# Interpolation with Radial Basis Functions

In [10]:
import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt

In [15]:
plot_x = np.linspace(-3, 3, 200)

In [147]:
np.random.seed(20)
centers = np.random.randn(10)*0.05 + np.linspace(-1.5, 1.5, 10)
centers = np.sort(centers)
centers

Out[147]:
array([-1.45580534, -1.15687342, -0.81545651, -0.6171631 , -0.2209083 ,
0.19465148,  0.54697347,  0.78440928,  1.19182151,  1.52032072])

In [176]:
radius = 0.3

In [177]:
def radial_basis_function(x, i):


Out[177]:
[<matplotlib.lines.Line2D at 0x7fa4884d7898>]

In [178]:
def f(x): return x**3 - 3*x

pt.plot(plot_x, f(plot_x))

Out[178]:
[<matplotlib.lines.Line2D at 0x7fa4884aa8d0>]


Let's build a Vandermonde matrix at the centers:

In [179]:
nodes = centers

V = np.array([
for i in range(len(centers))
]).T


Find the coefficients:

In [180]:
coeffs = la.solve(V, f(nodes))


Find the interpolant:

In [185]:
interpolant = 0
for i in range(len(centers)):
interpolant += coeffs[i] * radial_basis_function(plot_x, i)

pt.figure(figsize=(8,8))
pt.ylim([-5,5])
pt.plot(plot_x, interpolant, label="Interpolant")
pt.plot(plot_x, f(plot_x), label="$f$")
pt.plot(centers, f(centers), "o")
pt.legend(loc="best")

Out[185]:
<matplotlib.legend.Legend at 0x7fa488263400>

• Play around with the radius of the RBFs
• Play with node placement