.. 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 ` 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 v0.25.0 ============================ Inexact Gauss Newton ============================ # beta phi_d phi_m f |proj(x-g)-x| LS Comment ----------------------------------------------------------------------------- 0 1.86e+02 2.00e+05 0.00e+00 2.00e+05 1 1.86e+02 9.38e+04 6.90e-01 9.39e+04 2.47e+06 0 2 1.86e+02 6.41e+04 2.60e+00 6.46e+04 1.70e+05 0 3 1.86e+02 3.69e+04 9.50e+00 3.87e+04 1.16e+05 0 Skip BFGS 4 1.86e+02 2.50e+04 9.51e+00 2.67e+04 1.04e+05 0 5 1.86e+02 1.75e+04 1.51e+01 2.03e+04 1.96e+05 0 6 1.86e+02 8.96e+03 2.35e+01 1.33e+04 9.99e+04 0 7 1.86e+02 7.24e+03 2.38e+01 1.17e+04 1.22e+05 0 8 1.86e+02 6.35e+03 2.51e+01 1.10e+04 4.66e+04 0 9 1.86e+02 4.86e+03 2.71e+01 9.92e+03 1.55e+05 0 10 1.86e+02 2.73e+03 3.15e+01 8.60e+03 1.20e+05 0 11 1.86e+02 2.36e+03 3.27e+01 8.45e+03 1.01e+05 0 12 1.86e+02 2.18e+03 3.34e+01 8.40e+03 1.03e+05 0 13 1.86e+02 2.12e+03 3.26e+01 8.19e+03 9.40e+04 0 Skip BFGS 14 1.86e+02 1.94e+03 3.30e+01 8.08e+03 1.19e+05 0 15 1.86e+02 1.68e+03 3.29e+01 7.81e+03 8.71e+04 0 Skip BFGS 16 1.86e+02 1.68e+03 3.24e+01 7.71e+03 2.68e+04 0 17 1.86e+02 1.31e+03 3.37e+01 7.59e+03 2.68e+04 0 18 1.86e+02 1.30e+03 3.37e+01 7.58e+03 2.28e+04 0 Skip BFGS 19 1.86e+02 1.24e+03 3.40e+01 7.57e+03 2.27e+04 0 20 1.86e+02 1.18e+03 3.42e+01 7.55e+03 4.16e+04 0 Skip BFGS 21 1.86e+02 1.20e+03 3.40e+01 7.53e+03 2.31e+04 0 22 1.86e+02 1.24e+03 3.37e+01 7.51e+03 3.78e+04 0 Skip BFGS 23 1.86e+02 1.14e+03 3.42e+01 7.50e+03 3.87e+04 0 24 1.86e+02 1.08e+03 3.45e+01 7.49e+03 4.22e+04 0 Skip BFGS 25 1.86e+02 1.12e+03 3.42e+01 7.49e+03 4.40e+04 0 26 1.86e+02 1.13e+03 3.42e+01 7.48e+03 3.73e+04 0 Skip BFGS 27 1.86e+02 1.10e+03 3.43e+01 7.48e+03 3.25e+04 0 28 1.86e+02 1.06e+03 3.45e+01 7.47e+03 3.79e+04 0 Skip BFGS 29 1.86e+02 1.06e+03 3.45e+01 7.47e+03 4.33e+04 0 30 1.86e+02 1.06e+03 3.44e+01 7.47e+03 4.17e+04 0 Skip BFGS 31 1.86e+02 1.06e+03 3.45e+01 7.47e+03 4.21e+04 0 32 1.86e+02 9.81e+02 3.41e+01 7.33e+03 4.17e+04 0 Skip BFGS 33 1.86e+02 9.81e+02 3.41e+01 7.33e+03 7.66e+03 0 34 1.86e+02 9.76e+02 3.41e+01 7.33e+03 7.63e+03 0 35 1.86e+02 9.91e+02 3.40e+01 7.33e+03 1.54e+03 0 Skip BFGS 36 1.86e+02 9.93e+02 3.40e+01 7.33e+03 1.60e+03 0 37 1.86e+02 9.93e+02 3.40e+01 7.33e+03 1.34e+03 0 Skip BFGS 38 1.86e+02 9.93e+02 3.40e+01 7.33e+03 1.33e+03 0 39 1.86e+02 9.92e+02 3.40e+01 7.33e+03 1.34e+03 0 Skip BFGS 40 1.86e+02 9.93e+02 3.40e+01 7.33e+03 1.38e+03 0 41 1.86e+02 9.94e+02 3.40e+01 7.33e+03 1.50e+03 0 Skip BFGS 42 1.86e+02 9.92e+02 3.40e+01 7.33e+03 1.28e+03 0 43 1.86e+02 9.91e+02 3.40e+01 7.33e+03 4.06e+03 0 44 1.86e+02 9.90e+02 3.41e+01 7.33e+03 6.15e+02 0 Skip BFGS 45 1.86e+02 9.89e+02 3.41e+01 7.33e+03 2.90e+03 0 46 1.86e+02 9.91e+02 3.40e+01 7.33e+03 3.22e+03 0 Skip BFGS 47 1.86e+02 9.89e+02 3.41e+01 7.33e+03 2.94e+03 0 48 1.86e+02 9.88e+02 3.41e+01 7.33e+03 3.33e+03 0 49 1.86e+02 9.87e+02 3.41e+01 7.33e+03 1.06e+03 0 Skip BFGS 50 1.86e+02 9.88e+02 3.41e+01 7.33e+03 1.05e+03 0 ------------------------- STOP! ------------------------- 1 : |fc-fOld| = 3.0352e-03 <= tolF*(1+|f0|) = 2.0000e+04 1 : |xc-x_last| = 1.0995e-03 <= tolX*(1+|x0|) = 1.0000e-01 0 : |proj(x-g)-x| = 1.0573e+03 <= tolG = 1.0000e-01 0 : |proj(x-g)-x| = 1.0573e+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 34.446 seconds) **Estimated memory usage:** 319 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 ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_inv_1_inversion_lsq.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_inv_1_inversion_lsq.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_