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mp5: annotated 4.2 solution

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Claudio Maggioni (maggicl) 2020-11-29 23:11:28 +01:00
parent 9deb8a3238
commit 5bbd92b438
3 changed files with 9 additions and 44 deletions
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mp5/Project_5_Maggioni_Claudio.pdf
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mp5/Project_5_Maggioni_Claudio.tex
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@ -72,7 +72,6 @@ second derivative we have:
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}
@ -98,8 +97,10 @@ The plot of the squared residual 2-norms over all iterations can be found in Fig
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
\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]
@ -118,9 +119,14 @@ 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.}
Plots already rendered.
\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?}
\textit{pcg} better for many, myCG better for one thanks to cost of ichol.
\end{document}

41
mp5/Project_5_Maggioni_Claudio/code_template.asv
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@ -1,41 +0,0 @@
close all;
clear; clc;
%% Load Default Img Data
load('blur_data/B.mat');
B=double(B);
load('blur_data/A.mat');
A=double(A);
ciao = A;
% Show Image
figure
im_l=min(min(B));
im_u=max(max(B));
imshow(B,[im_l,im_u])
title('Blured Image')
% Vectorize the image (row by row)
b=B';
b=b(:);
%IL = ichol(A, struct('type', 'nofill', 'diagcomp', 0));
y = IL \ b;
x0 = IL' \ y;
[x, rvec] = myCG(A, b, diag(IL), 200, 1e-6);
semilogy(rvec);
[X2,flag,~,~,rvec2] = pcg(A, b, 1e-6, 200);
%% Validate Test values
load('test_data/A_test.mat');
load('test_data/x_test_exact.mat');
load('test_data/b_test.mat');
%res=||x^*-A^{-1}b||
res=x_test_exact-inv(A_test)*b_test;
norm(res);
%(Now do it with your CG and Matlab's PCG routine!!!)