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 v0.25.2.dev41+g7d721d035
================================================= Projected GNCG =================================================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS   iter_CG   CG |Ax-b|/|b|  CG |Ax-b|   Comment
-----------------------------------------------------------------------------------------------------------------
1.89e+06  3.68e+03  1.03e-09  3.68e+03                         0           inf          inf
1.89e+06  1.98e+03  3.22e-04  2.59e+03    1.96e+01      0      7        9.73e-04     4.48e+00
9.47e+05  1.40e+03  7.67e-04  2.12e+03    1.92e+01      0      9        2.58e-04     2.16e-01
4.73e+05  8.46e+02  1.60e-03  1.60e+03    1.87e+01      0      9        6.75e-04     4.21e-01
2.37e+05  4.35e+02  2.82e-03  1.10e+03    1.76e+01      0      10       7.34e-04     3.27e-01
1.18e+05  1.92e+02  4.24e-03  6.95e+02    1.62e+01      0      11       9.00e-04     2.64e-01
5.92e+04  7.64e+01  5.59e-03  4.07e+02    1.42e+01      0      13       4.95e-04     8.83e-02
2.96e+04  3.05e+01  6.64e-03  2.27e+02    1.26e+01      0      18       8.42e-04     8.56e-02
1.48e+04  1.45e+01  7.37e-03  1.24e+02    9.92e+00      0      28       6.90e-04     3.82e-02
Reached starting chifact with l2-norm regularization: Start IRLS steps...
irls_threshold 1.2302890347507078
1.48e+04  2.26e+01  9.08e-03  1.57e+02    1.43e+01      0      29       7.19e-04     2.48e-02
1.21e+04  2.43e+01  1.03e-02  1.49e+02    7.70e+00      0      29       7.00e-04     8.16e-03
9.51e+03  2.43e+01  1.14e-02  1.33e+02    2.94e+00      0      29       6.27e-04     6.29e-03
7.48e+03  2.37e+01  1.24e-02  1.16e+02    3.16e+00      0      29       5.21e-04     4.65e-03
5.95e+03  2.25e+01  1.30e-02  1.00e+02    4.66e+00      0      28       6.79e-04     5.65e-03
4.87e+03  2.09e+01  1.33e-02  8.58e+01    5.27e+00      0      24       8.03e-04     6.53e-03
4.87e+03  2.11e+01  1.27e-02  8.28e+01    6.16e+00      0      26       8.81e-04     5.73e-03
4.87e+03  2.09e+01  1.18e-02  7.86e+01    6.72e+00      0      24       5.51e-04     3.95e-03
4.87e+03  2.04e+01  1.09e-02  7.35e+01    7.33e+00      0      22       6.83e-04     5.49e-03
4.87e+03  1.95e+01  9.81e-03  6.73e+01    7.93e+00      0      22       6.26e-04     5.51e-03
4.87e+03  1.82e+01  8.76e-03  6.09e+01    8.40e+00      0      22       5.84e-04     5.43e-03
4.87e+03  1.70e+01  7.77e-03  5.49e+01    8.94e+00      0      21       5.96e-04     6.02e-03
7.73e+03  1.93e+01  6.24e-03  6.75e+01    1.61e+01      0      18       8.21e-04     2.96e-02
7.73e+03  1.87e+01  5.51e-03  6.14e+01    1.06e+01      0      18       7.61e-04     1.28e-02
7.73e+03  1.78e+01  4.84e-03  5.53e+01    1.12e+01      0      19       7.87e-04     1.53e-02
1.21e+04  1.98e+01  3.99e-03  6.80e+01    1.74e+01      0      17       7.16e-04     5.64e-02
1.21e+04  1.88e+01  3.41e-03  5.99e+01    1.15e+01      0      20       5.51e-04     1.70e-02
1.21e+04  1.74e+01  2.93e-03  5.27e+01    1.25e+01      0      18       6.41e-04     2.73e-02
1.90e+04  1.87e+01  2.33e-03  6.30e+01    1.79e+01      0      15       7.90e-04     1.30e-01
1.90e+04  1.75e+01  1.93e-03  5.42e+01    1.15e+01      0      21       5.79e-04     3.58e-02
2.98e+04  1.90e+01  1.54e-03  6.50e+01    1.81e+01      0      15       5.02e-04     1.28e-01
2.98e+04  1.81e+01  1.29e-03  5.66e+01    1.18e+01      0      22       7.84e-04     4.95e-02
2.98e+04  1.74e+01  1.11e-03  5.05e+01    1.24e+01      0      23       3.68e-04     3.60e-02
4.70e+04  1.86e+01  9.01e-04  6.09e+01    1.92e+01      0      19       8.74e-04     3.90e-01
4.70e+04  1.84e+01  7.67e-04  5.44e+01    1.20e+01      0      21       2.85e-04     3.74e-02
4.70e+04  1.81e+01  6.45e-04  4.84e+01    1.19e+01      0      22       5.77e-04     7.65e-02
4.70e+04  1.80e+01  5.38e-04  4.32e+01    1.18e+01      1      20       9.42e-04     1.32e-01
7.31e+04  1.90e+01  4.32e-04  5.06e+01    1.86e+01      0      18       5.86e-04     4.44e-01
7.31e+04  1.92e+01  3.63e-04  4.57e+01    1.22e+01      0      20       8.22e-04     1.50e-01
7.31e+04  1.92e+01  3.03e-04  4.13e+01    1.21e+01      0      21       7.20e-04     1.25e-01
7.31e+04  1.92e+01  2.52e-04  3.77e+01    1.22e+01      0      22       7.34e-04     1.30e-01
7.31e+04  1.92e+01  2.10e-04  3.45e+01    1.19e+01      0      23       6.03e-04     1.09e-01
7.31e+04  1.92e+01  1.74e-04  3.19e+01    1.18e+01      0      24       5.87e-04     1.07e-01
7.31e+04  1.91e+01  1.44e-04  2.96e+01    1.16e+01      0      24       4.38e-04     8.12e-02
7.31e+04  1.90e+01  1.20e-04  2.78e+01    1.12e+01      0      25       7.23e-04     1.55e-01
7.31e+04  1.90e+01  9.98e-05  2.63e+01    1.10e+01      0      27       3.99e-04     7.55e-02
7.31e+04  1.90e+01  8.27e-05  2.51e+01    1.07e+01      0      24       9.02e-04     1.73e-01
7.31e+04  1.91e+01  6.89e-05  2.41e+01    1.06e+01      0      25       5.89e-04     1.13e-01
7.31e+04  1.91e+01  5.74e-05  2.33e+01    1.06e+01      0      26       9.71e-04     1.87e-01
7.31e+04  1.91e+01  4.79e-05  2.26e+01    1.07e+01      0      27       8.21e-04     1.59e-01
Reach maximum number of IRLS cycles: 40
------------------------- STOP! -------------------------
: |fc-fOld| = 5.9581e-03 <= tolF*(1+|f0|) = 3.6789e+02
: |xc-x_last| = 2.2182e-02 <= tolX*(1+|x0|) = 1.0010e-01
: |proj(x-g)-x|    = 1.0718e+01 <= tolG          = 1.0000e-01
: |proj(x-g)-x|    = 1.0718e+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 21.493 seconds)

Estimated memory usage: 329 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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