Notebook for Exercise sheet 2

Important: run the code cell below once when you just opened this notebook! It will make sure that plotting is enabled.

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# some setup
%matplotlib inline
import numpy as np # makes numpy routines and data types available as np.[name ouf routine or data type]
import matplotlib.pyplot as plt # makes plotting command available as plot.[name of command]

H 2 (Dirichlet and Fejér kernel)

Plot the Dirichlet and Fejér kernel for various values of n. Also generate a plot where you plot both kernels in one diagram.

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# Code for Dirichlet plots
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#code for Fejer plots

H3 (Interpolation with kernels)

This exercise is about interpolation with the Gauss kernel $k(x,y)=\exp(-\gamma\|x-y\|_2^2)$.

a) You are given a set of sampling points $X = \{x_1,\dots,x_n\}$ with $x_i \in \mathbb R^2$ and function values at these points $f|_X = \{f(x_1),\dots,f(x_n)\}$. Write some code which solves $$ A_{X,X}\alpha = f|_X, $$ where $A_{X,X} = (k(x_i,x_j))_{i,j=1,\dots,n}$ is the kernel matrix. You can use any of the linear algebra routines that come along with numpy. Also write a subroutine which evaluates the interpolant $s_{X,f|_X} = \sum_{j=1}^n \alpha_j k(x_j,\cdot)$ at a given set of points $\{y_1,\dots,y_m\}$.

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# your code goes here

b) For each of the following type of sampling points, write a subroutine which generates, for given $m \in \mathbb N$, $m^2$ sampling points:

  • uniform grid points in $[0,1]^2$
  • tensor-product Chebyshev points on $[0,1]^2$ given by $$ (x_i,x_j) = \left( \cos\left(\frac{2i-1}{2m} \pi\right), \cos\left(\frac{2j-1}{2m} \pi\right) \right), \quad i,j=1,\dots,m $$
  • Halton points

For the Halton sequence you can use the ghalton package. It is not in the conda package index, but in the pip package index.

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# Uniform grid points
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# tensor-product Chebyshev points
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# Halton points

c) Use your code to interpolate the function $$ f(x,y) = 3(1-x)^2 \exp(-x^2 - (y+1)^2)\

  • 10(x/5 - x^3 - y^5) \exp(-x^2-y^2)\
  • \exp(-(x+1)^2 - y^2)/3 $$ on $[0,1]$ at $m^2$ uniform grid points, tensor-product Chebyshev points and Halton points for $m=10,20,30,40$ and Gauss parameter $\gamma = 1,2$. Generate plots which show the pointwise error $$ |f(y) - s_{X,f|X}(y)| $$ for each of type of sampling points. To this end, evaluate $f$ and $s{X,f|_X}$ on a uniform grid with mesh size $1/1000$.
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# your code goes here

d) For each the sampling points used in c) plot the power function $$ P_X^2(x) = k(x,x) - k(x,\cdot)(x) $$ Again, use a uniform grid with mesh size $1/1000$ to generate the plots. Compare the plots with the plots obtained in c).

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# your code goes here