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]:

    
xt = np.arange(-5,6,1)
yt = np.zeros_like(xt)
yt.fill(5)
ft = np.zeros_like(xt)

yl = np.arange(-5,5,1)
xl = np.zeros_like(yl)
xl.fill(-5)
fl = np.zeros_like(yl)

yr = yl
xr = np.zeros_like(xl)
xr.fill(5)
fr = fl

xb = np.arange(-4,5,1)
yb = np.zeros_like(xb)
yb.fill(-5)
fb = np.zeros_like(xb)


x = np.hstack((0,xt,xr,xb,xl))
y = np.hstack((0,yt,yr,yb,yl))
f = np.hstack((1,ft,fr,fb,fl))

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



In [4]:

    
plt.scatter(x, y);



In [5]:

    
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 [6]:

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

Fnew = griddata( (x,y), f, (Xnew,Ynew), method="cubic")



In [7]:

    
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 [8]:

    
plt.contour(Xnew, Ynew, Fnew)
plt.xlabel("X")
plt.ylabel("Y")
plt.title("f (x,y)", fontsize= 14)
ax = plt.gca()
ax.spines['right'].set_color('none')
ax.spines['top'].set_color('none')
plt.show()



In [ ]:

    
assert True # leave this to grade the plot