Note
Go to the end to download the full example code.
Sparse Inversion with Iteratively Re-Weighted Least-Squares#
Least-squares inversion produces smooth models which may not be an accurate representation of the true model. Here we demonstrate the basics of inverting for sparse and/or blocky models. Here, we used the iteratively reweighted least-squares approach. For this tutorial, we focus on the following:
Defining the forward problem
Defining the inverse problem (data misfit, regularization, optimization)
Defining the paramters for the IRLS algorithm
Specifying directives for the inversion
Recovering a set of model parameters which explains the observations
import numpy as np
import matplotlib.pyplot as plt
from discretize import TensorMesh
from simpeg import (
simulation,
maps,
data_misfit,
directives,
optimization,
regularization,
inverse_problem,
inversion,
)
# sphinx_gallery_thumbnail_number = 3
Defining the Model and Mapping#
Here we generate a synthetic model and a mappig which goes from the model space to the row space of our linear operator.
nParam = 100 # Number of model paramters
# A 1D mesh is used to define the row-space of the linear operator.
mesh = TensorMesh([nParam])
# Creating the true model
true_model = np.zeros(mesh.nC)
true_model[mesh.cell_centers_x > 0.3] = 1.0
true_model[mesh.cell_centers_x > 0.45] = -0.5
true_model[mesh.cell_centers_x > 0.6] = 0
# Mapping from the model space to the row space of the linear operator
model_map = maps.IdentityMap(mesh)
# Plotting the true model
fig = plt.figure(figsize=(8, 5))
ax = fig.add_subplot(111)
ax.plot(mesh.cell_centers_x, true_model, "b-")
ax.set_ylim([-2, 2])
(-2.0, 2.0)
Defining the Linear Operator#
Here we define the linear operator with dimensions (nData, nParam). In practive, you may have a problem-specific linear operator which you would like to construct or load here.
# Number of data observations (rows)
nData = 20
# Create the linear operator for the tutorial. The columns of the linear operator
# represents a set of decaying and oscillating functions.
jk = np.linspace(1.0, 60.0, nData)
p = -0.25
q = 0.25
def g(k):
return np.exp(p * jk[k] * mesh.cell_centers_x) * np.cos(
np.pi * q * jk[k] * mesh.cell_centers_x
)
G = np.empty((nData, nParam))
for i in range(nData):
G[i, :] = g(i)
# Plot the columns of G
fig = plt.figure(figsize=(8, 5))
ax = fig.add_subplot(111)
for i in range(G.shape[0]):
ax.plot(G[i, :])
ax.set_title("Columns of matrix G")
Text(0.5, 1.0, 'Columns of matrix G')
Defining the Simulation#
The simulation defines the relationship between the model parameters and predicted data.
Predict Synthetic Data#
Here, we use the true model to create synthetic data which we will subsequently invert.
# Standard deviation of Gaussian noise being added
std = 0.02
np.random.seed(1)
# Create a SimPEG data object
data_obj = sim.make_synthetic_data(true_model, noise_floor=std, add_noise=True)
Define the Inverse Problem#
The inverse problem is defined by 3 things:
Data Misfit: a measure of how well our recovered model explains the field data
Regularization: constraints placed on the recovered model and a priori information
Optimization: the numerical approach used to solve the inverse problem
# Define the data misfit. Here the data misfit is the L2 norm of the weighted
# residual between the observed data and the data predicted for a given model.
# Within the data misfit, the residual between predicted and observed data are
# normalized by the data's standard deviation.
dmis = data_misfit.L2DataMisfit(simulation=sim, data=data_obj)
# Define the regularization (model objective function). Here, 'p' defines the
# the norm of the smallness term and 'q' defines the norm of the smoothness
# term.
reg = regularization.Sparse(mesh, mapping=model_map)
reg.reference_model = np.zeros(nParam)
p = 0.0
q = 0.0
reg.norms = [p, q]
# Define how the optimization problem is solved.
opt = optimization.ProjectedGNCG(
maxIter=100, lower=-2.0, upper=2.0, maxIterLS=20, maxIterCG=30, tolCG=1e-4
)
# Here we define the inverse problem that is to be solved
inv_prob = inverse_problem.BaseInvProblem(dmis, reg, opt)
/home/vsts/work/1/s/simpeg/optimization.py:1535: FutureWarning:
InexactCG.tolCG has been deprecated, please use InexactCG.cg_atol. It will be removed in version 0.26.0 of SimPEG.
