PN vs QN

Copyright (C) 2026 Andreas Kloeckner

MIT License

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Based on Cubature, Approximation, and Isotropy in the Hypercube. (Trefethen)

import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as plt

import modepy as mp
import modepy.tools as mp_tools

shape = mp.Hypercube(2)
space = mp.QN(2, 60)

nodes = mp.edge_clustered_nodes_for_space(space, shape)
basis = mp.basis_for_space(space, shape)

plt.plot(nodes[0], nodes[1], "o")

[<matplotlib.lines.Line2D at 0x7f2816c50c20>]

vdm = mp.vandermonde(basis.functions, nodes)

r = la.norm(nodes, 2, axis=0)
gauss_coeffs = la.solve(vdm, np.exp(-(r/0.2)**2))

coeffs_reshaped = mp_tools.reshape_array_for_tensor_product_space(space, gauss_coeffs, axis=0)

plt.imshow(np.log10(1e-15+np.abs(coeffs_reshaped)))
plt.colorbar()

<matplotlib.colorbar.Colorbar at 0x7f281657d160>

plt.imshow(np.log10(1e-15+np.abs(coeffs_reshaped[::2, ::2])))
plt.colorbar()

<matplotlib.colorbar.Colorbar at 0x7f28166252b0>