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])
plot inv 2 inversion irls
(-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")
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:

  1. Data Misfit: a measure of how well our recovered model explains the field data

  2. Regularization: constraints placed on the recovered model and a priori information

  3. 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.

/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 v0.25.0
================================================= Projected GNCG =================================================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS   iter_CG   CG |Ax-b|/|b|  CG |Ax-b|   Comment
-----------------------------------------------------------------------------------------------------------------
   0  2.82e+06  3.73e+03  1.02e-09  3.73e+03                         0           inf          inf
   1  2.82e+06  2.31e+03  1.92e-04  2.85e+03    1.96e+01      0      7        8.21e-04     3.88e+00
   2  1.41e+06  1.75e+03  4.83e-04  2.43e+03    1.91e+01      0      8        6.70e-04     6.72e-01
   3  7.05e+05  1.16e+03  1.09e-03  1.92e+03    1.90e+01      0      9        4.01e-04     3.00e-01
   4  3.53e+05  6.51e+02  2.10e-03  1.39e+03    1.82e+01      0      10       5.05e-04     2.77e-01
   5  1.76e+05  3.15e+02  3.44e-03  9.21e+02    1.69e+01      0      11       9.57e-04     3.62e-01
   6  8.81e+04  1.36e+02  4.84e-03  5.63e+02    1.52e+01      0      13       9.97e-04     2.39e-01
   7  4.41e+04  5.84e+01  6.05e-03  3.25e+02    1.37e+01      0      17       7.86e-04     1.11e-01
   8  2.20e+04  2.89e+01  6.96e-03  1.82e+02    1.23e+01      0      24       5.98e-04     4.71e-02
   9  1.10e+04  1.76e+01  7.66e-03  1.02e+02    1.04e+01      0      30       1.77e-03     7.49e-02
Reached starting chifact with l2-norm regularization: Start IRLS steps...
irls_threshold 1.2341811332047237
  10  1.10e+04  2.41e+01  9.41e-03  1.28e+02    1.51e+01      0      30       7.34e-04     1.95e-02
  11  8.70e+03  2.50e+01  1.07e-02  1.18e+02    4.29e+00      0      30       2.09e-03     1.78e-02
  12  6.74e+03  2.48e+01  1.18e-02  1.05e+02    2.35e+00      0      30       1.68e-03     1.28e-02
  13  5.24e+03  2.42e+01  1.28e-02  9.10e+01    3.40e+00      0      30       1.64e-03     1.13e-02
  14  4.13e+03  2.31e+01  1.35e-02  7.88e+01    4.47e+00      0      30       9.79e-04     6.30e-03
  15  3.34e+03  2.18e+01  1.39e-02  6.80e+01    4.88e+00      0      28       6.64e-04     4.18e-03
  16  3.34e+03  2.20e+01  1.32e-02  6.61e+01    5.77e+00      0      30       8.50e-04     5.24e-03
  17  3.34e+03  2.20e+01  1.23e-02  6.30e+01    6.07e+00      0      30       4.91e-04     3.16e-03
  18  3.34e+03  2.17e+01  1.12e-02  5.89e+01    6.40e+00      0      29       6.62e-04     4.47e-03
  19  3.34e+03  2.14e+01  1.01e-02  5.50e+01    6.99e+00      0      25       4.88e-04     3.61e-03
  20  3.34e+03  2.10e+01  8.95e-03  5.08e+01    7.27e+00      0      23       6.75e-04     5.16e-03
  21  3.34e+03  2.03e+01  7.84e-03  4.65e+01    7.76e+00      0      25       4.38e-04     3.64e-03
  22  3.34e+03  1.96e+01  6.72e-03  4.20e+01    8.37e+00      0      25       9.03e-04     8.46e-03
  23  3.34e+03  1.90e+01  5.71e-03  3.81e+01    8.17e+00      0      24       8.75e-04     8.01e-03
