Inversion: Linear ProblemΒΆ
Here we go over the basics of creating a linear problem and inversion.
Out:
Efficiency Warning: Interpolation will be slow, use setup.py!
python setup.py build_ext --inplace
SimPEG.InvProblem will set Regularization.mref to m0.
SimPEG.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.
***Done using same Solver and solverOpts as the problem***
SimPEG.DataMisfit.l2_DataMisfit assigning default eps of 1e-5 * ||dobs||
model has any nan: 0
============================ Inexact Gauss Newton ============================
# beta phi_d phi_m f |proj(x-g)-x| LS Comment
-----------------------------------------------------------------------------
x0 has any nan: 0
0 2.07e+01 7.99e+04 0.00e+00 7.99e+04 3.59e+05 0
1 2.07e+01 3.36e+04 9.61e-01 3.36e+04 1.59e+05 0
2 2.07e+01 2.13e+04 2.83e+00 2.13e+04 8.15e+04 0
3 2.07e+01 7.48e+03 1.24e+01 7.74e+03 7.59e+04 0 Skip BFGS
4 2.07e+01 5.85e+03 1.23e+01 6.10e+03 7.69e+04 0
5 2.07e+01 5.41e+03 1.46e+01 5.72e+03 7.42e+04 0
6 2.07e+01 5.17e+03 1.49e+01 5.47e+03 8.20e+04 0
7 2.07e+01 5.00e+03 1.53e+01 5.31e+03 8.04e+04 0
8 2.07e+01 4.78e+03 1.53e+01 5.10e+03 7.04e+04 0
9 2.07e+01 4.29e+03 1.72e+01 4.65e+03 8.46e+04 0
10 2.07e+01 3.98e+03 1.76e+01 4.34e+03 7.22e+04 0
11 2.07e+01 3.67e+03 1.90e+01 4.06e+03 7.78e+04 0
12 2.07e+01 3.61e+03 1.90e+01 4.01e+03 8.30e+04 0
13 2.07e+01 3.57e+03 1.94e+01 3.97e+03 7.71e+04 0
14 2.07e+01 3.06e+02 2.14e+01 7.50e+02 4.88e+04 0 Skip BFGS
15 2.07e+01 2.00e+02 2.10e+01 6.34e+02 7.19e+03 0
16 2.07e+01 1.65e+02 2.11e+01 6.02e+02 6.74e+03 0
17 2.07e+01 1.58e+02 2.07e+01 5.86e+02 1.02e+04 0 Skip BFGS
18 2.07e+01 1.42e+02 2.10e+01 5.77e+02 5.33e+03 0
19 2.07e+01 1.19e+02 2.12e+01 5.58e+02 4.27e+03 0 Skip BFGS
20 2.07e+01 1.06e+02 2.15e+01 5.52e+02 4.05e+03 0
21 2.07e+01 1.03e+02 2.16e+01 5.51e+02 4.43e+03 0 Skip BFGS
22 2.07e+01 1.02e+02 2.16e+01 5.50e+02 4.10e+03 0
23 2.07e+01 1.00e+02 2.17e+01 5.50e+02 4.02e+03 0 Skip BFGS
24 2.07e+01 1.02e+02 2.16e+01 5.50e+02 4.04e+03 0
25 2.07e+01 9.82e+01 2.16e+01 5.46e+02 2.44e+03 0 Skip BFGS
26 2.07e+01 9.42e+01 2.17e+01 5.43e+02 3.97e+03 0
27 2.07e+01 6.40e+01 2.21e+01 5.22e+02 5.13e+03 0 Skip BFGS
28 2.07e+01 6.21e+01 2.21e+01 5.20e+02 4.99e+03 0
29 2.07e+01 6.18e+01 2.21e+01 5.20e+02 4.95e+03 0 Skip BFGS
30 2.07e+01 6.20e+01 2.21e+01 5.20e+02 4.98e+03 0
31 2.07e+01 4.82e+01 2.24e+01 5.12e+02 3.50e+03 0 Skip BFGS
32 2.07e+01 4.81e+01 2.22e+01 5.07e+02 3.08e+03 0
33 2.07e+01 4.06e+01 2.21e+01 4.98e+02 2.24e+03 0 Skip BFGS
34 2.07e+01 3.80e+01 2.21e+01 4.96e+02 1.45e+03 0
35 2.07e+01 3.26e+01 2.22e+01 4.92e+02 2.43e+03 0
36 2.07e+01 3.12e+01 2.22e+01 4.91e+02 1.83e+03 0 Skip BFGS
37 2.07e+01 3.17e+01 2.22e+01 4.91e+02 1.82e+03 0
38 2.07e+01 3.25e+01 2.21e+01 4.91e+02 1.92e+03 0
39 2.07e+01 3.23e+01 2.21e+01 4.91e+02 1.76e+03 0
40 2.07e+01 3.14e+01 2.22e+01 4.91e+02 1.64e+03 0
41 2.07e+01 3.14e+01 2.22e+01 4.91e+02 1.60e+03 0 Skip BFGS
42 2.07e+01 3.11e+01 2.22e+01 4.91e+02 1.79e+03 0
43 2.07e+01 3.18e+01 2.22e+01 4.91e+02 2.03e+03 0 Skip BFGS
44 2.07e+01 3.12e+01 2.22e+01 4.91e+02 2.20e+03 0