/home/vsts/work/1/s/simpeg/optimization.py:1061: FutureWarning:
InexactCG.maxIterCG has been deprecated, please use InexactCG.cg_maxiter. It will be removed in version 0.26.0 of SimPEG.
Define Inversion Directives#
Here we define any directiveas that are carried out during the inversion. This includes the cooling schedule for the trade-off parameter (beta), stopping criteria for the inversion and saving inversion results at each iteration.
# Add sensitivity weights but don't update at each beta
sensitivity_weights = directives.UpdateSensitivityWeights(every_iteration=False)
# Reach target misfit for L2 solution, then use IRLS until model stops changing.
IRLS = directives.UpdateIRLS(max_irls_iterations=40, f_min_change=1e-4)
# Defining a starting value for the trade-off parameter (beta) between the data
# misfit and the regularization.
starting_beta = directives.BetaEstimate_ByEig(beta0_ratio=1e0)
# Update the preconditionner
update_Jacobi = directives.UpdatePreconditioner()
# Save output at each iteration
saveDict = directives.SaveOutputEveryIteration(save_txt=False)
# Define the directives as a list
directives_list = [
sensitivity_weights,
IRLS,
starting_beta,
update_Jacobi,
saveDict,
]
/home/vsts/work/1/s/simpeg/directives/_directives.py:1865: FutureWarning:
SaveEveryIteration.save_txt has been deprecated, please use SaveEveryIteration.on_disk. It will be removed in version 0.26.0 of SimPEG.
/home/vsts/work/1/s/simpeg/directives/_directives.py:1866: FutureWarning:
SaveEveryIteration.save_txt has been deprecated, please use SaveEveryIteration.on_disk. It will be removed in version 0.26.0 of SimPEG.
Setting a Starting Model and Running the Inversion#
To define the inversion object, we need to define the inversion problem and the set of directives. We can then run the inversion.
# Here we combine the inverse problem and the set of directives
inv = inversion.BaseInversion(inv_prob, directives_list)
# Starting model
starting_model = 1e-4 * np.ones(nParam)
# Run inversion
recovered_model = inv.run(starting_model)
Running inversion with SimPEG v0.24.1.dev37+gf5e5be547
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 1.75e+06 3.68e+03 1.02e-09 3.68e+03 0 inf inf
1 1.75e+06 1.87e+03 3.64e-04 2.51e+03 1.96e+01 0 18 1.16e-08 5.52e-05
2 8.77e+05 1.29e+03 8.42e-04 2.03e+03 1.91e+01 0 17 6.84e-08 5.63e-05
3 4.39e+05 7.66e+02 1.69e-03 1.51e+03 1.86e+01 0 18 8.02e-08 4.83e-05
4 2.19e+05 3.87e+02 2.91e-03 1.02e+03 1.74e+01 0 21 6.37e-08 2.68e-05
5 1.10e+05 1.69e+02 4.29e-03 6.39e+02 1.65e+01 0 25 2.61e-07 7.12e-05
6 5.48e+04 6.66e+01 5.57e-03 3.72e+02 1.34e+01 0 30 7.87e-07 1.29e-04
7 2.74e+04 2.57e+01 6.59e-03 2.06e+02 1.13e+01 0 30 3.86e-05 3.60e-03
8 1.37e+04 1.08e+01 7.33e-03 1.11e+02 9.54e+00 0 30 8.20e-04 4.16e-02
Reached starting chifact with l2-norm regularization: Start IRLS steps...
