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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)
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,
]
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.dev17+g4a94b2ccf
simpeg.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.
***Done using the default solver Pardiso and no solver_opts.***
model has any nan: 0
=============================== Projected GNCG ===============================
# beta phi_d phi_m f |proj(x-g)-x| LS Comment
-----------------------------------------------------------------------------
x0 has any nan: 0
0 1.70e+06 3.70e+03 1.01e-09 3.70e+03 1.98e+01 0
1 8.50e+05 1.85e+03 3.84e-04 2.18e+03 1.92e+01 0
2 4.25e+05 1.26e+03 8.88e-04 1.64e+03 1.87e+01 0 Skip BFGS
3 2.13e+05 7.29e+02 1.78e-03 1.11e+03 1.81e+01 0 Skip BFGS
4 1.06e+05 3.58e+02 3.00e-03 6.77e+02 1.67e+01 0 Skip BFGS
5 5.32e+04 1.55e+02 4.32e-03 3.85e+02 1.49e+01 0 Skip BFGS
6 2.66e+04 6.41e+01 5.49e-03 2.10e+02 1.34e+01 0 Skip BFGS
7 1.33e+04 2.96e+01 6.37e-03 1.14e+02 1.05e+01 0 Skip BFGS
Reached starting chifact with l2-norm regularization: Start IRLS steps...
irls_threshold 1.2117646240428526
8 1.33e+04 1.77e+01 9.51e-03 1.44e+02 1.54e+01 0 Skip BFGS
9 1.05e+04 2.42e+01 1.01e-02 1.30e+02 5.33e+00 0
10 8.08e+03 2.52e+01 1.10e-02 1.14e+02 2.60e+00 0 Skip BFGS
11 6.24e+03 2.52e+01 1.17e-02 9.83e+01 3.75e+00 0 Skip BFGS
12 4.89e+03 2.45e+01 1.22e-02 8.40e+01 4.86e+00 0 Skip BFGS
13 3.93e+03 2.33e+01 1.22e-02 7.12e+01 5.37e+00 0 Skip BFGS
14 3.93e+03 2.16e+01 1.18e-02 6.79e+01 5.27e+00 0
15 3.93e+03 2.14e+01 1.09e-02 6.43e+01 5.91e+00 0
16 3.93e+03 2.11e+01 9.89e-03 6.00e+01 6.33e+00 0
17 3.93e+03 2.06e+01 8.87e-03 5.55e+01 6.67e+00 0
18 3.93e+03 1.99e+01 7.91e-03 5.10e+01 7.32e+00 0
19 3.93e+03 1.91e+01 6.99e-03 4.66e+01 7.94e+00 0
20 3.93e+03 1.84e+01 6.19e-03 4.28e+01 9.04e+00 0
21 6.15e+03 1.78e+01 5.59e-03 5.22e+01 1.42e+01 0
22 6.15e+03 1.92e+01 4.52e-03 4.70e+01 1.10e+01 0
23 6.15e+03 1.87e+01 3.80e-03 4.20e+01 1.10e+01 0
24 9.62e+03 1.77e+01 3.10e-03 4.75e+01 1.54e+01 0
25 9.62e+03 1.84e+01 2.32e-03 4.07e+01 1.06e+01 0
26 1.51e+04 1.76e+01 1.91e-03 4.64e+01 1.70e+01 0 Skip BFGS
27 1.51e+04 1.84e+01 1.48e-03 4.08e+01 1.08e+01 0
28 1.51e+04 1.82e+01 1.23e-03 3.69e+01 1.14e+01 0
29 1.51e+04 1.81e+01 1.05e-03 3.39e+01 1.16e+01 0
30 1.51e+04 1.81e+01 8.90e-04 3.15e+01 1.18e+01 0
31 1.51e+04 1.81e+01 7.61e-04 2.96e+01 1.14e+01 0 Skip BFGS
32 1.51e+04 1.81e+01 6.51e-04 2.79e+01 1.13e+01 0 Skip BFGS
33 1.51e+04 1.82e+01 5.56e-04 2.65e+01 1.12e+01 0 Skip BFGS
34 1.51e+04 1.82e+01 4.75e-04 2.54e+01 1.12e+01 0 Skip BFGS
35 1.51e+04 1.82e+01 4.07e-04 2.44e+01 1.13e+01 0 Skip BFGS
36 1.51e+04 1.83e+01 3.48e-04 2.35e+01 1.15e+01 0 Skip BFGS
37 1.51e+04 1.83e+01 2.99e-04 2.28e+01 1.17e+01 0 Skip BFGS
38 1.51e+04 1.83e+01 2.56e-04 2.22e+01 1.60e+01 2 Skip BFGS
39 1.51e+04 1.84e+01 2.09e-04 2.16e+01 1.15e+01 0
40 1.51e+04 1.84e+01 1.74e-04 2.11e+01 1.13e+01 0 Skip BFGS
41 1.51e+04 1.85e+01 1.45e-04 2.06e+01 1.10e+01 0 Skip BFGS
42 1.51e+04 1.85e+01 1.22e-04 2.03e+01 1.10e+01 0 Skip BFGS
43 1.51e+04 1.85e+01 1.03e-04 2.00e+01 1.36e+01 1 Skip BFGS
44 1.51e+04 1.85e+01 8.49e-05 1.98e+01 1.14e+01 0
45 1.51e+04 1.85e+01 7.09e-05 1.96e+01 1.16e+01 0 Skip BFGS
46 1.51e+04 1.86e+01 5.92e-05 1.95e+01 1.11e+01 0
47 1.51e+04 1.86e+01 4.96e-05 1.93e+01 1.11e+01 0
Reach maximum number of IRLS cycles: 40
------------------------- STOP! -------------------------
1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.7016e+02
1 : |xc-x_last| = 7.4292e-02 <= tolX*(1+|x0|) = 1.0010e-01
0 : |proj(x-g)-x| = 1.1107e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 1.1107e+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 43.906 seconds)
Estimated memory usage: 292 MB