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 v0.26.1.dev2+g3562108fd
================================================= Projected GNCG =================================================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS   iter_CG   CG |Ax-b|/|b|  CG |Ax-b|   Comment
-----------------------------------------------------------------------------------------------------------------
1.71e+06  3.70e+03  1.02e-09  3.70e+03                         0           inf          inf
1.71e+06  1.91e+03  3.73e-04  2.55e+03    1.96e+01      0      8        3.95e-04     1.84e+00
8.55e+05  1.32e+03  8.72e-04  2.06e+03    1.91e+01      0      9        2.38e-04     1.91e-01
4.27e+05  7.78e+02  1.77e-03  1.53e+03    1.85e+01      0      9        7.82e-04     4.64e-01
2.14e+05  3.89e+02  3.05e-03  1.04e+03    1.78e+01      0      10       8.55e-04     3.57e-01
1.07e+05  1.68e+02  4.49e-03  6.47e+02    1.55e+01      0      13       6.26e-04     1.70e-01
5.34e+04  6.54e+01  5.80e-03  3.75e+02    1.34e+01      0      16       5.01e-04     8.19e-02
2.67e+04  2.58e+01  6.81e-03  2.08e+02    1.11e+01      0      20       5.22e-04     4.83e-02
1.34e+04  1.24e+01  7.49e-03  1.12e+02    9.20e+00      0      28       7.98e-04     4.00e-02
Reached starting chifact with l2-norm regularization: Start IRLS steps...
irls_threshold 1.269979284493072
1.34e+04  1.92e+01  9.24e-03  1.43e+02    1.37e+01      0      30       1.67e-03     5.07e-02
1.34e+04  2.49e+01  1.02e-02  1.61e+02    1.28e+01      0      27       8.72e-04     1.84e-02
1.04e+04  2.46e+01  1.13e-02  1.42e+02    3.03e+00      0      29       4.39e-04     5.00e-03
8.10e+03  2.38e+01  1.22e-02  1.23e+02    3.14e+00      0      27       8.40e-04     8.40e-03
6.44e+03  2.25e+01  1.28e-02  1.05e+02    4.28e+00      0      27       5.51e-04     4.97e-03
5.27e+03  2.10e+01  1.31e-02  8.99e+01    4.48e+00      0      24       8.84e-04     7.49e-03
5.27e+03  2.12e+01  1.24e-02  8.68e+01    1.01e+01      0      21       9.66e-04     1.13e-02
5.27e+03  2.12e+01  1.17e-02  8.27e+01    7.42e+00      0      20       8.87e-04     7.24e-03
5.27e+03  2.10e+01  1.08e-02  7.79e+01    7.55e+00      0      20       5.25e-04     4.54e-03
5.27e+03  2.07e+01  9.87e-03  7.27e+01    7.67e+00      0      18       6.54e-04     5.82e-03
5.27e+03  2.02e+01  8.94e-03  6.73e+01    7.69e+00      0      18       6.10e-04     5.61e-03
5.27e+03  1.94e+01  8.00e-03  6.16e+01    8.11e+00      0      17       6.21e-04     6.26e-03
5.27e+03  1.83e+01  7.04e-03  5.55e+01    8.31e+00      0      17       6.89e-04     7.44e-03
5.27e+03  1.69e+01  6.12e-03  4.92e+01    8.68e+00      0      23       6.89e-04     7.84e-03
8.40e+03  1.82e+01  4.67e-03  5.74e+01    1.38e+01      0      20       5.58e-04     1.92e-02
8.40e+03  1.67e+01  4.05e-03  5.08e+01    9.12e+00      0      21       9.76e-04     1.50e-02
1.34e+04  1.78e+01  3.14e-03  6.00e+01    1.41e+01      0      18       9.90e-04     4.61e-02
2.09e+04  2.00e+01  2.52e-03  7.28e+01    1.42e+01      0      18       3.10e-04     2.13e-02
2.09e+04  1.94e+01  2.21e-03  6.57e+01    1.00e+01      0      19       6.22e-04     1.75e-02
2.09e+04  1.79e+01  1.89e-03  5.75e+01    1.08e+01      0      21       7.32e-04     2.02e-02
3.27e+04  1.98e+01  1.47e-03  6.79e+01    1.44e+01      0      19       9.19e-04     9.12e-02
3.27e+04  1.89e+01  1.28e-03  6.07e+01    1.17e+01      0      22       5.99e-04     2.93e-02
3.27e+04  1.74e+01  1.09e-03  5.32e+01    1.22e+01      0      23       4.22e-04     2.10e-02
5.14e+04  1.93e+01  8.77e-04  6.45e+01    1.52e+01      0      22       2.82e-04     5.24e-02
5.14e+04  1.80e+01  7.42e-04  5.62e+01    1.23e+01      0      22       9.72e-04     1.01e-01
8.00e+04  1.97e+01  5.91e-04  6.69e+01    1.52e+01      0      22       4.28e-04     1.02e-01
8.00e+04  1.83e+01  4.99e-04  5.82e+01    1.24e+01      0      24       4.08e-04     2.26e-02
8.00e+04  1.67e+01  4.32e-04  5.12e+01    1.27e+01      0      25       4.49e-04     2.96e-02
1.28e+05  1.80e+01  3.46e-04  6.23e+01    1.53e+01      0      23       9.03e-04     2.31e-01
1.28e+05  1.69e+01  2.89e-04  5.39e+01    1.24e+01      0      25       5.35e-04     4.44e-02
2.04e+05  1.84e+01  2.31e-04  6.55e+01    1.53e+01      0      24       3.44e-04     9.29e-02
2.04e+05  1.75e+01  1.97e-04  5.76e+01    1.20e+01      0      25       5.86e-04     3.33e-02
3.20e+05  1.90e+01  1.55e-04  6.87e+01    1.58e+01      0      25       5.40e-04     1.53e-01
3.20e+05  1.82e+01  1.34e-04  6.11e+01    1.21e+01      0      24       3.09e-04     1.60e-01
3.20e+05  1.68e+01  1.14e-04  5.34e+01    1.23e+01      0      27       3.33e-04     2.23e-02
5.11e+05  1.81e+01  9.07e-05  6.44e+01    1.59e+01      0      26       2.97e-04     8.47e-02
5.11e+05  1.73e+01  7.82e-05  5.72e+01    1.21e+01      0      26       8.53e-04     5.25e-02
8.07e+05  1.88e+01  6.23e-05  6.90e+01    1.59e+01      0      25       7.80e-04     2.15e-01
8.07e+05  1.81e+01  5.39e-05  6.16e+01    1.21e+01      0      26       9.99e-04     6.47e-02
8.07e+05  1.68e+01  4.59e-05  5.38e+01    1.21e+01      0      28       7.51e-04     4.87e-02
Reach maximum number of IRLS cycles: 40
------------------------- STOP! -------------------------
: |fc-fOld| = 1.3614e-01 <= tolF*(1+|f0|) = 3.7021e+02
: |xc-x_last| = 9.9349e-02 <= tolX*(1+|x0|) = 1.0010e-01
: |proj(x-g)-x|    = 1.2147e+01 <= tolG          = 1.0000e-01
: |proj(x-g)-x|    = 1.2147e+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 27.325 seconds)

Estimated memory usage: 343 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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