  24  3.34e+03  1.87e+01  4.71e-03  3.44e+01    7.30e+00      0      27       7.18e-04     5.94e-03
  25  3.34e+03  1.86e+01  3.98e-03  3.19e+01    7.73e+00      0      28       8.48e-04     7.69e-03
  26  3.34e+03  1.87e+01  3.45e-03  3.02e+01    8.67e+00      0      29       8.28e-04     9.67e-03
  27  3.34e+03  1.88e+01  2.98e-03  2.87e+01    8.48e+00      0      30       1.16e-03     1.37e-02
  28  3.34e+03  1.89e+01  2.59e-03  2.76e+01    8.33e+00      0      30       7.61e-04     8.07e-03
  29  3.34e+03  1.90e+01  2.30e-03  2.67e+01    8.12e+00      2      30       2.35e-03     2.42e-02
  30  3.34e+03  1.92e+01  1.97e-03  2.57e+01    1.33e+01      0      21       8.57e-04     1.83e-02
  31  3.34e+03  1.92e+01  1.71e-03  2.49e+01    7.84e+00      0      21       9.05e-04     8.92e-03
  32  3.34e+03  1.92e+01  1.51e-03  2.43e+01    7.79e+00      0      29       7.32e-04     7.12e-03
  33  3.34e+03  1.92e+01  1.34e-03  2.36e+01    7.70e+00      0      30       1.83e-03     1.81e-02
  34  3.34e+03  1.91e+01  1.19e-03  2.31e+01    7.57e+00      2      30       1.47e-03     1.49e-02
  35  3.34e+03  1.90e+01  1.08e-03  2.26e+01    1.13e+01      0      30       1.74e-03     3.23e-02
  36  3.34e+03  1.87e+01  9.95e-04  2.20e+01    8.05e+00      0      30       3.74e-03     4.32e-02
  37  3.34e+03  1.83e+01  9.15e-04  2.14e+01    8.90e+00      0      30       7.90e-03     1.01e-01
  38  3.34e+03  1.79e+01  8.31e-04  2.07e+01    8.82e+00      0      30       1.78e-02     2.67e-01
  39  5.19e+03  1.82e+01  5.93e-04  2.12e+01    1.66e+01      0      24       6.69e-04     4.22e-02
  40  5.19e+03  1.81e+01  5.35e-04  2.09e+01    9.15e+00      0      30       8.85e-03     1.99e-01
  41  5.19e+03  1.80e+01  4.83e-04  2.05e+01    8.87e+00      0      30       1.88e-02     4.24e-01
  42  8.08e+03  1.83e+01  3.46e-04  2.11e+01    1.37e+01      0      30       6.10e-03     5.90e-01
  43  8.08e+03  1.84e+01  3.12e-04  2.09e+01    1.06e+01      0      30       2.84e-02     1.00e+00
  44  8.08e+03  1.82e+01  2.82e-04  2.05e+01    1.03e+01      0      30       7.60e-02     2.60e+00
  45  8.08e+03  1.80e+01  2.49e-04  2.00e+01    9.99e+00      0      30       5.19e-02     1.96e+00
  46  1.26e+04  1.80e+01  1.76e-04  2.02e+01    1.45e+01      0      30       2.29e-03     3.25e-01
  47  1.26e+04  1.79e+01  1.55e-04  1.99e+01    1.13e+01      0      30       1.54e-02     1.04e+00
  48  1.96e+04  1.80e+01  1.14e-04  2.02e+01    1.44e+01      0      29       9.03e-04     1.73e-01
  49  3.05e+04  1.82e+01  8.73e-05  2.08e+01    1.48e+01      0      30       5.18e-02     1.14e+01
Reach maximum number of IRLS cycles: 40
------------------------- STOP! -------------------------
1 : |fc-fOld| = 1.2493e-01 <= tolF*(1+|f0|) = 3.7297e+02
1 : |xc-x_last| = 5.1156e-02 <= tolX*(1+|x0|) = 1.0010e-01
0 : |proj(x-g)-x|    = 1.4822e+01 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 1.4822e+01 <= 1e3*eps       = 1.0000e-02
0 : maxIter   =     100    <= iter          =     49
------------------------- 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)
  • plot inv 2 inversion irls
  • plot inv 2 inversion irls
Text(865.1527777777777, 0.5, '$\\phi_m$')

Total running time of the script: (0 minutes 30.819 seconds)

Estimated memory usage: 319 MB

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