45 2.07e+01 3.12e+01 2.22e+01 4.91e+02 2.00e+03 0
46 2.07e+01 2.60e+01 2.24e+01 4.89e+02 2.45e+03 0 Skip BFGS
47 2.07e+01 2.81e+01 2.22e+01 4.89e+02 2.37e+03 0
48 2.07e+01 2.82e+01 2.22e+01 4.88e+02 1.57e+03 0 Skip BFGS
49 2.07e+01 2.74e+01 2.23e+01 4.88e+02 1.61e+03 0
50 2.07e+01 2.57e+01 2.22e+01 4.86e+02 5.35e+02 0 Skip BFGS
51 2.07e+01 2.55e+01 2.22e+01 4.86e+02 5.51e+02 0
52 2.07e+01 2.56e+01 2.22e+01 4.86e+02 4.65e+02 0
53 2.07e+01 2.55e+01 2.22e+01 4.86e+02 2.44e+02 0 Skip BFGS
54 2.07e+01 2.55e+01 2.22e+01 4.86e+02 2.59e+02 0
55 2.07e+01 2.55e+01 2.22e+01 4.86e+02 2.73e+02 0 Skip BFGS
56 2.07e+01 2.55e+01 2.22e+01 4.86e+02 2.68e+02 0
57 2.07e+01 2.56e+01 2.22e+01 4.86e+02 2.68e+02 0 Skip BFGS
58 2.07e+01 2.56e+01 2.22e+01 4.86e+02 2.58e+02 0
59 2.07e+01 2.55e+01 2.22e+01 4.86e+02 6.63e+02 0 Skip BFGS
60 2.07e+01 2.54e+01 2.22e+01 4.86e+02 6.61e+02 0
------------------------- STOP! -------------------------
1 : |fc-fOld| = 1.7534e-03 <= tolF*(1+|f0|) = 7.9851e+03
1 : |xc-x_last| = 1.4427e-03 <= tolX*(1+|x0|) = 1.0000e-01
0 : |proj(x-g)-x| = 6.6098e+02 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 6.6098e+02 <= 1e3*eps = 1.0000e-02
1 : maxIter = 60 <= iter = 60
------------------------- DONE! -------------------------
from __future__ import print_function
import numpy as np
from SimPEG import Mesh
from SimPEG import Problem
from SimPEG import Survey
from SimPEG import DataMisfit
from SimPEG import Directives
from SimPEG import Optimization
from SimPEG import Regularization
from SimPEG import InvProblem
from SimPEG import Inversion
import matplotlib.pyplot as plt
def run(N=100, plotIt=True):
np.random.seed(1)
mesh = Mesh.TensorMesh([N])
nk = 20
jk = np.linspace(1., 60., nk)
p = -0.25
q = 0.25
def g(k):
return (
np.exp(p*jk[k]*mesh.vectorCCx) *
np.cos(np.pi*q*jk[k]*mesh.vectorCCx)
)
G = np.empty((nk, mesh.nC))
for i in range(nk):
G[i, :] = g(i)
mtrue = np.zeros(mesh.nC)
mtrue[mesh.vectorCCx > 0.3] = 1.
mtrue[mesh.vectorCCx > 0.45] = -0.5
mtrue[mesh.vectorCCx > 0.6] = 0
prob = Problem.LinearProblem(mesh, G=G)
survey = Survey.LinearSurvey()
survey.pair(prob)
survey.makeSyntheticData(mtrue, std=0.01)
M = prob.mesh
reg = Regularization.Tikhonov(mesh, alpha_s=1., alpha_x=1.)
dmis = DataMisfit.l2_DataMisfit(survey)
opt = Optimization.InexactGaussNewton(maxIter=60)
invProb = InvProblem.BaseInvProblem(dmis, reg, opt)
directives = [
Directives.BetaEstimate_ByEig(beta0_ratio=1e-2),
Directives.TargetMisfit()
]
inv = Inversion.BaseInversion(invProb, directiveList=directives)
m0 = np.zeros_like(survey.mtrue)
mrec = inv.run(m0)
if plotIt:
fig, axes = plt.subplots(1, 2, figsize=(12*1.2, 4*1.2))
for i in range(prob.G.shape[0]):
axes[0].plot(prob.G[i, :])
axes[0].set_title('Columns of matrix G')
axes[1].plot(M.vectorCCx, survey.mtrue, 'b-')
axes[1].plot(M.vectorCCx, mrec, 'r-')
axes[1].legend(('True Model', 'Recovered Model'))
axes[1].set_ylim([-2, 2])
return prob, survey, mesh, mrec
if __name__ == '__main__':
run()
plt.show()
Total running time of the script: ( 0 minutes 6.885 seconds)