Interpolation Exercise 2



In [1]:

    
%matplotlib inline
import matplotlib.pyplot as plt
import seaborn as sns
import numpy as np
sns.set_style('white')



In [13]:

    
from scipy.interpolate import griddata
from scipy.interpolate import interp2d

Sparse 2d interpolation

In this example the values of a scalar field $f(x,y)$ are known at a very limited set of points in a square domain:

The square domain covers the region $x\in[-5,5]$ and $y\in[-5,5]$.
The values of $f(x,y)$ are zero on the boundary of the square at integer spaced points.
The value of $f$ is known at a single interior point: $f(0,0)=1.0$.
The function $f$ is not known at any other points.

Create arrays x, y, f:

x should be a 1d array of the x coordinates on the boundary and the 1 interior point.
y should be a 1d array of the y coordinates on the boundary and the 1 interior point.
f should be a 1d array of the values of f at the corresponding x and y coordinates.

You might find that np.hstack is helpful.



In [9]:

    
x=np.array([5,5,5,5,5,5,4,3,2,1,0,-1,-2,-3,-4,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-4,-3,-2,-1,0,1,2,3,4,5,5,5,5,5,0])
y=np.array([0,1,2,3,4,5,5,5,5,5,5,5,5,5,5,5,4,3,2,1,0,-1,-2,-3,-4,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-4,-3,-2,-1,0])
f=np.zeros(len(x))
f[len(x)-1]=1.0

The following plot should show the points on the boundary and the single point in the interior:



In [10]:

    
plt.scatter(x, y);



In [11]:

    
assert x.shape==(41,)
assert y.shape==(41,)
assert f.shape==(41,)
assert np.count_nonzero(f)==1

Use meshgrid and griddata to interpolate the function $f(x,y)$ on the entire square domain:

xnew and ynew should be 1d arrays with 100 points between $[-5,5]$.
Xnew and Ynew should be 2d versions of xnew and ynew created by meshgrid.
Fnew should be a 2d array with the interpolated values of $f(x,y)$ at the points (Xnew,Ynew).
Use cubic spline interpolation.



In [15]:

    
xnew=np.linspace(-5,5,100)
ynew=np.linspace(-5,5,100)
Xnew,Ynew=np.meshgrid(xnew,ynew)
fnew=interp2d(x,y,f,kind='cubic')
Fnew=fnew(xnew,ynew)



In [16]:

    
assert xnew.shape==(100,)
assert ynew.shape==(100,)
assert Xnew.shape==(100,100)
assert Ynew.shape==(100,100)
assert Fnew.shape==(100,100)

Plot the values of the interpolated scalar field using a contour plot. Customize your plot to make it effective and beautiful.



In [20]:

    
plt.contourf(xnew,ynew,Fnew)
plt.set_cmap('RdBu')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Cubic Interpolation')
plt.tick_params(direction='out')
plt.colorbar()









    Out[20]:





<matplotlib.colorbar.Colorbar at 0x7f0a80570240>



In [ ]:

    
assert True # leave this to grade the plot