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(on_disk=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 v0.26.0
================================================= Projected GNCG =================================================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS   iter_CG   CG |Ax-b|/|b|  CG |Ax-b|   Comment
-----------------------------------------------------------------------------------------------------------------
1.73e+06  3.73e+03  1.03e-09  3.73e+03                         0           inf          inf
1.73e+06  1.95e+03  3.64e-04  2.58e+03    1.96e+01      0      8        4.19e-04     2.07e+00
8.63e+05  1.36e+03  8.57e-04  2.10e+03    1.90e+01      0      9        2.90e-04     2.37e-01
4.31e+05  8.14e+02  1.76e-03  1.57e+03    1.87e+01      0      9        7.19e-04     4.34e-01
2.16e+05  4.15e+02  3.06e-03  1.08e+03    1.76e+01      0      10       8.80e-04     3.76e-01
1.08e+05  1.84e+02  4.55e-03  6.75e+02    1.58e+01      0      12       5.99e-04     1.67e-01
5.39e+04  7.44e+01  5.95e-03  3.95e+02    1.38e+01      0      16       6.31e-04     1.07e-01
2.70e+04  3.03e+01  7.06e-03  2.21e+02    1.27e+01      0      20       8.59e-04     8.28e-02
1.35e+04  1.46e+01  7.85e-03  1.20e+02    1.08e+01      0      29       6.29e-04     3.30e-02
Reached starting chifact with l2-norm regularization: Start IRLS steps...
irls_threshold 1.223917956826682
1.35e+04  2.34e+01  9.64e-03  1.53e+02    1.51e+01      0      28       9.79e-04     3.19e-02
1.08e+04  2.50e+01  1.10e-02  1.44e+02    6.09e+00      0      30       1.13e-03     1.18e-02
8.39e+03  2.46e+01  1.22e-02  1.27e+02    2.88e+00      0      29       9.38e-04     8.96e-03
6.55e+03  2.36e+01  1.31e-02  1.10e+02    4.24e+00      0      29       6.21e-04     5.41e-03
5.23e+03  2.18e+01  1.38e-02  9.40e+01    4.86e+00      0      29       4.49e-04     3.72e-03
5.23e+03  2.25e+01  1.33e-02  9.19e+01    7.11e+00      0      29       4.83e-04     3.55e-03
4.29e+03  1.98e+01  1.31e-02  7.60e+01    5.79e+00      0      19       5.88e-04     5.66e-03
4.29e+03  1.88e+01  1.19e-02  6.99e+01    6.48e+00      0      26       6.82e-04     5.12e-03
4.29e+03  1.79e+01  1.07e-02  6.39e+01    7.01e+00      0      23       8.37e-04     6.54e-03
6.69e+03  2.15e+01  8.62e-03  7.91e+01    1.47e+01      0      18       9.39e-04     2.72e-02
6.69e+03  2.12e+01  7.82e-03  7.35e+01    9.52e+00      0      18       4.47e-04     5.37e-03
6.69e+03  2.02e+01  6.93e-03  6.65e+01    9.55e+00      0      18       6.30e-04     7.55e-03
6.69e+03  1.88e+01  6.01e-03  5.91e+01    9.56e+00      0      18       8.95e-04     1.13e-02
6.69e+03  1.73e+01  5.30e-03  5.27e+01    1.06e+01      0      19       6.62e-04     1.00e-02
1.06e+04  1.81e+01  4.19e-03  6.24e+01    1.40e+01      0      18       4.84e-04     2.26e-02
1.06e+04  1.59e+01  3.44e-03  5.21e+01    1.01e+01      0      19       7.69e-04     1.53e-02
1.72e+04  1.61e+01  2.64e-03  6.16e+01    1.46e+01      0      17       5.11e-04     3.31e-02
2.78e+04  1.74e+01  2.00e-03  7.31e+01    1.50e+01      0      17       8.57e-04     7.52e-02
4.38e+04  1.94e+01  1.55e-03  8.72e+01    1.64e+01      0      17       9.78e-04     1.13e-01
4.38e+04  1.89e+01  1.27e-03  7.47e+01    1.22e+01      0      19       7.60e-04     4.87e-02
4.38e+04  1.82e+01  1.05e-03  6.43e+01    1.26e+01      0      20       5.32e-04     2.86e-02
4.38e+04  1.77e+01  8.92e-04  5.68e+01    1.27e+01      0      20       7.28e-04     5.21e-02
6.87e+04  1.96e+01  7.28e-04  6.96e+01    1.64e+01      0      19       7.89e-04     1.61e-01
6.87e+04  1.93e+01  6.08e-04  6.11e+01    1.23e+01      0      21       5.11e-04     3.18e-02
6.87e+04  1.85e+01  5.26e-04  5.46e+01    1.33e+01      0      21       4.93e-04     6.06e-02
6.87e+04  1.72e+01  4.64e-04  4.91e+01    1.30e+01      0      21       5.15e-04     6.45e-02
1.09e+05  1.83e+01  3.85e-04  6.01e+01    1.57e+01      0      20       9.37e-04     3.11e-01
1.09e+05  1.75e+01  3.27e-04  5.31e+01    1.23e+01      0      21       3.91e-04     2.48e-02
1.70e+05  1.89e+01  2.69e-04  6.47e+01    1.58e+01      0      20       6.30e-04     2.29e-01
1.70e+05  1.82e+01  2.27e-04  5.70e+01    1.21e+01      0      21       3.75e-04     2.47e-02
1.70e+05  1.71e+01  1.99e-04  5.10e+01    1.26e+01      0      21       6.71e-04     6.70e-02
2.70e+05  1.81e+01  1.65e-04  6.28e+01    1.55e+01      0      20       7.12e-04     3.00e-01
2.70e+05  1.73e+01  1.42e-04  5.56e+01    1.26e+01      0      21       6.10e-04     4.64e-02
4.27e+05  1.86e+01  1.17e-04  6.86e+01    1.58e+01      0      20       8.68e-04     3.69e-01
4.27e+05  1.77e+01  9.99e-05  6.03e+01    1.33e+01      0      21       4.70e-04     4.02e-02
6.67e+05  1.92e+01  8.15e-05  7.35e+01    1.63e+01      0      20       5.31e-04     2.35e-01
6.67e+05  1.83e+01  6.80e-05  6.37e+01    1.26e+01      0      20       6.89e-04     1.36e-01
6.67e+05  1.70e+01  5.83e-05  5.59e+01    1.28e+01      0      22       3.56e-04     3.48e-02
1.06e+06  1.81e+01  4.74e-05  6.84e+01    1.54e+01      0      21       4.33e-04     1.84e-01
1.06e+06  1.72e+01  3.97e-05  5.94e+01    1.23e+01      0      22       4.79e-04     3.15e-02
Reach maximum number of IRLS cycles: 40
------------------------- STOP! -------------------------
: |fc-fOld| = 7.9873e-02 <= tolF*(1+|f0|) = 3.7305e+02
: |xc-x_last| = 5.4526e-02 <= tolX*(1+|x0|) = 1.0010e-01
: |proj(x-g)-x|    = 1.2346e+01 <= tolG          = 1.0000e-01
: |proj(x-g)-x|    = 1.2346e+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 25.932 seconds)

Estimated memory usage: 331 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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