.. 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_2_inversion_irls.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_2_inversion_irls.py: Sparse Inversion with Iteratively Re-Weighted Least-Squares =========================================================== Least-squares inversion produces smooth models which may not be an accurate representation of the true model. Here we demonstrate the basics of inverting for sparse and/or blocky models. Here, we used the iteratively reweighted least-squares approach. For this tutorial, we focus on the following: - Defining the forward problem - Defining the inverse problem (data misfit, regularization, optimization) - Defining the paramters for the IRLS algorithm - Specifying directives for the inversion - Recovering a set of model parameters which explains the observations .. GENERATED FROM PYTHON SOURCE LINES 18-37 .. 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 38-44 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 44-65 .. 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_2_inversion_irls_001.png :alt: plot inv 2 inversion irls :srcset: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_2_inversion_irls_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 66-73 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 73-104 .. 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_2_inversion_irls_002.png :alt: Columns of matrix G :srcset: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_2_inversion_irls_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 105-111 Defining the Simulation ----------------------- The simulation defines the relationship between the model parameters and predicted data. .. GENERATED FROM PYTHON SOURCE LINES 111-115 .. code-block:: Python sim = simulation.LinearSimulation(mesh, G=G, model_map=model_map) .. GENERATED FROM PYTHON SOURCE LINES 116-122 Predict Synthetic Data ---------------------- Here, we use the true model to create synthetic data which we will subsequently invert. .. GENERATED FROM PYTHON SOURCE LINES 122-130 .. code-block:: Python # Standard deviation of Gaussian noise being added std = 0.02 np.random.seed(1) # Create a SimPEG data object data_obj = sim.make_synthetic_data(true_model, noise_floor=std, add_noise=True) .. GENERATED FROM PYTHON SOURCE LINES 131-140 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 140-164 .. 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). Here, 'p' defines the # the norm of the smallness term and 'q' defines the norm of the smoothness # term. reg = regularization.Sparse(mesh, mapping=model_map) reg.reference_model = np.zeros(nParam) p = 0.0 q = 0.0 reg.norms = [p, q] # Define how the optimization problem is solved. opt = optimization.ProjectedGNCG( maxIter=100, lower=-2.0, upper=2.0, maxIterLS=20, cg_maxiter=30, cg_rtol=1e-3 ) # Here we define the inverse problem that is to be solved inv_prob = inverse_problem.BaseInvProblem(dmis, reg, opt) .. GENERATED FROM PYTHON SOURCE LINES 165-172 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 172-199 .. code-block:: Python # Add sensitivity weights but don't update at each beta sensitivity_weights = directives.UpdateSensitivityWeights(every_iteration=False) # Reach target misfit for L2 solution, then use IRLS until model stops changing. IRLS = directives.UpdateIRLS(max_irls_iterations=40, f_min_change=1e-4) # Defining a starting value for the trade-off parameter (beta) between the data # misfit and the regularization. starting_beta = directives.BetaEstimate_ByEig(beta0_ratio=1e0) # Update the preconditionner update_Jacobi = directives.UpdatePreconditioner() # Save output at each iteration saveDict = directives.SaveOutputEveryIteration(save_txt=False) # Define