irls_threshold 1.210108083793358
9 1.37e+04 1.83e+01 9.02e-03 1.42e+02 1.45e+01 0 30 4.60e-04 1.44e-02
10 1.37e+04 2.44e+01 9.87e-03 1.60e+02 1.34e+01 0 30 1.57e-04 3.29e-03
11 1.08e+04 2.43e+01 1.10e-02 1.42e+02 3.18e+00 0 30 5.00e-04 5.55e-03
12 8.46e+03 2.34e+01 1.19e-02 1.24e+02 3.37e+00 0 30 2.43e-04 2.37e-03
13 6.79e+03 2.19e+01 1.26e-02 1.07e+02 4.59e+00 0 30 2.69e-04 2.40e-03
14 6.79e+03 2.35e+01 1.24e-02 1.07e+02 9.68e+00 0 30 3.43e-05 3.48e-04
15 5.43e+03 2.07e+01 1.26e-02 8.91e+01 6.36e+00 0 30 3.15e-05 3.54e-04
16 5.43e+03 2.02e+01 1.17e-02 8.39e+01 8.08e+00 0 30 4.36e-05 3.93e-04
17 5.43e+03 1.94e+01 1.08e-02 7.83e+01 8.54e+00 0 30 2.75e-05 2.61e-04
18 5.43e+03 1.83e+01 9.85e-03 7.19e+01 8.80e+00 0 30 1.30e-05 1.28e-04
19 5.43e+03 1.71e+01 8.85e-03 6.52e+01 9.20e+00 0 30 1.25e-05 1.31e-04
20 8.61e+03 2.06e+01 7.17e-03 8.23e+01 1.68e+01 0 27 1.80e-06 7.10e-05
21 8.61e+03 2.01e+01 6.44e-03 7.55e+01 1.13e+01 0 28 2.63e-06 4.42e-05
22 8.61e+03 1.87e+01 5.64e-03 6.73e+01 1.16e+01 0 29 1.31e-06 2.33e-05
23 8.61e+03 1.72e+01 4.89e-03 5.93e+01 1.18e+01 0 28 4.60e-06 9.06e-05
24 1.36e+04 1.98e+01 3.99e-03 7.41e+01 1.91e+01 0 26 1.33e-06 9.46e-05
25 1.36e+04 1.85e+01 3.44e-03 6.54e+01 1.26e+01 0 28 1.98e-06 5.48e-05
26 1.36e+04 1.78e+01 2.97e-03 5.83e+01 1.39e+01 0 29 9.77e-07 3.24e-05
27 2.12e+04 1.87e+01 2.35e-03 6.86e+01 1.98e+01 0 25 5.44e-07 6.14e-05
28 2.12e+04 1.79e+01 1.99e-03 6.03e+01 1.48e+01 0 26 1.63e-06 7.87e-05
29 3.31e+04 1.91e+01 1.57e-03 7.12e+01 1.92e+01 0 26 2.73e-07 4.16e-05
30 3.31e+04 1.90e+01 1.32e-03 6.27e+01 1.56e+01 0 27 5.59e-07 3.58e-05
31 3.31e+04 1.87e+01 1.08e-03 5.45e+01 1.85e+01 0 23 1.23e-06 8.33e-05
32 3.31e+04 1.87e+01 8.95e-04 4.83e+01 1.47e+01 0 24 6.83e-07 3.72e-05
33 3.31e+04 1.86e+01 7.41e-04 4.32e+01 1.46e+01 0 25 3.92e-07 2.91e-05
34 3.31e+04 1.86e+01 6.19e-04 3.91e+01 1.46e+01 0 25 1.59e-06 7.52e-05
35 3.31e+04 1.86e+01 5.16e-04 3.57e+01 1.45e+01 0 26 1.70e-06 8.31e-05
36 3.31e+04 1.86e+01 4.30e-04 3.29e+01 1.60e+01 0 25 8.43e-07 4.08e-05
37 3.31e+04 1.87e+01 3.60e-04 3.06e+01 1.47e+01 0 26 1.59e-06 8.26e-05
38 3.31e+04 1.87e+01 3.01e-04 2.86e+01 1.48e+01 0 29 6.79e-07 3.62e-05
39 3.31e+04 1.87e+01 2.52e-04 2.70e+01 1.48e+01 0 30 1.28e-06 6.97e-05
