#!/usr/bin/env python
# coding: utf-8

# # Interpolation with Radial Basis Functions
# 
# Copyright (C) 2020 Andreas Kloeckner
# 
# <details>
# <summary>MIT License</summary>
# Permission is hereby granted, free of charge, to any person obtaining a copy
# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights
# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
# copies of the Software, and to permit persons to whom the Software is
# furnished to do so, subject to the following conditions:
# 
# The above copyright notice and this permission notice shall be included in
# all copies or substantial portions of the Software.
# 
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
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# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
# THE SOFTWARE.
# </details>

# 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


# 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))


# In[6]:


def f(x): return x**3 - 3*x

pt.plot(plot_x, f(plot_x))


# 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]:


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")


# * Play around with the radius of the RBFs
# * Play with node placement