Python package containing the tools necessary for radial basis function (RBF) applications. Applications include interpolating/smoothing scattered data and solving PDEs over irregular domains. The complete documentation for this package can be found here.
- Efficient functions to evaluate RBFs and their analytically derived derivatives
- Regularized RBF interpolants (including smoothing splines) for noisy, scattered, data
- An algorithm for generating Radial Basis Function Finite Difference (RBF-FD) weights
- RBF-FD Filtering for denoising BIG, scattered data
- Node and stencil generation algorithms for solving PDEs over irregular domains
- Halton sequence generator
- Computational geometry functions for 1, 2, and 3 spatial dimensions
'''
In this example we generate synthetic scattered data with added noise
and then fit it with a smoothed interpolant. See rbf.filter for
smoothing large data sets.
'''
import numpy as np
from rbf.interpolate import RBFInterpolant
import matplotlib.pyplot as plt
np.random.seed(1)
# create noisy data
x_obs = np.random.random((100,2)) # observation points
u_obs = np.sin(2*np.pi*x_obs[:,0])*np.cos(2*np.pi*x_obs[:,1])
u_obs += np.random.normal(0.0,0.2,100)
# create smoothed interpolant
I = RBFInterpolant(x_obs,u_obs,penalty=0.001)
# create interpolation points
x_itp = np.random.random((10000,2))
u_itp = I(x_itp)
plt.tripcolor(x_itp[:,0],x_itp[:,1],u_itp,vmin=-1.1,vmax=1.1,cmap='viridis')
plt.scatter(x_obs[:,0],x_obs[:,1],s=100,c=u_obs,vmin=-1.1,vmax=1.1,cmap='viridis')
plt.xlim((0.05,0.95))
plt.ylim((0.05,0.95))
plt.colorbar()
plt.tight_layout()
plt.savefig('../figures/interpolate.a.png')
plt.show()The above code will produce this plot, which shows the observations as scatter points and the smoothed interpolant as the color field.
'''
In this example we solve the Poisson equation with a constant forcing
term using the spectral RBF method.
'''
import numpy as np
from rbf.basis import phs3
from rbf.domain import circle
from rbf.nodes import menodes
import matplotlib.pyplot as plt
# define the problem domain
vert = np.array([[0.762,0.057],[0.492,0.247],[0.225,0.06 ],[0.206,0.056],
[0.204,0.075],[0.292,0.398],[0.043,0.609],[0.036,0.624],
[0.052,0.629],[0.373,0.63 ],[0.479,0.953],[0.49 ,0.966],
[0.503,0.952],[0.611,0.629],[0.934,0.628],[0.95 ,0.622],
[0.941,0.607],[0.692,0.397],[0.781,0.072],[0.779,0.055]])
smp = np.array([[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],
[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],
[16,17],[17,18],[18,19],[19,0]])
N = 500 # total number of nodes
nodes,smpid = menodes(N,vert,smp) # generate nodes
boundary, = (smpid>=0).nonzero() # identify boundary nodes
interior, = (smpid==-1).nonzero() # identify interior nodes
# create left-hand-side matrix and right-hand-side vector
A = np.empty((N,N))
A[interior] = phs3(nodes[interior],nodes,diff=[2,0])
A[interior] += phs3(nodes[interior],nodes,diff=[0,2])
A[boundary,:] = phs3(nodes[boundary],nodes)
d = np.empty(N)
d[interior] = -100.0
d[boundary] = 0.0
# Solve the PDE
coeff = np.linalg.solve(A,d) # solve for the RBF coefficients
itp = menodes(10000,vert,smp)[0] # interpolation points
soln = phs3(itp,nodes).dot(coeff) # evaluate at the interp points
fig,ax = plt.subplots()
p = ax.scatter(itp[:,0],itp[:,1],s=20,c=soln,edgecolor='none',cmap='viridis')
ax.set_aspect('equal')
ax.plot(nodes[:,0],nodes[:,1],'ko',markersize=4)
ax.set_xlim((0.025,0.975))
ax.set_ylim((0.03,0.98))
plt.colorbar(p,ax=ax)
plt.tight_layout()
plt.savefig('../figures/basis.a.png')
plt.show()The above code will produce this plot, which shows the collocation nodes as black points and the interpolated solution as the color field.
