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 [2]:

    
from scipy.interpolate import griddata

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 [3]:

    
x = np.empty((1,),dtype=int)
x[0] = 0
for i in range(-4,5):
    x = np.hstack((x,np.array((i,i))))

x = np.hstack((x,np.array([-5]*11)))
x = np.hstack((x,np.array([5]*11)))



In [4]:

    
y = np.empty((1,),dtype=int)
y[0]=0
y = np.hstack((y,np.array((5,-5)*9)))

for i in range(-5,6):
    y = np.hstack((y,np.array((i))))
for i in range(-5,6):
    y = np.hstack((y,np.array((i))))



In [5]:

    
f=np.zeros_like(y)
f[0]=1

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



In [6]:

    
plt.scatter(x, y);



In [7]:

    
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 [8]:

    
xnew = np.linspace(-5, 5, 100)
ynew = np.linspace(-5, 5, 100)
Xnew, Ynew = np.meshgrid(xnew, ynew)

Fnew = griddata((x,y), f, (Xnew, Ynew), method='cubic', fill_value=0.0)



In [9]:

    
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 [10]:

    
plt.contourf(Xnew,Ynew,Fnew,cmap='jet');
plt.colorbar(shrink=.8);
plt.xlabel('x value');
plt.ylabel('y value');
plt.title('Contour Plot of Interpolated Function');



In [11]:

    
assert True # leave this to grade the plot