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, maxIterCG=30, tolCG=1e-4
)

# 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,
]

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 v0.23.0

                    simpeg.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.
                    ***Done using the default solver Pardiso and no solver_opts.***

model has any nan: 0
=============================== Projected GNCG ===============================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS    Comment
-----------------------------------------------------------------------------
x0 has any nan: 0
2.53e+06  3.70e+03  1.03e-09  3.70e+03    1.99e+01      0
1.26e+06  2.17e+03  2.26e-04  2.46e+03    1.93e+01      0
6.31e+05  1.60e+03  5.55e-04  1.95e+03    1.91e+01      0   Skip BFGS
3.16e+05  1.03e+03  1.20e-03  1.41e+03    1.86e+01      0   Skip BFGS
1.58e+05  5.70e+02  2.24e-03  9.23e+02    1.71e+01      0   Skip BFGS
7.89e+04  2.72e+02  3.56e-03  5.53e+02    1.44e+01      0   Skip BFGS
3.95e+04  1.17e+02  4.91e-03  3.11e+02    1.28e+01      0   Skip BFGS
1.97e+04  5.07e+01  6.06e-03  1.70e+02    1.04e+01      0   Skip BFGS
9.86e+03  2.59e+01  6.92e-03  9.41e+01    8.92e+00      0   Skip BFGS
Reached starting chifact with l2-norm regularization: Start IRLS steps...
irls_threshold 1.2048896961458564
9.86e+03  1.71e+01  1.03e-02  1.18e+02    1.37e+01      0   Skip BFGS
6.30e+03  2.26e+01  1.08e-02  9.08e+01    5.78e+00      0
4.03e+03  2.10e+01  1.20e-02  6.92e+01    6.80e+00      0
2.58e+03  1.91e+01  1.28e-02  5.20e+01    6.92e+00      0   Skip BFGS
3.20e+03  1.72e+01  1.30e-02  5.89e+01    7.85e+00      0   Skip BFGS
3.97e+03  1.87e+01  1.24e-02  6.78e+01    1.06e+01      0
4.93e+03  2.09e+01  1.14e-02  7.70e+01    1.18e+01      0
3.11e+03  2.34e+01  1.03e-02  5.53e+01    8.46e+00      0
1.96e+03  2.00e+01  9.60e-03  3.88e+01    8.09e+00      0
2.46e+03  1.68e+01  8.86e-03  3.87e+01    6.92e+00      0
3.09e+03  1.69e+01  7.42e-03  3.98e+01    8.53e+00      0
3.84e+03  1.72e+01  6.19e-03  4.09e+01    9.59e+00      0
4.73e+03  1.75e+01  5.25e-03  4.23e+01    9.88e+00      0
5.78e+03  1.77e+01  4.35e-03  4.28e+01    1.02e+01      0
7.02e+03  1.79e+01  3.58e-03  4.31e+01    1.11e+01      0
8.53e+03  1.82e+01  3.05e-03  4.42e+01    1.26e+01      0
1.04e+04  1.85e+01  2.58e-03  4.52e+01    1.31e+01      0
1.26e+04  1.88e+01  2.16e-03  4.60e+01    1.34e+01      0
1.53e+04  1.91e+01  1.80e-03  4.67e+01    1.34e+01      0
1.86e+04  1.94e+01  1.51e-03  4.74e+01    1.33e+01      0
2.26e+04  1.98e+01  1.26e-03  4.83e+01    1.34e+01      0
2.74e+04  2.02e+01  1.04e-03  4.87e+01    1.34e+01      0
3.33e+04  2.07e+01  8.63e-04  4.95e+01    1.35e+01      0
4.05e+04  2.12e+01  7.11e-04  5.00e+01    1.35e+01      0
4.92e+04  2.16e+01  5.82e-04  5.03e+01    1.34e+01      0
3.17e+04  2.20e+01  4.84e-04  3.74e+01    1.16e+01      0   Skip BFGS
2.05e+04  1.98e+01  4.69e-04  2.94e+01    1.09e+01      0
1.32e+04  1.89e+01  4.32e-04  2.46e+01    1.30e+01      1   Skip BFGS
1.63e+04  1.74e+01  4.10e-04  2.41e+01    1.26e+01      0
2.01e+04  1.75e+01  3.37e-04  2.42e+01    1.28e+01      0
2.45e+04  1.78e+01  2.74e-04  2.45e+01    1.26e+01      0   Skip BFGS
3.00e+04  1.81e+01  2.22e-04  2.47e+01    1.26e+01      0   Skip BFGS
3.66e+04  1.84e+01  1.80e-04  2.50e+01    1.27e+01      0
4.47e+04  1.86e+01  1.46e-04  2.52e+01    1.28e+01      0
5.46e+04  1.89e+01  1.19e-04  2.54e+01    1.29e+01      0
6.66e+04  1.90e+01  9.76e-05  2.55e+01    1.29e+01      0
8.14e+04  1.92e+01  8.02e-05  2.57e+01    1.29e+01      0
9.94e+04  1.92e+01  6.62e-05  2.58e+01    1.29e+01      0
1.21e+05  1.93e+01  5.48e-05  2.60e+01    1.29e+01      0
1.48e+05  1.94e+01  4.54e-05  2.61e+01    1.29e+01      0
Reach maximum number of IRLS cycles: 40
------------------------- STOP! -------------------------
: |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.6961e+02
: |xc-x_last| = 5.3318e-03 <= tolX*(1+|x0|) = 1.0010e-01
: |proj(x-g)-x|    = 1.2922e+01 <= tolG          = 1.0000e-01
: |proj(x-g)-x|    = 1.2922e+01 <= 1e3*eps       = 1.0000e-02
: 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)

Text(865.1527777777777, 0.5, '$\\phi_m$')

Total running time of the script: (0 minutes 39.751 seconds)

Estimated memory usage: 288 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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