Interpolation Exercise 2



In [73]:

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



In [74]:

    
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 [75]:

    
x = np.array([-5,-5,-5,-5,-5,-5,-5,-5,-5,-5])
x = np.append(x,range(-5,6))
x = np.append(x,[5,5,5,5,5,5,5,5,5,5])
x = np.append(x,range(4,-5,-1))
x = np.append(x,0)
y = np.array(range(-5,6))
y = np.append(y,[5,5,5,5,5,5,5,5,5,5])
y = np.append(y,range(4,-6,-1))
y = np.append(y,[-5,-5,-5,-5,-5,-5,-5,-5,-5])
y = np.append(y,0)

f = np.zeros(40)
f = np.append(f,1)

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



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plt.scatter(x, y);



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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 [78]:

    
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 [79]:

    
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 [80]:

    
plt.figure(figsize=(10,7))
plt.contourf(xnew,ynew,Fnew,cmap='gist_rainbow');
plt.colorbar();
plt.title('contour plot of scaler field f(x,y)');



In [81]:

    
assert True # leave this to grade the plot