Kernel in the Source and Target Variables ("Anterpolation and Interpolation")¶

Copyright (C) 2026 Andreas Kloeckner

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In [52]:
import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as plt

Some Chebyshev utilities up front:

In [53]:
def Tk(k, x):
    return np.cos(k*np.arccos(x))

nnodes = 20
k = nnodes-1
nodes = np.cos(np.arange(k+1)/k*np.pi)

xplot = np.linspace(-1, 1, 1000)
plt.plot(xplot, Tk(k, xplot))
plt.plot(nodes, Tk(k, nodes), "x")
Out[53]:
[<matplotlib.lines.Line2D at 0x7fd6564a63c0>]
No description has been provided for this image

Now construct an interpolation toolkit with vdm(targets) and interp_to_targets(targets, nodes), and test it:

In [73]:
def vdm(targets):
    return Tk(np.arange(k+1), targets[:, None])
        
def interp_to_targets(targets, nodes):
    return vdm(targets) @ la.inv(vdm(nodes))

print(la.norm(np.sin(5*xplot) - interp_to_targets(xplot, nodes)@np.sin(5*nodes), np.inf))

1.330113796882415e-11
In [74]:
N = 1000
sources = np.random.rand(N) * 2 - 1
charges = np.random.randn(N)

targets = np.random.rand(N)*2 - 1 - 4

def interaction_mat(targets, sources):
    return np.log(np.abs(targets[:, None] - sources))

Try out target-space interpolation as a matrix identity:

In [75]:
tgt_nodes = nodes - 4
la.norm(
    interp_to_targets(targets+4, tgt_nodes+4) @ interaction_mat(tgt_nodes, sources)
    -
    interaction_mat(targets, sources),
    2
)
Out[75]:
np.float64(6.222583118946094e-13)

Now try out source interpolation ("anterpolation") as a matrix identity:

In [76]:
la.norm(
    interaction_mat(targets, nodes) @ interp_to_targets(sources, nodes).T
    -
    interaction_mat(targets, sources), 
    2)
Out[76]:
np.float64(6.806409517866512e-13)
In [ ]: