.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "content/user-guide/tutorials/02-linear_inversion/plot_inv_1_inversion_lsq.py"
.. LINE NUMBERS ARE GIVEN BELOW.
.. only:: html
.. note::
:class: sphx-glr-download-link-note
:ref:`Go to the end <sphx_glr_download_content_user-guide_tutorials_02-linear_inversion_plot_inv_1_inversion_lsq.py>`
to download the full example code.
.. rst-class:: sphx-glr-example-title
.. _sphx_glr_content_user-guide_tutorials_02-linear_inversion_plot_inv_1_inversion_lsq.py:
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
.. GENERATED FROM PYTHON SOURCE LINES 18-21
Import Modules
--------------
.. GENERATED FROM PYTHON SOURCE LINES 21-41
.. code-block:: Python
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
.. GENERATED FROM PYTHON SOURCE LINES 42-48
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.
.. GENERATED FROM PYTHON SOURCE LINES 48-69
.. code-block:: Python
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])
.. image-sg:: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_1_inversion_lsq_001.png
:alt: plot inv 1 inversion lsq
:srcset: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_1_inversion_lsq_001.png
:class: sphx-glr-single-img
.. rst-class:: sphx-glr-script-out
.. code-block:: none
(-2.0, 2.0)
.. GENERATED FROM PYTHON SOURCE LINES 70-77
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.
.. GENERATED FROM PYTHON SOURCE LINES 77-108
.. code-block:: Python
# 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")
.. image-sg:: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_1_inversion_lsq_002.png
:alt: Columns of matrix G
:srcset: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_1_inversion_lsq_002.png
:class: sphx-glr-single-img
.. rst-class:: sphx-glr-script-out
.. code-block:: none
Text(0.5, 1.0, 'Columns of matrix G')
.. GENERATED FROM PYTHON SOURCE LINES 109-115
Defining the Simulation
-----------------------
The simulation defines the relationship between the model parameters and
predicted data.
.. GENERATED FROM PYTHON SOURCE LINES 115-118
.. code-block:: Python
sim = simulation.LinearSimulation(mesh, G=G, model_map=model_map)
.. GENERATED FROM PYTHON SOURCE LINES 119-125
Predict Synthetic Data
----------------------
Here, we use the true model to create synthetic data which we will subsequently
invert.
.. GENERATED FROM PYTHON SOURCE LINES 125-133
.. code-block:: Python
# 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)
.. GENERATED FROM PYTHON SOURCE LINES 134-143
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
.. GENERATED FROM PYTHON SOURCE LINES 143-159
.. code-block:: Python
# 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)
.. GENERATED FROM PYTHON SOURCE LINES 160-167
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.
.. GENERATED FROM PYTHON SOURCE LINES 167-179
.. code-block:: Python
# 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]
.. GENERATED FROM PYTHON SOURCE LINES 180-186
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.
.. GENERATED FROM PYTHON SOURCE LINES 186-196
.. code-block:: Python
# 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)
.. rst-class:: sphx-glr-script-out
.. code-block:: none
Running inversion with SimPEG v0.23.1.dev27+g1148ef1e9
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 1.82e+02 2.00e+05 0.00e+00 2.00e+05 2.51e+06 0
1 1.82e+02 9.36e+04 6.91e-01 9.38e+04 1.75e+05 0
2 1.82e+02 6.39e+04 2.59e+00 6.44e+04 1.14e+05 0
3 1.82e+02 3.76e+04 9.17e+00 3.93e+04 1.02e+05 0 Skip BFGS
4 1.82e+02 2.58e+04 9.20e+00 2.75e+04 2.09e+05 0
5 1.82e+02 1.83e+04 1.46e+01 2.10e+04 9.92e+04 0
6 1.82e+02 9.14e+03 2.35e+01 1.34e+04 1.22e+05 0
7 1.82e+02 7.27e+03 2.37e+01 1.16e+04 4.73e+04 0
8 1.82e+02 6.33e+03 2.52e+01 1.09e+04 1.63e+05 0
9 1.82e+02 5.03e+03 2.68e+01 9.90e+03 1.07e+05 0
10 1.82e+02 2.55e+03 3.28e+01 8.52e+03 1.04e+05 0
