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])
plot inv 2 inversion irls
(-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")
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:

  1. Data Misfit: a measure of how well our recovered model explains the field data

  2. Regularization: constraints placed on the recovered model and a priori information

  3. 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 v0.24.0

                    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
   0  1.69e+06  3.73e+03  1.02e-09  3.73e+03    1.99e+01      0
   1  8.46e+05  1.82e+03  3.96e-04  2.16e+03    1.93e+01      0
   2  4.23e+05  1.23e+03  9.01e-04  1.61e+03    1.89e+01      0   Skip BFGS
   3  2.12e+05  7.12e+02  1.77e-03  1.09e+03    1.80e+01      0   Skip BFGS
   4  1.06e+05  3.51e+02  2.97e-03  6.65e+02    1.57e+01      0   Skip BFGS
   5  5.29e+04  1.52e+02  4.27e-03  3.78e+02    1.37e+01      0   Skip BFGS
   6  2.64e+04  6.18e+01  5.44e-03  2.06e+02    1.28e+01      0   Skip BFGS
   7  1.32e+04  2.61e+01  6.37e-03  1.10e+02    1.17e+01      0   Skip BFGS
Reached starting chifact with l2-norm regularization: Start IRLS steps...
irls_threshold 1.1707247707684598
   8  1.32e+04  1.26e+01  9.65e-03  1.40e+02    1.59e+01      0   Skip BFGS
   9  1.32e+04  2.02e+01  1.01e-02  1.54e+02    1.41e+01      0
  10  7.98e+03  2.63e+01  1.06e-02  1.11e+02    9.67e+00      0   Skip BFGS
  11  4.81e+03  2.13e+01  1.18e-02  7.80e+01    9.73e+00      0
  12  6.16e+03  1.62e+01  1.26e-02  9.41e+01    1.51e+01      0   Skip BFGS
  13  7.89e+03  2.08e+01  1.19e-02  1.15e+02    1.56e+01      0
  14  4.67e+03  2.76e+01  1.09e-02  7.83e+01    1.16e+01      0
  15  2.77e+03  2.09e+01  1.08e-02  5.08e+01    1.13e+01      0
  16  3.86e+03  1.37e+01  1.08e-02  5.54e+01    1.22e+01      0
  17  5.13e+03  1.51e+01  9.18e-03  6.22e+01    1.27e+01      0
  18  6.37e+03  1.72e+01  7.70e-03  6.63e+01    1.22e+01      0
  19  7.91e+03  1.89e+01  6.52e-03  7.05e+01    1.23e+01      0
  20  9.83e+03  2.04e+01  5.49e-03  7.44e+01    1.28e+01      0
  21  1.22e+04  2.18e+01  4.60e-03  7.79e+01    1.35e+01      0
  22  7.76e+03  2.29e+01  3.85e-03  5.28e+01    1.17e+01      0
  23  4.93e+03  1.86e+01  3.66e-03  3.67e+01    8.88e+00      0
  24  6.23e+03  1.66e+01  3.36e-03  3.76e+01    1.89e+01      0   Skip BFGS
  25  8.34e+03  1.49e+01  2.75e-03  3.79e+01    1.32e+01      0
  26  1.11e+04  1.50e+01  2.25e-03  4.00e+01    1.35e+01      0
  27  1.47e+04  1.52e+01  1.83e-03  4.22e+01    1.40e+01      0
  28  1.93e+04  1.56e+01  1.50e-03  4.45e+01    1.44e+01      0
  29  2.47e+04  1.61e+01  1.21e-03  4.61e+01    1.47e+01      0
  30  3.12e+04  1.67e+01  9.61e-04  4.67e+01    1.49e+01      0
  31  3.86e+04  1.73e+01  7.89e-04  4.77e+01    1.50e+01      0   Skip BFGS
  32  4.70e+04  1.78e+01  6.48e-04  4.83e+01    1.50e+01      0
  33  5.72e+04  1.82e+01  5.32e-04  4.86e+01    1.50e+01      0
  34  6.97e+04  1.84e+01  4.35e-04  4.88e+01    1.52e+01      0
  35  8.50e+04  1.86e+01  3.48e-04  4.81e+01    1.48e+01      0
  36  1.04e+05  1.85e+01  2.74e-04  4.68e+01    1.47e+01      0
  37  1.26e+05  1.83e+01  2.22e-04  4.63e+01    1.45e+01      0
  38  1.54e+05  1.81e+01  1.79e-04  4.56e+01    1.44e+01      0
  39  1.87e+05  1.81e+01  1.44e-04  4.51e+01    1.46e+01      0
  40  2.28e+05  1.83e+01  1.18e-04  4.53e+01    1.47e+01      0
  41  2.78e+05  1.86e+01  9.95e-05  4.63e+01    1.50e+01      0   Skip BFGS
  42  3.39e+05  1.90e+01  8.32e-05  4.71e+01    1.50e+01      0
  43  4.12e+05  1.93e+01  6.96e-05  4.81e+01    1.50e+01      0
  44  5.02e+05  1.98e+01  5.84e-05  4.91e+01    1.50e+01      0
  45  6.12e+05  2.02e+01  4.91e-05  5.03e+01    1.50e+01      0
  46  7.46e+05  2.06e+01  4.15e-05  5.16e+01    1.50e+01      0
  47  9.09e+05  2.08e+01  3.52e-05  5.28e+01    1.51e+01      0
Reach maximum number of IRLS cycles: 40
------------------------- STOP! -------------------------
1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.7318e+02
1 : |xc-x_last| = 5.4526e-02 <= tolX*(1+|x0|) = 1.0010e-01
0 : |proj(x-g)-x|    = 1.5134e+01 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 1.5134e+01 <= 1e3*eps       = 1.0000e-02
0 : 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)
  • plot inv 2 inversion irls
  • plot inv 2 inversion irls
Text(865.1527777777777, 0.5, '$\\phi_m$')

Total running time of the script: (0 minutes 41.839 seconds)

Estimated memory usage: 289 MB

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