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.

sim = simulation.LinearSimulation(mesh, G=G, model_map=model_map)

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, cg_maxiter=30, cg_rtol=1e-3
)

# 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,
]

/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.
  self.save_txt = save_txt
/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.
  on_disk = self.save_txt

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 v0.25.2.dev8+g3c268dda4
================================================= Projected GNCG =================================================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS   iter_CG   CG |Ax-b|/|b|  CG |Ax-b|   Comment
-----------------------------------------------------------------------------------------------------------------
1.72e+06  3.74e+03  1.02e-09  3.74e+03                         0           inf          inf
1.72e+06  1.93e+03  3.74e-04  2.57e+03    1.96e+01      0      8        3.81e-04     1.85e+00
8.58e+05  1.33e+03  8.74e-04  2.08e+03    1.90e+01      0      9        2.66e-04     2.19e-01
4.29e+05  7.91e+02  1.77e-03  1.55e+03    1.86e+01      0      9        7.51e-04     4.52e-01
2.14e+05  4.01e+02  3.05e-03  1.06e+03    1.76e+01      0      10       8.53e-04     3.61e-01
1.07e+05  1.77e+02  4.50e-03  6.60e+02    1.62e+01      0      12       7.71e-04     2.12e-01
5.36e+04  7.26e+01  5.84e-03  3.86e+02    1.41e+01      0      13       9.27e-04     1.53e-01
2.68e+04  3.10e+01  6.90e-03  2.16e+02    1.26e+01      0      20       9.78e-04     9.18e-02
1.34e+04  1.61e+01  7.65e-03  1.19e+02    1.06e+01      0      29       8.28e-04     4.22e-02
Reached starting chifact with l2-norm regularization: Start IRLS steps...
irls_threshold 1.23745792815413
1.34e+04  2.40e+01  9.43e-03  1.50e+02    1.51e+01      0      29       9.02e-04     2.87e-02
1.06e+04  2.51e+01  1.08e-02  1.39e+02    5.43e+00      0      30       1.15e-03     1.19e-02
8.22e+03  2.47e+01  1.19e-02  1.23e+02    2.61e+00      0      30       1.02e-03     9.52e-03
6.40e+03  2.38e+01  1.29e-02  1.07e+02    3.80e+00      0      29       7.49e-04     6.18e-03
5.09e+03  2.23e+01  1.36e-02  9.16e+01    4.83e+00      0      29       5.85e-04     4.61e-03
4.19e+03  2.05e+01  1.39e-02  7.86e+01    5.08e+00      0      22       9.68e-04     7.26e-03
4.19e+03  2.06e+01  1.31e-02  7.56e+01    6.56e+00      0      29       6.59e-04     4.50e-03
4.19e+03  2.03e+01  1.22e-02  7.16e+01    6.99e+00      0      25       6.51e-04     4.79e-03
4.19e+03  1.97e+01  1.11e-02  6.65e+01    7.25e+00      0      23       8.04e-04     6.19e-03
4.19e+03  1.89e+01  1.00e-02  6.09e+01    7.67e+00      0      23       8.54e-04     6.97e-03
4.19e+03  1.81e+01  8.94e-03  5.56e+01    8.24e+00      0      20       9.85e-04     8.67e-03
4.19e+03  1.72e+01  7.86e-03  5.02e+01    8.56e+00      0      21       5.63e-04     5.22e-03
6.62e+03  1.92e+01  6.40e-03  6.16e+01    1.42e+01      0      16       8.84e-04     3.25e-02
6.62e+03  1.84e+01  5.59e-03  5.54e+01    1.07e+01      0      18       8.61e-04     1.33e-02