the directives as a list directives_list = [ sensitivity_weights, IRLS, starting_beta, update_Jacobi, saveDict, ] .. rst-class:: sphx-glr-script-out .. code-block:: none /home/vsts/work/1/s/simpeg/directives/_directives.py:1865: FutureWarning: SaveEveryIteration.save_txt has been deprecated, please use SaveEveryIteration.on_disk. It will be removed in version 0.26.0 of SimPEG. /home/vsts/work/1/s/simpeg/directives/_directives.py:1866: FutureWarning: SaveEveryIteration.save_txt has been deprecated, please use SaveEveryIteration.on_disk. It will be removed in version 0.26.0 of SimPEG. .. GENERATED FROM PYTHON SOURCE LINES 200-206 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 206-216 .. 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 = 1e-4 * np.ones(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 ================================================= Projected GNCG ================================================= # beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment ----------------------------------------------------------------------------------------------------------------- 0 2.78e+06 3.78e+03 1.01e-09 3.78e+03 0 inf inf 1 2.78e+06 2.32e+03 2.01e-04 2.88e+03 1.96e+01 0 7 8.45e-04 4.07e+00 2 1.39e+06 1.75e+03 5.04e-04 2.45e+03 1.92e+01 0 8 5.66e-04 5.78e-01 3 6.94e+05 1.14e+03 1.13e-03 1.93e+03 1.89e+01 0 9 3.90e-04 2.96e-01 4 3.47e+05 6.40e+02 2.16e-03 1.39e+03 1.83e+01 0 10 4.67e-04 2.57e-01 5 1.74e+05 3.07e+02 3.50e-03 9.14e+02 1.72e+01 0 12 4.43e-04 1.68e-01 6 8.68e+04 1.33e+02 4.88e-03 5.57e+02 1.56e+01 0 14 5.87e-04 1.40e-01 7 4.34e+04 5.75e+01 6.07e-03 3.21e+02 1.39e+01 0 17 7.73e-04 1.09e-01 8 2.17e+04 2.90e+01 6.96e-03 1.80e+02 1.28e+01 0 22 9.47e-04 7.39e-02 9 1.08e+04 1.85e+01 7.63e-03 1.01e+02 1.13e+01 0 29 8.41e-04 3.53e-02 Reached starting chifact with l2-norm regularization: Start IRLS steps... irls_threshold 1.189250648941873 10 1.08e+04 2.54e+01 9.33e-03 1.27e+02 1.51e+01 0 30 1.33e-03 3.38e-02 11 8.35e+03 2.62e+01 1.06e-02 1.15e+02 3.16e+00 0 30 2.20e-03 1.83e-02 12 6.31e+03 2.57e+01 1.16e-02 9.92e+01 3.78e+00 0 30 1.60e-03 1.29e-02 13 4.82e+03 2.45e+01 1.25e-02 8.46e+01 4.78e+00 0 30 1.22e-03 9.28e-03 14 3.78e+03 2.27e+01 1.29e-02 7.13e+01 5.20e+00 0 28 6.97e-04 5.17e-03 15 3.08e+03 2.06e+01 1.26e-02 5.93e+01 5.30e+00 0 30 1.18e-03 8.44e-03 16 3.08e+03 1.99e+01 1.14e-02 5.49e+01 5.06e+00 0 30 2.87e-03 1.53e-02 17 3.08e+03 1.91e+01 1.00e-02 5.00e+01 5.24e+00 0 30 2.60e-03 1.43e-02 18 3.08e+03 1.82e+01 8.68e-03 4.50e+01 5.37e+00 0 30 2.17e-03 1.25e-02 19 3.08e+03 1.75e+01 7.42e-03 4.03e+01 5.63e+00 0 30 5.89e-04 3.57e-03 20 4.84e+03 1.91e+01 5.84e-03 4.74e+01 1.26e+01 0 21 7.52e-04 1.77e-02 21 4.84e+03 1.89e+01 4.98e-03 4.30e+01 7.65e+00 0 19 8.30e-04 7.45e-03 22 4.84e+03 1.84e+01 4.16e-03 3.86e+01 7.54e+00 0 21 6.46e-04 5.60e-03 23 4.84e+03 1.79e+01 3.50e-03 3.48e+01 8.03e+00 0 22 7.62e-04 7.51e-03 24 7.54e+03 1.86e+01 2.70e-03 3.90e+01 1.28e+01 0 20 7.05e-04 2.29e-02 25 7.54e+03 1.84e+01 2.27e-03 3.55e+01 8.18e+00 0 20 8.98e-04 1.09e-02 26 7.54e+03 1.84e+01 1.96e-03 3.32e+01 9.47e+00 0 23 6.76e-04 9.05e-03 27 7.54e+03 1.84e+01 1.71e-03 3.13e+01 9.03e+00 0 24 2.32e-04 3.21e-03 28 7.54e+03 1.83e+01 1.49e-03 2.96e+01 9.09e+00 0 21 4.03e-04 6.71e-03 29 7.54e+03 1.82e+01 1.30e-03 2.80e+01 9.15e+00 0 20 8.91e-04 1.54e-02 30 7.54e+03 1.80e+01 1.13e-03 2.66e+01 8.99e+00 0 18 8.99e-04 1.49e-02 31 7.54e+03 1.79e+01 9.83e-04 2.53e+01 8.95e+00 0 20 7.03e-04 1.29e-02 32 1.18e+04 1.84e+01 7.93e-04 2.77e+01 1.40e+01 0 18 8.36e-04 6.90e-02 33 1.18e+04 1.84e+01 6.85e-04 2.64e+01 9.42e+00 0 22 8.72e-04 2.31e-02 34 1.18e+04 1.83e+01 5.92e-04 2.53e+01 9.52e+00 0 23 5.98e-04 1.74e-02 35 1.18e+04 1.83e+01 5.10e-04 2.43e+01 9.63e+00 0 26 8.60e-04 