.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "content/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_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/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_1_inversion_lsq_001.png :alt: plot inv 1 inversion lsq :srcset: /content/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/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_1_inversion_lsq_002.png :alt: Columns of matrix G :srcset: /content/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 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 1.84e+02 2.00e+05 0.00e+00 2.00e+05 2.49e+06 0 1 1.84e+02 9.35e+04 7.07e-01 9.36e+04 1.67e+05 0 2 1.84e+02 6.37e+04 2.65e+00 6.42e+04 1.15e+05 0 3 1.84e+02 3.57e+04 1.00e+01 3.75e+04 1.07e+05 0 Skip BFGS 4 1.84e+02 2.38e+04 1.00e+01 2.56e+04 1.87e+05 0 5 1.84e+02 1.66e+04 1.58e+01 1.95e+04 1.00e+05 0 6 1.84e+02 8.97e+03 2.36e+01 1.33e+04 1.24e+05 0 7 1.84e+02 7.22e+03 2.41e+01 1.17e+04 4.68e+04 0 8 1.84e+02 6.36e+03 2.54e+01 1.10e+04 1.51e+05 0 9 1.84e+02 4.64e+03 2.81e+01 9.81e+03 1.24e+05 0 10 1.84e+02 3.22e+03 3.04e+01 8.82e+03 1.18e+05 0 11 1.84e+02 2.45e+03 3.29e+01 8.51e+03 1.13e+05 0 12 1.84e+02 2.23e+03 3.29e+01 8.29e+03 9.62e+04 0 13 1.84e+02 1.98e+03 3.40e+01 8.24e+03 9.51e+04 0 14 1.84e+02 1.47e+03 3.49e+01 7.89e+03 4.20e+04 0 Skip BFGS 15 1.84e+02 1.30e+03 3.51e+01 7.77e+03 4.97e+04 0 16 1.84e+02 1.27e+03 3.47e+01 7.65e+03 3.24e+04 0 Skip BFGS 17 1.84e+02 1.13e+03 3.52e+01 7.61e+03 5.23e+04 0 18 1.84e+02 1.08e+03 3.50e+01 7.52e+03 3.06e+04 0 Skip BFGS 19 1.84e+02 1.09e+03 3.49e+01 7.52e+03 2.41e+04 0 20 1.84e+02 1.07e+03 3.50e+01 7.50e+03 2.75e+04 0 Skip BFGS 21 1.84e+02 1.10e+03 3.47e+01 7.50e+03 2.28e+04 0 22 1.84e+02 1.13e+03 3.45e+01 7.48e+03 2.47e+04 0 Skip BFGS 23 1.84e+02 1.13e+03 3.45e+01 7.48e+03 2.34e+04 0 24 1.84e+02 1.22e+03 3.39e+01 7.47e+03 2.68e+04 0 Skip BFGS 25 1.84e+02 1.13e+03 3.44e+01 7.46e+03 2.34e+04 0 26 1.84e+02 1.05e+03 3.47e+01 7.45e+03 2.15e+04 0 Skip BFGS 27 1.84e+02 1.06e+03 3.47e+01 7.45e+03 2.11e+04 0 28 1.84e+02 1.05e+03 3.47e+01 7.44e+03 2.17e+04 0 Skip BFGS 29 1.84e+02 1.05e+03 3.47e+01 7.44e+03 2.09e+04 0 30 1.84e+02 1.10e+03 3.41e+01 7.38e+03 1.94e+04 0 Skip BFGS 31 1.84e+02 1.10e+03 3.41e+01 7.38e+03 1.84e+04 0 32 1.84e+02 1.10e+03 3.41e+01 7.38e+03 1.90e+04 0 Skip BFGS 33 1.84e+02 1.10e+03 3.41e+01 7.38e+03 1.71e+04 0 34 1.84e+02 1.09e+03 3.41e+01 7.38e+03 1.82e+04 0 35 1.84e+02 1.11e+03 3.40e+01 7.38e+03 1.62e+04 0 36 1.84e+02 1.03e+03 3.45e+01 7.37e+03 9.02e+03 0 37 1.84e+02 1.03e+03 3.45e+01 7.37e+03 8.65e+03 0 Skip BFGS 38 1.84e+02 1.03e+03 3.45e+01 7.37e+03 9.00e+03 0 39 1.84e+02 1.03e+03 3.44e+01 7.37e+03 9.57e+03 0 Skip BFGS 40 1.84e+02 1.03e+03 3.45e+01 7.37e+03 9.25e+03 0 41 1.84e+02 1.03e+03 3.45e+01 7.37e+03 8.97e+03 0 Skip BFGS 42 1.84e+02 1.02e+03 3.45e+01 7.37e+03 9.02e+03 0 43 1.84e+02 1.03e+03 3.44e+01 7.37e+03 8.84e+03 0 44 1.84e+02 1.03e+03 3.44e+01 7.37e+03 8.87e+03 0 Skip BFGS 45 1.84e+02 1.03e+03 3.44e+01 7.37e+03 9.48e+03 0 46 1.84e+02 1.03e+03 3.44e+01 7.37e+03 9.90e+03 0 Skip BFGS 47 1.84e+02 1.03e+03 3.44e+01 7.37e+03 9.65e+03 0 48 1.84e+02 1.03e+03 3.44e+01 7.37e+03 9.62e+03 0 Skip BFGS 49 1.84e+02 1.03e+03 3.44e+01 7.37e+03 9.67e+03 0 50 1.84e+02 1.03e+03 3.44e+01 7.37e+03 1.93e+03 0 ------------------------- STOP! ------------------------- 1 : |fc-fOld| = 4.2751e-02 <= tolF*(1+|f0|) = 2.0000e+04 1 : |xc-x_last| = 3.6829e-03 <= tolX*(1+|x0|) = 1.0000e-01 0 : |proj(x-g)-x| = 1.9348e+03 <= tolG = 1.0000e-01 0 : |proj(x-g)-x| = 1.9348e+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/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_1_inversion_lsq_003.png :alt: plot inv 1 inversion lsq :srcset: /content/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 21.231 seconds) **Estimated memory usage:** 8 MB .. _sphx_glr_download_content_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 ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_