6.62e+03  1.71e+01  4.80e-03  4.89e+01    1.09e+01      0      21       8.81e-04     1.60e-02
1.05e+04  1.77e+01  3.74e-03  5.70e+01    1.44e+01      0      18       7.19e-04     4.75e-02
1.64e+04  1.87e+01  2.85e-03  6.55e+01    1.50e+01      0      16       5.70e-04     5.13e-02
1.64e+04  1.75e+01  2.31e-03  5.54e+01    1.04e+01      0      19       5.09e-04     1.60e-02
2.58e+04  1.84e+01  1.81e-03  6.52e+01    1.55e+01      0      17       3.97e-04     5.18e-02
2.58e+04  1.75e+01  1.50e-03  5.62e+01    1.04e+01      0      19       5.20e-04     2.88e-02
4.06e+04  1.85e+01  1.19e-03  6.68e+01    1.61e+01      0      18       8.67e-04     1.45e-01
4.06e+04  1.80e+01  9.99e-04  5.86e+01    1.08e+01      0      20       8.21e-04     3.46e-02
4.06e+04  1.75e+01  8.39e-04  5.15e+01    1.11e+01      0      21       3.31e-04     1.56e-02
6.38e+04  1.85e+01  6.76e-04  6.16e+01    1.58e+01      0      21       7.07e-04     1.57e-01
6.38e+04  1.82e+01  5.72e-04  5.47e+01    1.10e+01      0      22       9.13e-04     5.30e-02
6.38e+04  1.78e+01  4.82e-04  4.85e+01    1.13e+01      0      24       2.70e-04     1.66e-02
9.96e+04  1.86e+01  3.92e-04  5.77e+01    1.59e+01      0      24       4.01e-04     1.11e-01
9.96e+04  1.83e+01  3.31e-04  5.13e+01    1.08e+01      0      24       5.47e-04     3.60e-02
9.96e+04  1.78e+01  2.79e-04  4.56e+01    1.10e+01      0      25       7.83e-04     5.50e-02
1.56e+05  1.84e+01  2.29e-04  5.40e+01    1.48e+01      0      25       2.72e-04     8.62e-02
1.56e+05  1.82e+01  1.92e-04  4.80e+01    1.06e+01      0      25       4.93e-04     3.26e-02
1.56e+05  1.78e+01  1.60e-04  4.27e+01    1.08e+01      0      27       7.34e-04     5.47e-02
2.43e+05  1.82e+01  1.31e-04  5.00e+01    1.36e+01      0      25       2.54e-04     8.48e-02
2.43e+05  1.79e+01  1.09e-04  4.45e+01    1.04e+01      0      25       3.23e-04     5.43e-02
3.79e+05  1.84e+01  8.90e-05  5.21e+01    1.40e+01      0      25       6.15e-04     2.71e-01
3.79e+05  1.81e+01  7.54e-05  4.67e+01    1.06e+01      0      26       4.06e-04     2.98e-02
3.79e+05  1.77e+01  6.37e-05  4.18e+01    1.06e+01      0      27       8.86e-04     6.74e-02
5.93e+05  1.81e+01  5.25e-05  4.92e+01    1.35e+01      0      25       7.28e-04     2.51e-01
5.93e+05  1.78e+01  4.41e-05  4.40e+01    1.03e+01      0      27       8.02e-04     5.48e-02
9.26e+05  1.83e+01  3.61e-05  5.18e+01    1.36e+01      0      26       4.59e-04     1.55e-01
Reach maximum number of IRLS cycles: 40
------------------------- STOP! -------------------------
: |fc-fOld| = 6.9326e-02 <= tolF*(1+|f0|) = 3.7435e+02
: |xc-x_last| = 5.4188e-02 <= tolX*(1+|x0|) = 1.0010e-01
: |proj(x-g)-x|    = 1.3629e+01 <= tolG          = 1.0000e-01
: |proj(x-g)-x|    = 1.3629e+01 <= 1e3*eps       = 1.0000e-02
: 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 26.244 seconds)

Estimated memory usage: 333 MB

Download Jupyter notebook: plot_inv_2_inversion_irls.ipynb

Download Python source code: plot_inv_2_inversion_irls.py

Download zipped: plot_inv_2_inversion_irls.zip

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