2.74e-02 36 1.18e+04 1.83e+01 4.38e-04 2.34e+01 9.74e+00 0 27 8.01e-04 2.75e-02 37 1.18e+04 1.83e+01 3.74e-04 2.27e+01 9.98e+00 0 26 9.13e-04 3.33e-02 38 1.18e+04 1.83e+01 3.17e-04 2.21e+01 1.00e+01 0 26 5.06e-04 1.93e-02 39 1.18e+04 1.84e+01 2.68e-04 2.15e+01 9.92e+00 0 26 4.40e-04 1.67e-02 40 1.18e+04 1.84e+01 2.26e-04 2.11e+01 9.70e+00 0 26 8.38e-04 3.06e-02 41 1.18e+04 1.84e+01 1.92e-04 2.07e+01 9.73e+00 0 27 5.27e-04 1.99e-02 42 1.18e+04 1.84e+01 1.64e-04 2.03e+01 9.85e+00 0 27 7.39e-04 2.96e-02 43 1.18e+04 1.84e+01 1.41e-04 2.01e+01 1.10e+01 0 28 3.08e-04 1.29e-02 44 1.18e+04 1.84e+01 1.21e-04 1.98e+01 9.91e+00 1 28 4.48e-04 1.86e-02 45 1.18e+04 1.84e+01 1.04e-04 1.96e+01 1.16e+01 3 29 4.09e-04 2.69e-02 46 1.18e+04 1.83e+01 9.10e-05 1.94e+01 1.45e+01 0 26 4.44e-04 4.76e-02 47 1.18e+04 1.83e+01 7.45e-05 1.91e+01 1.05e+01 2 30 2.89e-01 1.40e+01 48 1.18e+04 1.77e+01 8.14e-05 1.87e+01 1.70e+01 0 27 5.21e-04 4.78e-02 49 1.84e+04 1.77e+01 5.03e-05 1.86e+01 1.56e+01 5 30 6.52e-01 1.35e+02 Reach maximum number of IRLS cycles: 40 ------------------------- STOP! ------------------------- 1 : |fc-fOld| = 7.3062e-03 <= tolF*(1+|f0|) = 3.7851e+02 1 : |xc-x_last| = 1.4987e-02 <= tolX*(1+|x0|) = 1.0010e-01 0 : |proj(x-g)-x| = 1.5637e+01 <= tolG = 1.0000e-01 0 : |proj(x-g)-x| = 1.5637e+01 <= 1e3*eps = 1.0000e-02 0 : maxIter = 100 <= iter = 49 ------------------------- DONE! ------------------------- .. GENERATED FROM PYTHON SOURCE LINES 217-220 Plotting Results ---------------- .. GENERATED FROM PYTHON SOURCE LINES 220-261 .. code-block:: Python fig, ax = plt.subplots(1, 2, figsize=(12 * 1.2, 4 * 1.2)) # True versus recovered model ax[0].plot(mesh.cell_centers_x, true_model, "k-") ax[0].plot(mesh.cell_centers_x, inv_prob.l2model, "b-") ax[0].plot(mesh.cell_centers_x, recovered_model, "r-") ax[0].legend(("True Model", "Recovered L2 Model", "Recovered Sparse Model")) ax[0].set_ylim([-2, 2]) # Observed versus predicted data ax[1].plot(data_obj.dobs, "k-") ax[1].plot(inv_prob.dpred, "ko") ax[1].legend(("Observed Data", "Predicted Data")) # Plot convergence fig = plt.figure(figsize=(9, 5)) ax = fig.add_axes([0.2, 0.1, 0.7, 0.85]) ax.plot(saveDict.phi_d, "k", lw=2) twin = ax.twinx() twin.plot(saveDict.phi_m, "k--", lw=2) ax.plot( np.r_[IRLS.metrics.start_irls_iter, IRLS.metrics.start_irls_iter], np.r_[0, np.max(saveDict.phi_d)], "k:", ) ax.text( IRLS.metrics.start_irls_iter, 0.0, "IRLS Start", va="bottom", ha="center", rotation="vertical", size=12, bbox={"facecolor": "white"}, ) ax.set_ylabel(r"$\phi_d$", size=16, rotation=0) ax.set_xlabel("Iterations", size=14) twin.set_ylabel(r"$\phi_m$", size=16, rotation=0) .. rst-class:: sphx-glr-horizontal * .. image-sg:: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_2_inversion_irls_003.png :alt: plot inv 2 inversion irls :srcset: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_2_inversion_irls_003.png :class: sphx-glr-multi-img * .. image-sg:: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_2_inversion_irls_004.png :alt: plot inv 2 inversion irls :srcset: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_2_inversion_irls_004.png :class: sphx-glr-multi-img .. rst-class:: sphx-glr-script-out .. code-block:: none Text(865.1527777777777, 0.5, '$\\phi_m$') .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 37.230 seconds) **Estimated memory usage:** 319 MB .. _sphx_glr_download_content_user-guide_tutorials_02-linear_inversion_plot_inv_2_inversion_irls.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_2_inversion_irls.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_inv_2_inversion_irls.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_inv_2_inversion_irls.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_