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Matrix numerical analysis: Need explanations

  1. Apr 30, 2009 #1
    1. The problem statement, all variables and given/known data

    Given that [tex]\Omega \subset \mathbb{R}^2[/tex] and [tex](x,y) \in [0,1] \times [0,1][/tex].

    The goal is to find [tex]u:(x,y) \in \Omega \longmapsto u(x,y) \in \mathbb{R}[/tex].

    2. Relevant equations

    Details:

    [tex]
    (P1) \left\{ \begin{array}{rlc}
    -\frac{\partial}{\partial x^2} u(x,y) - \frac{\partial}{\partial y^2} u(x,y) &=\quad f(x,y), &

    \forall (x,y) \in \Omega \\
    u(x=0,y)&=\quad 0 \\
    u(x,y=0)&=\quad 0 \\
    u(x=1,y)&=\quad y^2 \\
    u(x,y=1)&=\quad x^2 \end{array}\right
    [/tex]

    Where [tex]f[/tex] is given as:

    [tex]
    f(x,y)=-2\sin(4\pi x)-2x^2-2y^2+(4\pi)^2y(y-1)\sin(4\pi x)
    [/tex]

    We are, supposedly, using finite difference method to approach numerically the solution of [tex]

    (P1)[/tex].

    As advised by the author, one begins by discretizing [tex]\Omega[/tex] with [tex]N=n^2[/tex] points attributed regularly and numbered accordingly, like in the example below:

    http://www.chokleong.com/ESIEE/images/exemple_de_discretisation.png [Broken]

    Notice that the point [tex]1[/tex] has [tex](\frac{1}{n+1},\frac{1}{n+1})[/tex] as its coordinate, etc...

    He, then, searches one equation (can be solved numerically) satisfied by approximated values of [tex]u[/tex] at points [tex]1,2,3,[/tex] etc... He gathers these values in a vector with the size of [tex]N=n^2[/tex]:

    [tex]
    U=\begin{pmatrix}
    u_1\\
    u_2\\
    \vdots\\
    u_N\end{pmatrix} \in \mathbb{R}^N
    [/tex]

    He introduces the [tex]N \times N[/tex] matrix below:

    [tex]
    A_N=
    \begin{pmatrix}
    A & B & 0 & \cdots & 0 \\
    B^T & \ddots & \ddots & \ddots & \vdots\\
    0 & \ddots & A & \ddots & 0 \\
    \vdots & \ddots & \ddots & \ddots & B\\
    0 & \cdots & 0 & B^T & A \end {pmatrix} \in \mathbb{R}^{N\times N}
    [/tex]

    Where [tex]A[/tex] and [tex]B[/tex] are both [tex]n \times n[/tex] matrices given as:

    [tex]
    A=
    \begin{pmatrix}
    4 & -1 & 0 & \cdots & 0\\
    -1& \ddots & \ddots & \ddots & \vdots\\
    0 & \ddots & 4 & \ddots & 0\\
    \vdots & \ddots & \ddots & \ddots & -1\\
    0 & \cdots & 0 & -1 & 4\end{pmatrix}
    \qquad,\quad
    B=
    \begin{pmatrix}
    -1 & 0 & \cdots & \cdots & 0\\
    0 & \ddots & \ddots & & \vdots \\
    \vdots & \ddots & -1 & \ddots & \vdots \\
    0 & \ddots & \ddots & \ddots & 0\\
    -1 & 0 & \cdots & 0 & -1\end{pmatrix}
    [/tex]

    Questions:
    1) Explain why [tex](P1)[/tex] can be approached by [tex](P2)[/tex] below:

    [tex](P2)[/tex] Find [tex]U \in \mathbb{R}^N[/tex] such that [tex]\frac{1}{(n+1)^2}A_N U = b[/tex] where the form of the vector [tex]b \in \mathbb{R}^N[/tex] will be given by the student.
    Hint: Begin with [tex]n[/tex] small [tex]( n = 2, 3,\quad etc...)[/tex].

    2) By introducing [tex]X \in \mathbb{R}^N[/tex], a [tex]N[/tex] vector, explain why [tex](P2)[/tex] is equivalent to the least squares problem below:

    [tex](P3)[/tex] Find [tex]X \in \mathbb{R}^N[/tex] minimum of [tex]J(X)=\frac{1}{2}X^TA_N X-(n+1)^2X^Tb[/tex].

    That's it.

    3. The attempt at a solution

    There are a total of 5 questions to answer, but I am having problems with the above 2 questions which I don't have a clue of.

    If you know somethings, please help me. Otherwise, I couldn't sleep for a few days.

    Thanks.

    (Update: My biggest problem is that I don't know how to write the vectors [tex]U[/tex] and [tex]b[/tex] explicitly which prevent me from solving the problem. If someones know this matter better than me, I would be glad that you can shed some light for me on this matter.

    My biggest concern is what represent the vector [tex]b \in \mathbb{R}^N[/tex] actually? What could it possibly be? Any idea?)
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
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