Linear Least-Squares Inversion#

Here we demonstrate the basics of inverting data with SimPEG by considering a linear inverse problem. We formulate the inverse problem as a least-squares optimization problem. For this tutorial, we focus on the following:

Defining the forward problem

Defining the inverse problem (data misfit, regularization, optimization)

Specifying directives for the inversion

Recovering a set of model parameters which explains the observations

Import Modules#

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 = 0.01
np.random.seed(1)

# Create a SimPEG data object
data_obj = sim.make_synthetic_data(true_model, relative_error=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).
reg = regularization.WeightedLeastSquares(mesh, alpha_s=1.0, alpha_x=1.0)

# Define how the optimization problem is solved.
opt = optimization.InexactGaussNewton(maxIter=50)

# 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.

# Defining a starting value for the trade-off parameter (beta) between the data
# misfit and the regularization.
starting_beta = directives.BetaEstimate_ByEig(beta0_ratio=1e-4)

# Setting a stopping criteria for the inversion.
target_misfit = directives.TargetMisfit()

# The directives are defined as a list.
directives_list = [starting_beta, target_misfit]

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 = np.zeros(nParam)

# Run inversion
recovered_model = inv.run(starting_model)

Running inversion with SimPEG v0.25.0
============================ Inexact Gauss Newton ============================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS    Comment
-----------------------------------------------------------------------------
1.86e+02  2.00e+05  0.00e+00  2.00e+05
1.86e+02  9.38e+04  6.90e-01  9.39e+04    2.47e+06      0
1.86e+02  6.41e+04  2.60e+00  6.46e+04    1.70e+05      0
1.86e+02  3.69e+04  9.50e+00  3.87e+04    1.16e+05      0   Skip BFGS
1.86e+02  2.50e+04  9.51e+00  2.67e+04    1.04e+05      0
1.86e+02  1.75e+04  1.51e+01  2.03e+04    1.96e+05      0
1.86e+02  8.96e+03  2.35e+01  1.33e+04    9.99e+04      0
1.86e+02  7.24e+03  2.38e+01  1.17e+04    1.22e+05      0
1.86e+02  6.35e+03  2.51e+01  1.10e+04    4.66e+04      0
1.86e+02  4.86e+03  2.71e+01  9.92e+03    1.55e+05      0
1.86e+02  2.73e+03  3.15e+01  8.60e+03    1.20e+05      0
1.86e+02  2.36e+03  3.27e+01  8.45e+03    1.01e+05      0
1.86e+02  2.18e+03  3.34e+01  8.40e+03    1.03e+05      0
1.86e+02  2.12e+03  3.26e+01  8.19e+03    9.40e+04      0   Skip BFGS
1.86e+02  1.94e+03  3.30e+01  8.08e+03    1.19e+05      0
1.86e+02  1.68e+03  3.29e+01  7.81e+03    8.71e+04      0   Skip BFGS
1.86e+02  1.68e+03  3.24e+01  7.71e+03    2.68e+04      0
1.86e+02  1.31e+03  3.37e+01  7.59e+03    2.68e+04      0
1.86e+02  1.30e+03  3.37e+01  7.58e+03    2.28e+04      0   Skip BFGS
1.86e+02  1.24e+03  3.40e+01  7.57e+03    2.27e+04      0
1.86e+02  1.18e+03  3.42e+01  7.55e+03    4.16e+04      0   Skip BFGS
1.86e+02  1.20e+03  3.40e+01  7.53e+03    2.31e+04      0
1.86e+02  1.24e+03  3.37e+01  7.51e+03    3.78e+04      0   Skip BFGS
1.86e+02  1.14e+03  3.42e+01  7.50e+03    3.87e+04      0
1.86e+02  1.08e+03  3.45e+01  7.49e+03    4.22e+04      0   Skip BFGS
1.86e+02  1.12e+03  3.42e+01  7.49e+03    4.40e+04      0
1.86e+02  1.13e+03  3.42e+01  7.48e+03    3.73e+04      0   Skip BFGS
1.86e+02  1.10e+03  3.43e+01  7.48e+03    3.25e+04      0
1.86e+02  1.06e+03  3.45e+01  7.47e+03    3.79e+04      0   Skip BFGS
1.86e+02  1.06e+03  3.45e+01  7.47e+03    4.33e+04      0
1.86e+02  1.06e+03  3.44e+01  7.47e+03    4.17e+04      0   Skip BFGS
1.86e+02  1.06e+03  3.45e+01  7.47e+03    4.21e+04      0
1.86e+02  9.81e+02  3.41e+01  7.33e+03    4.17e+04      0   Skip BFGS
1.86e+02  9.81e+02  3.41e+01  7.33e+03    7.66e+03      0
1.86e+02  9.76e+02  3.41e+01  7.33e+03    7.63e+03      0
1.86e+02  9.91e+02  3.40e+01  7.33e+03    1.54e+03      0   Skip BFGS
1.86e+02  9.93e+02  3.40e+01  7.33e+03    1.60e+03      0
1.86e+02  9.93e+02  3.40e+01  7.33e+03    1.34e+03      0   Skip BFGS
1.86e+02  9.93e+02  3.40e+01  7.33e+03    1.33e+03      0
1.86e+02  9.92e+02  3.40e+01  7.33e+03    1.34e+03      0   Skip BFGS
1.86e+02  9.93e+02  3.40e+01  7.33e+03    1.38e+03      0
1.86e+02  9.94e+02  3.40e+01  7.33e+03    1.50e+03      0   Skip BFGS
1.86e+02  9.92e+02  3.40e+01  7.33e+03    1.28e+03      0
1.86e+02  9.91e+02  3.40e+01  7.33e+03    4.06e+03      0
1.86e+02  9.90e+02  3.41e+01  7.33e+03    6.15e+02      0   Skip BFGS
1.86e+02  9.89e+02  3.41e+01  7.33e+03    2.90e+03      0
1.86e+02  9.91e+02  3.40e+01  7.33e+03    3.22e+03      0   Skip BFGS
1.86e+02  9.89e+02  3.41e+01  7.33e+03    2.94e+03      0
1.86e+02  9.88e+02  3.41e+01  7.33e+03    3.33e+03      0
1.86e+02  9.87e+02  3.41e+01  7.33e+03    1.06e+03      0   Skip BFGS
1.86e+02  9.88e+02  3.41e+01  7.33e+03    1.05e+03      0
------------------------- STOP! -------------------------
: |fc-fOld| = 3.0352e-03 <= tolF*(1+|f0|) = 2.0000e+04
: |xc-x_last| = 1.0995e-03 <= tolX*(1+|x0|) = 1.0000e-01
: |proj(x-g)-x|    = 1.0573e+03 <= tolG          = 1.0000e-01
: |proj(x-g)-x|    = 1.0573e+03 <= 1e3*eps       = 1.0000e-02
: maxIter   =      50    <= iter          =     50
------------------------- DONE! -------------------------

Plotting Results#

# Observed versus predicted data
fig, ax = plt.subplots(1, 2, figsize=(12 * 1.2, 4 * 1.2))
ax[0].plot(data_obj.dobs, "b-")
ax[0].plot(inv_prob.dpred, "r-")
ax[0].legend(("Observed Data", "Predicted Data"))

# True versus recovered model
ax[1].plot(mesh.cell_centers_x, true_model, "b-")
ax[1].plot(mesh.cell_centers_x, recovered_model, "r-")
ax[1].legend(("True Model", "Recovered Model"))
ax[1].set_ylim([-2, 2])