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.dev9+g43b0120dd
================================================= Projected GNCG =================================================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS   iter_CG   CG |Ax-b|/|b|  CG |Ax-b|   Comment
-----------------------------------------------------------------------------------------------------------------
1.67e+06  3.76e+03  1.03e-09  3.76e+03                         0           inf          inf
1.67e+06  1.90e+03  3.86e-04  2.54e+03    1.96e+01      0      8        3.89e-04     2.11e+00
8.34e+05  1.32e+03  8.88e-04  2.06e+03    1.91e+01      0      9        2.70e-04     2.25e-01
4.17e+05  7.89e+02  1.79e-03  1.53e+03    1.84e+01      0      9        8.75e-04     5.27e-01
2.08e+05  4.08e+02  3.07e-03  1.05e+03    1.75e+01      0      10       7.73e-04     3.24e-01
1.04e+05  1.85e+02  4.56e-03  6.60e+02    1.67e+01      0      13       6.30e-04     1.71e-01
5.21e+04  7.74e+01  5.98e-03  3.89e+02    1.45e+01      0      16       3.03e-04     4.97e-02
2.60e+04  3.32e+01  7.14e-03  2.19e+02    1.22e+01      0      13       9.16e-04     8.57e-02
1.30e+04  1.74e+01  7.96e-03  1.21e+02    9.70e+00      0      23       9.81e-04     5.01e-02
Reached starting chifact with l2-norm regularization: Start IRLS steps...
irls_threshold 1.2268377598754505
1.30e+04  2.64e+01  9.83e-03  1.54e+02    1.45e+01      0      30       1.30e-03     4.11e-02
9.82e+03  2.65e+01  1.14e-02  1.38e+02    3.35e+00      0      30       1.28e-03     1.38e-02
7.40e+03  2.53e+01  1.27e-02  1.20e+02    3.78e+00      0      30       1.05e-03     1.04e-02
5.70e+03  2.38e+01  1.38e-02  1.02e+02    4.45e+00      0      29       8.82e-04     7.66e-03
4.53e+03  2.19e+01  1.44e-02  8.72e+01    4.60e+00      0      29       9.11e-04     7.43e-03
4.53e+03  2.24e+01  1.38e-02  8.49e+01    6.43e+00      0      29       5.11e-04     3.43e-03
3.72e+03  2.00e+01  1.36e-02  7.06e+01    4.80e+00      0      24       9.49e-04     7.95e-03
3.72e+03  1.92e+01  1.22e-02  6.48e+01    5.34e+00      0      29       5.95e-04     3.80e-03
3.72e+03  1.86e+01  1.09e-02  5.92e+01    5.79e+00      0      26       5.87e-04     3.86e-03
3.72e+03  1.79e+01  9.56e-03  5.35e+01    6.20e+00      0      26       7.85e-04     5.50e-03
5.80e+03  2.08e+01  7.55e-03  6.46e+01    1.41e+01      0      19       7.75e-04     1.93e-02
5.80e+03  2.07e+01  6.78e-03  6.00e+01    8.94e+00      0      19       9.49e-04     1.07e-02
5.80e+03  2.01e+01  5.97e-03  5.48e+01    8.72e+00      0      21       7.27e-04     7.29e-03
5.80e+03  1.97e+01  5.25e-03  5.01e+01    9.01e+00      0      22       7.72e-04     7.48e-03
5.80e+03  1.94e+01  4.82e-03  4.74e+01    1.09e+01      0      22       4.06e-04     5.67e-03
5.80e+03  1.99e+01  4.36e-03  4.52e+01    1.16e+01      0      23       3.98e-04     7.08e-03
5.80e+03  1.91e+01  3.83e-03  4.13e+01    1.80e+01      0      17       7.24e-04     6.00e-02
5.80e+03  1.86e+01  3.24e-03  3.74e+01    1.07e+01      0      21       9.79e-04     1.65e-02
5.80e+03  1.79e+01  2.70e-03  3.36e+01    1.09e+01      0      22       5.25e-04     8.56e-03
9.04e+03  1.88e+01  2.05e-03  3.74e+01    1.66e+01      0      17       9.01e-04     5.55e-02
9.04e+03  1.88e+01  1.75e-03  3.46e+01    1.28e+01      1      22       3.63e-04     1.05e-02
9.04e+03  1.92e+01  1.44e-03  3.22e+01    1.44e+01      0      17       6.79e-04     2.41e-02
9.04e+03  1.95e+01  1.22e-03  3.05e+01    1.16e+01      0      21       4.82e-04     1.47e-02
9.04e+03  1.97e+01  1.05e-03  2.91e+01    1.17e+01      0      21       5.06e-04     1.46e-02
9.04e+03  1.99e+01  9.04e-04  2.81e+01    1.18e+01      0      22       4.07e-04     1.17e-02
9.04e+03  2.01e+01  7.85e-04  2.72e+01    1.17e+01      0      23       8.51e-04     2.68e-02
9.04e+03  2.03e+01  6.86e-04  2.65e+01    1.17e+01      0      25       4.36e-04     1.51e-02
9.04e+03  2.04e+01  6.14e-04  2.59e+01    1.16e+01      1      26       8.34e-04     3.15e-02
9.04e+03  2.06e+01  5.29e-04  2.54e+01    1.23e+01      0      27       8.77e-04     5.03e-02
9.04e+03  2.08e+01  4.67e-04  2.50e+01    1.14e+01      0      22       5.02e-04     2.28e-02
9.04e+03  2.09e+01  4.26e-04  2.47e+01    1.51e+01      1      21       7.14e-04     3.96e-02
9.04e+03  2.09e+01  3.97e-04  2.45e+01    1.56e+01      3      22       5.34e-04     4.43e-02
9.04e+03  2.09e+01  3.51e-04  2.41e+01    1.45e+01      0      20       5.87e-04     6.54e-02
9.04e+03  2.07e+01  3.34e-04  2.37e+01    1.31e+01      0      20       5.56e-04     3.98e-02
9.04e+03  2.00e+01  3.32e-04  2.30e+01    1.34e+01      0      24       3.73e-04     2.70e-02
9.04e+03  1.89e+01  3.32e-04  2.19e+01    1.32e+01      0      25       4.67e-04     3.80e-02
9.04e+03  1.77e+01  2.85e-04  2.02e+01    1.36e+01      0      26       4.74e-04     4.39e-02
1.42e+04  1.74e+01  1.69e-04  1.98e+01    1.68e+01      0      26       9.42e-04     2.99e-01
2.23e+04  1.75e+01  1.15e-04  2.01e+01    1.66e+01      0      22       7.23e-04     2.89e-01
3.50e+04  1.78e+01  8.62e-05  2.08e+01    1.63e+01      0      26       6.76e-04     2.73e-01
Reach maximum number of IRLS cycles: 40
------------------------- STOP! -------------------------
: |fc-fOld| = 1.2167e-01 <= tolF*(1+|f0|) = 3.7601e+02
: |xc-x_last| = 4.4659e-02 <= tolX*(1+|x0|) = 1.0010e-01
: |proj(x-g)-x|    = 1.6348e+01 <= tolG          = 1.0000e-01
: |proj(x-g)-x|    = 1.6348e+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 30.064 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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