Note
Go to the end to download the full example code.
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.
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)
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 2.00e+05 0.00e+00 2.00e+05 2.46e+06 0
1 1.86e+02 9.37e+04 7.02e-01 9.39e+04 1.61e+05 0
2 1.86e+02 6.39e+04 2.69e+00 6.44e+04 1.23e+05 0
3 1.86e+02 3.67e+04 9.82e+00 3.85e+04 9.03e+04 0 Skip BFGS
4 1.86e+02 2.46e+04 9.78e+00 2.64e+04 1.50e+05 0
5 1.86e+02 1.72e+04 1.54e+01 2.01e+04 7.20e+04 0
6 1.86e+02 8.77e+03 2.38e+01 1.32e+04 8.85e+04 0
7 1.86e+02 6.97e+03 2.42e+01 1.15e+04 4.69e+04 0
8 1.86e+02 6.02e+03 2.57e+01 1.08e+04 1.26e+05 0
9 1.86e+02 5.07e+03 2.73e+01 1.01e+04 7.30e+04 0
10 1.86e+02 3.86e+03 2.96e+01 9.35e+03 1.01e+05 0
11 1.86e+02 3.09e+03 3.14e+01 8.91e+03 9.31e+04 0
12 1.86e+02 1.91e+03 3.37e+01 8.16e+03 7.74e+04 0 Skip BFGS
13 1.86e+02 1.98e+03 3.27e+01 8.05e+03 6.27e+04 0
14 1.86e+02 1.78e+03 3.31e+01 7.92e+03 1.22e+05 0
15 1.86e+02 1.53e+03 3.32e+01 7.68e+03 5.31e+04 0 Skip BFGS
16 1.86e+02 1.40e+03 3.37e+01 7.64e+03 5.15e+04 0
17 1.86e+02 1.33e+03 3.40e+01 7.63e+03 4.48e+04 0
18 1.86e+02 1.18e+03 3.46e+01 7.61e+03 3.53e+04 0 Skip BFGS
19 1.86e+02 1.26e+03 3.41e+01 7.59e+03 4.13e+04 0
20 1.86e+02 1.28e+03 3.36e+01 7.51e+03 4.60e+04 0 Skip BFGS
21 1.86e+02 1.18e+03 3.39e+01 7.48e+03 3.93e+04 0
22 1.86e+02 1.15e+03 3.41e+01 7.48e+03 4.13e+04 0 Skip BFGS
23 1.86e+02 1.17e+03 3.39e+01 7.47e+03 3.75e+04 0
24 1.86e+02 1.16e+03 3.39e+01 7.45e+03 2.45e+04 0 Skip BFGS
25 1.86e+02 1.15e+03 3.40e+01 7.45e+03 2.46e+04 0
26 1.86e+02 1.13e+03 3.40e+01 7.45e+03 2.33e+04 0 Skip BFGS
27 1.86e+02 1.13e+03 3.41e+01 7.45e+03 2.57e+04 0
28 1.86e+02 1.12e+03 3.41e+01 7.45e+03 2.61e+04 0 Skip BFGS
29 1.86e+02 1.12e+03 3.41e+01 7.45e+03 2.34e+04 0
30 1.86e+02 1.12e+03 3.41e+01 7.45e+03 2.49e+04 0
31 1.86e+02 1.01e+03 3.45e+01 7.42e+03 3.76e+04 0 Skip BFGS
32 1.86e+02 9.87e+02 3.46e+01 7.41e+03 1.28e+04 0
33 1.86e+02 9.90e+02 3.46e+01 7.41e+03 7.25e+03 0 Skip BFGS
34 1.86e+02 1.00e+03 3.45e+01 7.41e+03 8.28e+03 0
35 1.86e+02 1.00e+03 3.45e+01 7.41e+03 3.51e+03 0 Skip BFGS
36 1.86e+02 1.01e+03 3.45e+01 7.41e+03 2.65e+03 0
37 1.86e+02 1.00e+03 3.45e+01 7.41e+03 2.46e+03 0
38 1.86e+02 1.00e+03 3.45e+01 7.41e+03 2.70e+03 0 Skip BFGS
39 1.86e+02 1.00e+03 3.45e+01 7.41e+03 2.47e+03 0
40 1.86e+02 1.02e+03 3.44e+01 7.41e+03 3.93e+03 0 Skip BFGS
41 1.86e+02 1.01e+03 3.45e+01 7.41e+03 2.63e+03 0
42 1.86e+02 1.02e+03 3.44e+01 7.41e+03 3.50e+03 0 Skip BFGS
43 1.86e+02 1.02e+03 3.44e+01 7.41e+03 3.09e+03 0
44 1.86e+02 1.01e+03 3.44e+01 7.41e+03 1.52e+03 0
45 1.86e+02 1.02e+03 3.44e+01 7.40e+03 3.14e+03 0 Skip BFGS
46 1.86e+02 1.01e+03 3.44e+01 7.40e+03 1.53e+03 0
47 1.86e+02 1.01e+03 3.45e+01 7.40e+03 1.34e+03 0 Skip BFGS
48 1.86e+02 1.01e+03 3.45e+01 7.40e+03 1.47e+03 0
49 1.86e+02 1.01e+03 3.45e+01 7.40e+03 1.43e+03 0 Skip BFGS
50 1.86e+02 1.01e+03 3.45e+01 7.40e+03 1.44e+03 0
------------------------- STOP! -------------------------
1 : |fc-fOld| = 1.8914e-04 <= tolF*(1+|f0|) = 2.0000e+04
1 : |xc-x_last| = 4.3145e-04 <= tolX*(1+|x0|) = 1.0000e-01
0 : |proj(x-g)-x| = 1.4382e+03 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 1.4382e+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])
(-2.0, 2.0)
Total running time of the script: (0 minutes 27.241 seconds)
Estimated memory usage: 8 MB