Interpolation with Radial Basis Functions

Copyright (C) 2020 Andreas Kloeckner

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In [1]:
import numpy as np

import numpy.linalg as la

import matplotlib.pyplot as pt
In [2]:
plot_x = np.linspace(-3, 3, 200)
In [3]:
np.random.seed(20)

centers = np.random.randn(10)*0.05 + np.linspace(-1.5, 1.5, 10)

centers = np.sort(centers)

centers
Out[3]:
array([-1.45580534, -1.15687342, -0.81545651, -0.6171631 , -0.2209083 ,
        0.19465148,  0.54697347,  0.78440928,  1.19182151,  1.52032072])
In [4]:
radius = 0.3
In [5]:
def radial_basis_function(x, i):

    return np.exp(-(x-centers[i])**2/radius**2)



pt.plot(plot_x, radial_basis_function(plot_x, 3))
Out[5]:
[<matplotlib.lines.Line2D at 0x7feae0471ef0>]
In [6]:
def f(x): return x**3 - 3*x



pt.plot(plot_x, f(plot_x))
Out[6]:
[<matplotlib.lines.Line2D at 0x7feae0412f28>]

Let's build a Vandermonde matrix at the centers:

In [7]:
nodes = centers



V = np.array([

    radial_basis_function(nodes, i)

    for i in range(len(centers))

    ]).T

Find the coefficients:

In [8]:
#clear

coeffs = la.solve(V, f(nodes))

Find the interpolant:

In [9]:
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[9]:
<matplotlib.legend.Legend at 0x7feae03853c8>

  • Play around with the radius of the RBFs

  • Play with node placement