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.dev14+g41727cb54
================================================= 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  3.65e+03  1.03e-09  3.65e+03                         0           inf          inf
1.72e+06  1.88e+03  3.62e-04  2.50e+03    1.96e+01      0      8        4.14e-04     1.99e+00
8.59e+05  1.30e+03  8.48e-04  2.03e+03    1.90e+01      0      9        3.02e-04     2.46e-01
4.29e+05  7.74e+02  1.73e-03  1.52e+03    1.87e+01      0      9        7.80e-04     4.69e-01
2.15e+05  3.92e+02  2.98e-03  1.03e+03    1.72e+01      0      11       5.51e-04     2.34e-01
1.07e+05  1.75e+02  4.38e-03  6.45e+02    1.61e+01      0      13       6.80e-04     1.88e-01
5.37e+04  7.33e+01  5.68e-03  3.78e+02    1.43e+01      0      14       7.61e-04     1.27e-01
2.68e+04  3.26e+01  6.72e-03  2.13e+02    1.33e+01      0      15       9.85e-04     9.28e-02
1.34e+04  1.75e+01  7.48e-03  1.18e+02    1.18e+01      0      17       9.76e-04     4.99e-02
Reached starting chifact with l2-norm regularization: Start IRLS steps...
irls_threshold 1.1827997158077372
1.34e+04  2.66e+01  9.21e-03  1.50e+02    1.65e+01      0      30       9.88e-04     3.25e-02
1.01e+04  2.67e+01  1.06e-02  1.34e+02    3.08e+00      0      30       1.52e-03     1.63e-02
7.55e+03  2.55e+01  1.19e-02  1.15e+02    4.15e+00      0      30       1.10e-03     1.09e-02
5.79e+03  2.40e+01  1.28e-02  9.83e+01    4.97e+00      0      29       9.03e-04     7.95e-03
4.58e+03  2.21e+01  1.34e-02  8.36e+01    5.25e+00      0      29       7.60e-04     6.17e-03
3.79e+03  2.01e+01  1.35e-02  7.12e+01    5.07e+00      0      22       7.81e-04     5.82e-03
3.79e+03  1.97e+01  1.26e-02  6.75e+01    5.76e+00      0      29       9.38e-04     5.79e-03
3.79e+03  1.91e+01  1.16e-02  6.29e+01    6.11e+00      0      29       9.15e-04     5.98e-03
3.79e+03  1.83e+01  1.04e-02  5.76e+01    6.53e+00      0      27       7.49e-04     5.20e-03
3.79e+03  1.75e+01  9.23e-03  5.25e+01    7.53e+00      0      26       5.25e-04     4.15e-03
5.95e+03  1.97e+01  7.39e-03  6.36e+01    1.40e+01      0      17       9.75e-04     3.13e-02
5.95e+03  1.90e+01  6.61e-03  5.84e+01    1.10e+01      0      21       2.23e-04     3.20e-03
5.95e+03  1.82e+01  5.87e-03  5.31e+01    1.08e+01      0      21       4.85e-04     7.74e-03
5.95e+03  1.75e+01  5.08e-03  4.77e+01    1.06e+01      0      19       9.90e-04     1.59e-02
9.36e+03  1.93e+01  4.12e-03  5.79e+01    1.48e+01      0      16       7.37e-04     5.10e-02
9.36e+03  1.92e+01  3.60e-03  5.29e+01    1.18e+01      0      18       5.11e-04     1.46e-02
9.36e+03  1.90e+01  3.15e-03  4.85e+01    1.18e+01      0      18       7.17e-04     2.41e-02
9.36e+03  1.90e+01  2.74e-03  4.46e+01    1.16e+01      0      18       8.60e-04     3.11e-02
9.36e+03  1.90e+01  2.40e-03  4.14e+01    1.14e+01      0      19       8.11e-04     3.06e-02
9.36e+03  1.91e+01  2.10e-03  3.88e+01    1.13e+01      0      21       9.78e-04     3.93e-02
9.36e+03  1.95e+01  1.84e-03  3.67e+01    1.12e+01      0      22       4.00e-04     1.76e-02
9.36e+03  1.97e+01  1.59e-03  3.46e+01    1.71e+01      0      19       8.17e-04     5.87e-02
9.36e+03  1.99e+01  1.38e-03  3.27e+01    1.09e+01      0      22       5.92e-04     2.54e-02
9.36e+03  2.01e+01  1.20e-03  3.12e+01    1.10e+01      0      25       8.25e-04     3.40e-02
9.36e+03  2.02e+01  1.06e-03  3.01e+01    1.06e+01      1      28       4.68e-04     2.00e-02
9.36e+03  2.05e+01  9.20e-04  2.92e+01    1.23e+01      0      28       9.71e-04     6.77e-02
9.36e+03  2.08e+01  8.17e-04  2.84e+01    9.81e+00      0      30       3.16e-04     1.54e-02
9.36e+03  2.09e+01  7.32e-04  2.77e+01    9.85e+00      0      30       1.77e-03     8.85e-02
9.36e+03  2.08e+01  6.63e-04  2.70e+01    1.45e+01      0      29       6.68e-04     4.14e-02
9.36e+03  2.07e+01  6.06e-04  2.63e+01    9.89e+00      0      30       4.01e-03     2.35e-01
9.36e+03  2.03e+01  5.58e-04  2.55e+01    1.01e+01      0      30       3.78e-03     2.48e-01
9.36e+03  1.98e+01  5.13e-04  2.46e+01    1.01e+01      0      30       5.05e-04     3.75e-02
9.36e+03  1.92e+01  4.64e-04  2.35e+01    9.93e+00      0      30       3.80e-04     3.19e-02
9.36e+03  1.86e+01  3.98e-04  2.23e+01    9.70e+00      0      29       8.49e-04     7.64e-02
9.36e+03  1.82e+01  3.22e-04  2.12e+01    1.60e+01      0      28       6.72e-04     5.95e-02
9.36e+03  1.82e+01  2.48e-04  2.05e+01    9.26e+00      7      30       4.29e-02     3.51e+00
9.36e+03  1.82e+01  2.20e-04  2.02e+01    1.05e+01      6      30       5.05e-02     6.52e+00
9.36e+03  1.80e+01  1.93e-04  1.98e+01    1.20e+01      0      30       3.01e-03     5.13e-01
1.46e+04  1.83e+01  1.28e-04  2.01e+01    1.14e+01      0      30       1.29e-02     5.02e+00
1.46e+04  1.83e+01  1.07e-04  1.98e+01    1.54e+01      0      30       2.20e-02     4.07e+00
Reach maximum number of IRLS cycles: 40
------------------------- STOP! -------------------------
: |fc-fOld| = 1.1606e-01 <= tolF*(1+|f0|) = 3.6504e+02
: |xc-x_last| = 8.2990e-02 <= tolX*(1+|x0|) = 1.0010e-01
: |proj(x-g)-x|    = 1.5397e+01 <= tolG          = 1.0000e-01
: |proj(x-g)-x|    = 1.5397e+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(865.2777777777777, 0.5, '$\\phi_m$')

Total running time of the script: (0 minutes 27.136 seconds)

Estimated memory usage: 332 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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