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 [29]:

    
# YOUR CODE HERE
#raise NotImplementedError()
#I worked with James Amarel
x=np.empty((1,))
x[0]=0
y=np.empty((1,))
y[0]=0

#hstack acts like appending but for arrays (:
for i in range(-4,5):
    x=np.hstack((x,(i,i)))
x=np.hstack((x,np.array([-5]*11)))
x=np.hstack((x,np.array([5]*11)))

y=np.hstack((y,np.array([-5,5]*9)))

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

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

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



In [23]:

    
plt.scatter(x, y);



In [24]:

    
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 [25]:

    
# YOUR CODE HERE
#raise NotImplementedError()
#creates right sized arrays, turns them to meshgrid
xnew = np.linspace(-5,5,100)
ynew=np.linspace(-5,5,100)
Xnew, Ynew = np.meshgrid(xnew,ynew)
#uses griddata to interpolate the 2D data in meshgrid form
#I helped Orion with this part
Fnew = griddata((x,y), f, (Xnew,Ynew), method = 'cubic', fill_value=0.0)



In [26]:

    
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 [27]:

    
# YOUR CODE HERE
#raise NotImplementedError()
plt.contourf(Xnew, Ynew, Fnew, cmap="gist_ncar")
plt.colorbar(shrink=0.7)
plt.box(False)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Scalar Field, $f(x,y)$');



In [28]:

    
assert True # leave this to grade the plot