.. 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 Running inversion with SimPEG v0.22.0 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 1.85e+02 2.00e+05 0.00e+00 2.00e+05 2.48e+06 0 1 1.85e+02 9.39e+04 6.94e-01 9.40e+04 1.68e+05 0 2 1.85e+02 6.40e+04 2.65e+00 6.45e+04 1.17e+05 0 3 1.85e+02 3.67e+04 9.71e+00 3.85e+04 9.84e+04 0 Skip BFGS 4 1.85e+02 2.48e+04 9.68e+00 2.66e+04 1.76e+05 0 5 1.85e+02 1.74e+04 1.53e+01 2.02e+04 8.73e+04 0 6 1.85e+02 9.10e+03 2.35e+01 1.34e+04 1.14e+05 0 7 1.85e+02 7.38e+03 2.39e+01 1.18e+04 5.36e+04 0 8 1.85e+02 6.43e+03 2.53e+01 1.11e+04 1.37e+05 0 9 1.85e+02 2.94e+03 3.17e+01 8.79e+03 1.69e+05 0 Skip BFGS 10 1.85e+02 2.65e+03 3.07e+01 8.32e+03 5.77e+04 0 11 1.85e+02 2.26e+03 3.24e+01 8.24e+03 1.03e+05 0 12 1.85e+02 2.02e+03 3.28e+01 8.09e+03 6.98e+04 0 13 1.85e+02 1.92e+03 3.29e+01 8.01e+03 9.21e+04 0 Skip BFGS 14 1.85e+02 1.80e+03 3.32e+01 7.94e+03 7.21e+04 0 15 1.85e+02 1.73e+03 3.34e+01 7.91e+03 7.71e+04 0 Skip BFGS 16 1.85e+02 1.75e+03 3.32e+01 7.88e+03 7.76e+04 0 17 1.85e+02 1.11e+03 3.53e+01 7.63e+03 3.05e+04 0 Skip BFGS 18 1.85e+02 1.15e+03 3.48e+01 7.59e+03 5.26e+04 0 19 1.85e+02 1.10e+03 3.45e+01 7.48e+03 2.85e+04 0 Skip BFGS 20 1.85e+02 1.12e+03 3.43e+01 7.47e+03 3.39e+04 0 21 1.85e+02 1.14e+03 3.43e+01 7.47e+03 2.47e+04 0 22 1.85e+02 1.07e+03 3.41e+01 7.38e+03 2.03e+04 0 Skip BFGS 23 1.85e+02 1.14e+03 3.37e+01 7.38e+03 1.87e+04 0 24 1.85e+02 1.15e+03 3.37e+01 7.38e+03 1.81e+04 0 Skip BFGS 25 1.85e+02 1.14e+03 3.37e+01 7.38e+03 1.89e+04 0 26 1.85e+02 1.15e+03 3.37e+01 7.38e+03 1.76e+04 0 27 1.85e+02 1.14e+03 3.37e+01 7.38e+03 2.07e+04 0 Skip BFGS 28 1.85e+02 1.15e+03 3.37e+01 7.38e+03 1.76e+04 0 29 1.85e+02 1.14e+03 3.37e+01 7.37e+03 1.77e+04 0 Skip BFGS 30 1.85e+02 1.13e+03 3.37e+01 7.37e+03 1.77e+04 0 31 1.85e+02 1.13e+03 3.38e+01 7.37e+03 1.74e+04 0 Skip BFGS 32 1.85e+02 1.13e+03 3.37e+01 7.37e+03 1.77e+04 0 33 1.85e+02 1.11e+03 3.38e+01 7.36e+03 2.30e+04 0 Skip BFGS 34 1.85e+02 1.11e+03 3.38e+01 7.36e+03 1.54e+04 0 35 1.85e+02 1.06e+03 3.40e+01 7.34e+03 9.04e+03 0 Skip BFGS 36 1.85e+02 1.04e+03 3.41e+01 7.34e+03 3.69e+04 0 37 1.85e+02 9.84e+02 3.44e+01 7.34e+03 1.23e+04 0 Skip BFGS 38 1.85e+02 9.84e+02 3.44e+01 7.34e+03 1.31e+04 0 39 1.85e+02 9.90e+02 3.43e+01 7.34e+03 6.94e+03 0 Skip BFGS 40 1.85e+02 9.89e+02 3.43e+01 7.34e+03 1.07e+04 0 41 1.85e+02 9.95e+02 3.43e+01 7.34e+03 1.07e+04 0 Skip BFGS 42 1.85e+02 9.91e+02 3.43e+01 7.34e+03 7.73e+03 0 43 1.85e+02 9.88e+02 3.43e+01 7.34e+03 3.62e+03 0 Skip BFGS 44 1.85e+02 9.92e+02 3.43e+01 7.34e+03 8.05e+03 0 45 1.85e+02 9.93e+02 3.43e+01 7.34e+03 1.15e+04 0 Skip BFGS 46 1.85e+02 9.94e+02 3.43e+01 7.34e+03 1.18e+04 0 Skip BFGS 47 1.85e+02 9.94e+02 3.43e+01 7.34e+03 1.17e+04 0 48 1.85e+02 9.94e+02 3.43e+01 7.34e+03 1.15e+04 0 Skip BFGS 49 1.85e+02 9.94e+02 3.43e+01 7.34e+03 1.17e+04 0 50 1.85e+02 9.95e+02 3.43e+01 7.34e+03 1.05e+04 0 ------------------------- STOP! ------------------------- 1 : |fc-fOld| = 3.0090e-03 <= tolF*(1+|f0|) = 2.0000e+04 1 : |xc-x_last| = 8.0228e-04 <= tolX*(1+|x0|) = 1.0000e-01 0 : |proj(x-g)-x| = 1.0499e+04 <= tolG = 1.0000e-01 0 : |proj(x-g)-x| = 1.0499e+04 <= 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 29.288 seconds) **Estimated memory usage:** 9 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 `_