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 [4]:

    
# left, top, right, bottom
x = np.hstack((np.array([-5]*10), np.linspace(-5, 5, 10), np.array([5]*10), np.linspace(5, -5, 10), 0))
y = np.hstack((np.linspace(-5, 5, 10), np.array([5]*10), np.linspace(5, -5, 10), np.array([-5]*10), 0))
f = np.hstack(([0]*40, 1))
#raise NotImplementedError()

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



In [5]:

    
plt.scatter(x, y);



In [6]:

    
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 [9]:

    
xnew = np.linspace(-5.0, 5.0, 100)
ynew = np.linspace(-5.0, 5.0, 100)
Xnew, Ynew = np.meshgrid(xnew, ynew)

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



In [10]:

    
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 [11]:

    
plt.contour(Xnew, Ynew, Fnew)
#raise NotImplementedError()









    Out[11]:





<matplotlib.contour.QuadContourSet at 0x7f13847f0c18>



In [ ]:

    
assert True # leave this to grade the plot