# 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
from scipy.interpolate import interp1d
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 [94]:

xb=np.array([-5,-4,-3,-2,-1,0,1,2,3,4,5])
yb=np.array([-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5])
yt=np.array([5]*11)
yc=np.array(0)
x=np.hstack((xb,xb,yb[1:10],yt[1:10],yc))
y=np.hstack((yb,yt,xb[1:10],xb[1:10],yc))
f1=np.array([0]*40)
f2=[1]
f=np.hstack((f1,f2))



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 [113]:

# F=np.meshgrid(f,y)
xnew=np.linspace(-5,5,100)
ynew=xnew
Xnew,Ynew=np.meshgrid(xnew,ynew)
Fnew=griddata((x,y),f,(Xnew,Ynew),method='cubic') # worked with Jessi Pilgram




In [114]:

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 [133]:

plt.figure(figsize=(12,8))
plt.contourf(Xnew,Ynew,Fnew)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Sparse 2d Interpolation')
plt.colorbar(shrink=.9)




Out[133]:

<matplotlib.colorbar.Colorbar at 0x7fd26ecd4518>




In [134]:

plt.figure(figsize=(12,8))
plt.pcolor(Xnew,Ynew,Fnew)
plt.xlabel('x')
plt.ylabel('y')
plt.title('2d Sparse Interpolation')
plt.colorbar(shrink=.9)




Out[134]:

<matplotlib.colorbar.Colorbar at 0x7fd26e958898>




In [116]:

assert True # leave this to grade the plot