added exercises 25

exercises/Exercises 25.ipynb

+%% Cell type:markdown id: tags:
+
+# Exercises 25
+
+##problem 1
+
+Put together the stationary heat equation (poisson equation) with the explicit Euler method to be able to solve the *time dependent heat equation*.
+
+Given a rectangular domain $\Omega \subset \mathbb{R}^2 $ and a time dependent function
+$u(x,t) , x \in \Omega , t \in \mathbb{R}^{+}$. Then the time dependent heat equation is
+$$\frac{\partial u}{\partial t} - \alpha \triangle u = 0  \text{ in } \Omega $$
+where $\alpha \in \mathbb{R}$ is a constant . Given dirichlet boundary conditions
+$$u = R , \text{ on } \partial \Omega$$
+with a function $R: \partial \Omega \mapsto C(\partial \Omega)$. At time $t=0$ let
+$$u(x,0) = f(x), \forall x \in \Omega.$$
+with a arbitrary, but fixed initial function $f: \mathbb{R} \mapsto \mathbb{R}$.
+
+*Remark:* Try to write the code with a sparse matrix (but with small number of points per axis up to 50 a dense matrix is also fine)
+
+%% Cell type:markdown id: tags:
+
+## problem 2
+
+Lets look into a bit image transformation. Take the greyscale image `"images/Grayscale_8bits_palette.bmp"` and make a *edge detection* on it. *edge detection* works through applying a convolution of the (here grayscale values in every pixel) with the convolution-matrix
+
+ $$\left(\begin{array}
+-- 1 & - 1 &- 1 \\
+ - 1  &8 &- 1 \\
+     - 1 &- 1 &- 1 \end{array}\right)$$
+
+You can use repa stencils for this. Reading the image can also be done with
+
+```haskell
+import Data.Array.Repa.IO.BMP (readImageFromBMP)
+```
+
+You can either write the image again (and the show it with an imageviewer) or you can use
+the simple plot-interface
+
+```haskell
+import Graphics.Rendering.Plot
+import Data.Packed.Repa (repaToMatrix)
+```
+
+%% Cell type:markdown id: tags:
+
+## problem 3
+
+Remember the *Jacobi* method to solve an system of linear equations. Rewrite the Jacobi method that it works with *stencils* (so the forward operator is given as stencil).
+
+For this you also the need the componentwise version of the Jacobi method, which is given by
+
+$$x_i^{( k + 1 )} = \frac{1}{a_{ii}} ( b_i - \sum_{j = i} a_{ij} x_j^{( k )} ) , i = 1 , 2 , … , n $$
+
+where $a_{ij}$ are the component of the matrix $A$, $b$ is the right hand side and the $x$ starts from an initial guess.
+
+You can work along the line of the example of the cg-method from the lecture.
+
+For a testcase use the appropriate stencil for the 1d poisson equation.

solutions/Exercises 24_sol.ipynb

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