s?R!djc%*-~T>F{IAp!-V49u zJ>M?f+0p&Ve&^b(tb@XXH*S?wpLluByFcXQzt*Pum-(pnbTDGu?@pb)pYhG#QsXaF zU9Tm*J+iNR|DA7hULPblV-F;ypzWV0xgWQFc>m$_D`J8E;Ps9BuiUPDk^FY+HmvMZ 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