11 1.82e+02 2.38e+03 3.32e+01 8.41e+03 3.38e+05 0 Skip BFGS
12 1.82e+02 2.42e+03 3.25e+01 8.33e+03 1.09e+05 0
13 1.82e+02 2.35e+03 3.27e+01 8.31e+03 9.32e+04 0
14 1.82e+02 2.12e+03 3.30e+01 8.13e+03 6.72e+04 0
15 1.82e+02 1.92e+03 3.28e+01 7.90e+03 7.03e+04 0 Skip BFGS
16 1.82e+02 1.84e+03 3.30e+01 7.85e+03 7.38e+04 0
17 1.82e+02 1.32e+03 3.36e+01 7.44e+03 3.38e+04 0 Skip BFGS
18 1.82e+02 1.43e+03 3.29e+01 7.41e+03 1.30e+04 0
19 1.82e+02 1.29e+03 3.35e+01 7.39e+03 2.29e+04 0
20 1.82e+02 1.20e+03 3.39e+01 7.37e+03 2.68e+04 0 Skip BFGS
21 1.82e+02 1.25e+03 3.36e+01 7.35e+03 5.15e+04 0
22 1.82e+02 1.23e+03 3.35e+01 7.34e+03 5.39e+04 0 Skip BFGS
23 1.82e+02 1.20e+03 3.37e+01 7.33e+03 3.33e+04 0
24 1.82e+02 1.18e+03 3.38e+01 7.33e+03 3.51e+04 0 Skip BFGS
25 1.82e+02 1.19e+03 3.37e+01 7.33e+03 4.26e+04 0
26 1.82e+02 1.13e+03 3.41e+01 7.33e+03 8.80e+04 0 Skip BFGS
27 1.82e+02 1.18e+03 3.38e+01 7.32e+03 4.24e+04 0
28 1.82e+02 1.17e+03 3.37e+01 7.30e+03 2.12e+04 0 Skip BFGS
29 1.82e+02 1.12e+03 3.39e+01 7.29e+03 2.58e+04 0
30 1.82e+02 1.05e+03 3.42e+01 7.27e+03 3.23e+04 0 Skip BFGS
31 1.82e+02 1.05e+03 3.42e+01 7.27e+03 1.31e+04 0
32 1.82e+02 1.03e+03 3.43e+01 7.26e+03 9.97e+03 0 Skip BFGS
33 1.82e+02 1.03e+03 3.43e+01 7.26e+03 1.23e+04 0
34 1.82e+02 9.62e+02 3.46e+01 7.25e+03 4.13e+04 0 Skip BFGS
35 1.82e+02 9.98e+02 3.44e+01 7.25e+03 2.19e+04 0
36 1.82e+02 1.01e+03 3.43e+01 7.25e+03 8.91e+03 0 Skip BFGS
37 1.82e+02 1.01e+03 3.43e+01 7.25e+03 9.65e+03 0
38 1.82e+02 1.01e+03 3.43e+01 7.25e+03 1.43e+04 0 Skip BFGS
39 1.82e+02 1.01e+03 3.43e+01 7.25e+03 1.31e+04 0
40 1.82e+02 1.02e+03 3.42e+01 7.25e+03 1.12e+04 0
41 1.82e+02 1.02e+03 3.42e+01 7.25e+03 1.42e+04 0 Skip BFGS
42 1.82e+02 1.02e+03 3.42e+01 7.25e+03 1.20e+04 0
43 1.82e+02 1.03e+03 3.42e+01 7.25e+03 1.29e+04 0 Skip BFGS
44 1.82e+02 1.02e+03 3.42e+01 7.25e+03 1.20e+04 0
45 1.82e+02 1.02e+03 3.42e+01 7.25e+03 1.26e+04 0 Skip BFGS
46 1.82e+02 1.02e+03 3.42e+01 7.25e+03 1.21e+04 0
47 1.82e+02 1.03e+03 3.42e+01 7.25e+03 1.19e+04 0
48 1.82e+02 1.00e+03 3.43e+01 7.25e+03 1.03e+03 0
49 1.82e+02 1.00e+03 3.43e+01 7.25e+03 9.34e+02 0 Skip BFGS
50 1.82e+02 9.95e+02 3.44e+01 7.24e+03 2.77e+03 0
------------------------- STOP! -------------------------
1 : |fc-fOld| = 2.6387e-01 <= tolF*(1+|f0|) = 2.0000e+04
1 : |xc-x_last| = 9.1731e-03 <= tolX*(1+|x0|) = 1.0000e-01
0 : |proj(x-g)-x| = 2.7682e+03 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 2.7682e+03 <= 1e3*eps = 1.0000e-02
1 : maxIter = 50 <= iter = 50
------------------------- DONE! -------------------------
.. GENERATED FROM PYTHON SOURCE LINES 197-200
Plotting Results
----------------
.. GENERATED FROM PYTHON SOURCE LINES 200-212
.. code-block:: Python
# 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])
.. image-sg:: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_1_inversion_lsq_003.png
:alt: plot inv 1 inversion lsq
:srcset: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_1_inversion_lsq_003.png
:class: sphx-glr-single-img
.. rst-class:: sphx-glr-script-out
.. code-block:: none
(-2.0, 2.0)
.. rst-class:: sphx-glr-timing
**Total running time of the script:** (0 minutes 38.008 seconds)
**Estimated memory usage:** 295 MB
.. _sphx_glr_download_content_user-guide_tutorials_02-linear_inversion_plot_inv_1_inversion_lsq.py:
.. only:: html
.. container:: sphx-glr-footer sphx-glr-footer-example
.. container:: sphx-glr-download sphx-glr-download-jupyter
:download:`Download Jupyter notebook: plot_inv_1_inversion_lsq.ipynb <plot_inv_1_inversion_lsq.ipynb>`
.. container:: sphx-glr-download sphx-glr-download-python
:download:`Download Python source code: plot_inv_1_inversion_lsq.py <plot_inv_1_inversion_lsq.py>`
.. container:: sphx-glr-download sphx-glr-download-zip
:download:`Download zipped: plot_inv_1_inversion_lsq.zip <plot_inv_1_inversion_lsq.zip>`
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