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])
plot inv 1 inversion lsq
(-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 = 0.01
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:

  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).
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)
SimPEG.InvProblem will set Regularization.reference_model to m0.
SimPEG.InvProblem will set Regularization.reference_model to m0.
SimPEG.InvProblem will set Regularization.reference_model to m0.

                    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
============================ Inexact Gauss Newton ============================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS    Comment
-----------------------------------------------------------------------------
x0 has any nan: 0
   0  1.86e+02  1.00e+05  0.00e+00  1.00e+05    1.26e+06      0
   1  1.86e+02  4.68e+04  3.50e-01  4.68e+04    8.00e+04      0
   2  1.86e+02  3.18e+04  1.35e+00  3.21e+04    5.90e+04      0
   3  1.86e+02  1.75e+04  5.21e+00  1.85e+04    5.36e+04      0   Skip BFGS
   4  1.86e+02  1.16e+04  5.18e+00  1.25e+04    8.12e+04      0
   5  1.86e+02  8.05e+03  8.09e+00  9.55e+03    4.44e+04      0
   6  1.86e+02  4.39e+03  1.19e+01  6.59e+03    5.39e+04      0
   7  1.86e+02  3.33e+03  1.24e+01  5.65e+03    2.83e+04      0
   8  1.86e+02  2.87e+03  1.31e+01  5.30e+03    8.05e+04      0
   9  1.86e+02  2.38e+03  1.38e+01  4.95e+03    3.99e+04      0
  10  1.86e+02  1.60e+03  1.55e+01  4.48e+03    6.49e+04      0
  11  1.86e+02  1.32e+03  1.64e+01  4.37e+03    6.36e+04      0
  12  1.86e+02  1.19e+03  1.66e+01  4.28e+03    4.79e+04      0
  13  1.86e+02  1.01e+03  1.72e+01  4.21e+03    6.98e+04      0
  14  1.86e+02  8.37e+02  1.69e+01  3.98e+03    9.79e+03      0   Skip BFGS
  15  1.86e+02  7.72e+02  1.69e+01  3.92e+03    2.84e+04      0
  16  1.86e+02  5.85e+02  1.72e+01  3.79e+03    1.35e+04      0   Skip BFGS
  17  1.86e+02  6.35e+02  1.68e+01  3.77e+03    2.13e+04      0
  18  1.86e+02  6.40e+02  1.68e+01  3.77e+03    2.09e+04      0   Skip BFGS
  19  1.86e+02  6.35e+02  1.68e+01  3.77e+03    2.37e+04      0
  20  1.86e+02  6.09e+02  1.69e+01  3.76e+03    2.20e+04      0   Skip BFGS
  21  1.86e+02  6.04e+02  1.69e+01  3.76e+03    2.52e+04      0
  22  1.86e+02  5.58e+02  1.72e+01  3.75e+03    2.92e+04      0   Skip BFGS
  23  1.86e+02  5.70e+02  1.71e+01  3.75e+03    2.46e+04      0
  24  1.86e+02  5.91e+02  1.70e+01  3.75e+03    2.40e+04      0   Skip BFGS
  25  1.86e+02  5.87e+02  1.70e+01  3.75e+03    2.36e+04      0
  26  1.86e+02  5.89e+02  1.70e+01  3.75e+03    2.58e+04      0
  27  1.86e+02  5.87e+02  1.70e+01  3.75e+03    2.44e+04      0   Skip BFGS
  28  1.86e+02  5.89e+02  1.70e+01  3.75e+03    2.59e+04      0
  29  1.86e+02  5.77e+02  1.70e+01  3.74e+03    2.47e+04      0   Skip BFGS
  30  1.86e+02  5.77e+02  1.70e+01  3.74e+03    2.48e+04      0
  31  1.86e+02  5.27e+02  1.72e+01  3.72e+03    8.88e+03      0   Skip BFGS
  32  1.86e+02  5.26e+02  1.72e+01  3.72e+03    8.87e+03      0
  33  1.86e+02  5.43e+02  1.71e+01  3.72e+03    7.55e+03      0
  34  1.86e+02  5.31e+02  1.71e+01  3.72e+03    4.64e+03      0
  35  1.86e+02  5.32e+02  1.71e+01  3.71e+03    1.37e+03      0   Skip BFGS
  36  1.86e+02  5.27e+02  1.71e+01  3.71e+03    4.06e+03      0
  37  1.86e+02  5.15e+02  1.72e+01  3.71e+03    2.42e+03      0   Skip BFGS
  38  1.86e+02  5.17e+02  1.72e+01  3.71e+03    4.50e+03      0
  39  1.86e+02  5.12e+02  1.72e+01  3.71e+03    4.44e+03      0   Skip BFGS
  40  1.86e+02  5.12e+02  1.72e+01  3.71e+03    5.18e+03      0
  41  1.86e+02  5.11e+02  1.72e+01  3.71e+03    3.87e+03      0
  42  1.86e+02  5.09e+02  1.72e+01  3.71e+03    4.40e+03      0   Skip BFGS
  43  1.86e+02  5.10e+02  1.72e+01  3.71e+03    4.38e+03      0
  44  1.86e+02  5.10e+02  1.72e+01  3.71e+03    4.54e+03      0   Skip BFGS
  45  1.86e+02  5.10e+02  1.72e+01  3.71e+03    4.43e+03      0
  46  1.86e+02  5.10e+02  1.72e+01  3.71e+03    4.59e+03      0   Skip BFGS
  47  1.86e+02  5.10e+02  1.72e+01  3.71e+03    4.54e+03      0
  48  1.86e+02  5.11e+02  1.72e+01  3.71e+03    4.72e+03      0
  49  1.86e+02  5.10e+02  1.72e+01  3.71e+03    4.51e+02      0
  50  1.86e+02  5.09e+02  1.72e+01  3.71e+03    1.30e+03      0
------------------------- STOP! -------------------------
1 : |fc-fOld| = 9.9288e-03 <= tolF*(1+|f0|) = 1.0000e+04
1 : |xc-x_last| = 2.2611e-03 <= tolX*(1+|x0|) = 1.0000e-01
0 : |proj(x-g)-x|    = 1.3031e+03 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 1.3031e+03 <= 1e3*eps       = 1.0000e-02
1 : 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])
plot inv 1 inversion lsq
(-2.0, 2.0)

Total running time of the script: ( 0 minutes 23.719 seconds)

Estimated memory usage: 18 MB

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