40 3.31e+04 1.86e+01 2.10e-04 2.56e+01 1.49e+01 0 30 4.88e-06 2.72e-04
41 3.31e+04 1.86e+01 1.76e-04 2.44e+01 1.51e+01 0 30 1.10e-05 6.27e-04
42 3.31e+04 1.84e+01 1.47e-04 2.33e+01 1.50e+01 0 30 1.94e-05 1.14e-03
43 3.31e+04 1.83e+01 1.23e-04 2.24e+01 1.50e+01 0 30 8.29e-05 6.79e-03
44 3.31e+04 1.81e+01 1.03e-04 2.15e+01 1.49e+01 0 30 2.82e-04 1.71e-02
45 3.31e+04 1.79e+01 8.43e-05 2.07e+01 1.50e+01 0 30 9.25e-06 5.71e-04
46 5.17e+04 1.78e+01 6.70e-05 2.12e+01 1.73e+01 0 30 2.50e-06 7.20e-04
47 8.07e+04 1.77e+01 5.47e-05 2.21e+01 1.72e+01 0 30 1.35e-07 3.78e-05
48 1.26e+05 1.76e+01 4.47e-05 2.32e+01 1.71e+01 0 26 2.48e-07 6.93e-05
Reach maximum number of IRLS cycles: 40
------------------------- STOP! -------------------------
1 : |fc-fOld| = 1.8045e-01 <= tolF*(1+|f0|) = 3.6797e+02
1 : |xc-x_last| = 9.0348e-02 <= tolX*(1+|x0|) = 1.0010e-01
0 : |proj(x-g)-x| = 1.7115e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 1.7115e+01 <= 1e3*eps = 1.0000e-02
0 : maxIter = 100 <= iter = 48
------------------------- DONE! -------------------------
Plotting Results#
fig, ax = plt.subplots(1, 2, figsize=(12 * 1.2, 4 * 1.2))
# True versus recovered model
ax[0].plot(mesh.cell_centers_x, true_model, "k-")
ax[0].plot(mesh.cell_centers_x, inv_prob.l2model, "b-")
ax[0].plot(mesh.cell_centers_x, recovered_model, "r-")
ax[0].legend(("True Model", "Recovered L2 Model", "Recovered Sparse Model"))
ax[0].set_ylim([-2, 2])
# Observed versus predicted data
ax[1].plot(data_obj.dobs, "k-")
ax[1].plot(inv_prob.dpred, "ko")
ax[1].legend(("Observed Data", "Predicted Data"))
# Plot convergence
fig = plt.figure(figsize=(9, 5))
ax = fig.add_axes([0.2, 0.1, 0.7, 0.85])
ax.plot(saveDict.phi_d, "k", lw=2)
twin = ax.twinx()
twin.plot(saveDict.phi_m, "k--", lw=2)
ax.plot(
np.r_[IRLS.metrics.start_irls_iter, IRLS.metrics.start_irls_iter],
np.r_[0, np.max(saveDict.phi_d)],
"k:",
)
ax.text(
IRLS.metrics.start_irls_iter,
0.0,
"IRLS Start",
va="bottom",
ha="center",
rotation="vertical",
size=12,
bbox={"facecolor": "white"},
)
ax.set_ylabel(r"$\phi_d$", size=16, rotation=0)
ax.set_xlabel("Iterations", size=14)
twin.set_ylabel(r"$\phi_m$", size=16, rotation=0)
Text(865.1527777777777, 0.5, '$\\phi_m$')
Total running time of the script: (0 minutes 27.827 seconds)
Estimated memory usage: 298 MB