exercise 28

exercises/Exercises 28.ipynb

+%% Cell type:markdown id: tags:
+
+# Exercise: Minimal Surfaces
+
+Implement the gradient descent function
+
+```haskell
+gradientDescent :: Double -> Mesh -> [Mesh]
+gradientDescent stepsize mesh = [...]
+```
+
+and the $H^1$ version
+
+```haskell
+gradientDescentH1 :: Double -> Mesh -> [Mesh]
+gradientDescentH1 stepsize mesh = [...]
+```
+
+The minsurf code constructs the Laplace operator as an `AssocMatrix` without taking care of the boundary, and provides a function `boundary :: Mesh -> S.Set Int`  that returns the indices of the boundary vertices of the mesh.
+
+The gradient descent functions should construct the Laplace operator with appropriate boundary conditions as `GMatrix`. For the $H^1$ version, the equation should be solved using `cg :: GMatrix -> Matrix Double -> [CGState]`, where the `Matrix Double` parameter is the matrix of vertices.
+
+**Note**: Due to restriction in the way HMatrix handles sparse matrices -- the impossibility to specify the size of the matrix explicitly -- it may turn out to be difficult to use `mkSparse`. For the exercise, it is ok to convert the `AssocMatrix` to a `GMatrix` using `mkDense . toDense`.