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z@@VZ2#6E{KHwISh*xX)bdIdKQE&Iec?(H2S?Uoxrd1Z1E(z_5;4)9%0wi}dz$2oQB z`%&(w>Qftd?OkG=l#FNo^8W2ySg zVAIc_^Uq+}GNYL=lU)&J$OS8Ki7wZ#i6VQaa*7X;DHl8Lk^yrxhd z(v}m{zG)tWZ}gRYXVk@27uzMQ{}w;FTqeu#cMY+1Q(JB1ePxtfL82Vrb47DuAgZ1H gdnd{hq!&33eo7A<6d^J*Cl?nBA|<7mf;i%T0ZJWZ2><{9 diff --git a/Claudio_Maggioni_2/Claudio_Maggioni_2.tex b/Claudio_Maggioni_2/Claudio_Maggioni_2.tex index 3c0bba5..a35ee47 100755 --- a/Claudio_Maggioni_2/Claudio_Maggioni_2.tex +++ b/Claudio_Maggioni_2/Claudio_Maggioni_2.tex @@ -40,28 +40,55 @@ 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.} -Answer is a energy function does not exist. Since A is not symmetric -(even if it is pd), the minimizer used for the c.g. method -(i.e. $\frac12 x^T A x - b^T x$ won't work -since $x^T A x$ might be negative and thus the minimizer does not point to -the solution of $Ax = b$ necessairly +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.} -we already enforce x(1) = x(n) = 0, since b(1) = b(n) = 0 and thus -A(1, :) * x = b(0) = 0 and same for n can be solved only for x(1) = x(n) -= 0size(A, 1) +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. -The objective is therefore $\phi(x) = (1/2)x^T\overline{A}x - b^x$ with a and b -defined above, gradient is = $\overline{A}x - b$, hessian is $= \overline{A}$ + 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}.} See page 112 (133 for pdf) for the algorithm implementation -The solution of this task can be found in Section 1.3 of the script \texttt{main.m}. + The 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.} diff --git a/Claudio_Maggioni_2/main.m b/Claudio_Maggioni_2/main.m index 2bff263..a665818 100644 --- a/Claudio_Maggioni_2/main.m +++ b/Claudio_Maggioni_2/main.m @@ -4,10 +4,19 @@ % Note: exercises are not in the right order due to matlab constraints of % functions inside of scripts. +A = [1 0 0 0; + -1 2 -1 0; + 0 -1 2 1; + 0 0 0 1]; + +b = [0; 1/9; 1/9; 0]; + +A \ b + clear clc close all -plots = 1; +plots = 0; %% 1.4 - Solution for 1D Poisson for N=1000 using CG