Interpolation Exercise 2



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



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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.



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# YOUR CODE HERE
x_1 = np.arange(-5, 6, 1)
y_1 = 5 * np.ones(11)
f_1 = np.zeros(11)

x_2 = np.arange(-5, 6, 1)
y_2 = -5 * np.ones(11)
f_2 = np.zeros(11)

x_3 = 5 * np.ones(9)
y_3 = np.arange(-4, 5, 1)
f_3 = np.zeros(9)

x_4 = -5 * np.ones(9)
y_4 = np.arange(-4, 5, 1)
f_4 = np.zeros(9)

x_5 = np.array([0])
y_5 = np.array([0])
f_5 = np.array([1])

x = np.hstack((x_1, x_2, x_3, x_4, x_5))
y = np.hstack((y_1, y_2, y_3, y_4, x_5))
f = np.hstack((f_1, f_2, f_3, f_4, f_5))

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);
plt.grid(True)



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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.



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# YOUR CODE HERE
# raise NotImplementedError()
xnew = np.linspace(-5.0, 6.0, 100)
ynew = np.linspace(-5.0, 6.0, 100)
Xnew, Ynew = np.meshgrid(xnew, ynew)
Fnew = griddata((x,y), f, (Xnew, Ynew), method='cubic', fill_value=0.0)



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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.



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# YOUR CODE HERE
plt.contourf(Fnew, cmap='jet')
plt.title('2D Interpolation')









    Out[17]:





<matplotlib.text.Text at 0x7fec5e5c0828>



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assert True # leave this to grade the plot



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