.. 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_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_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/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_2_inversion_irls_001.png :alt: plot inv 2 inversion irls :srcset: /content/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/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_2_inversion_irls_002.png :alt: Columns of matrix G :srcset: /content/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, maxIterCG=30, tolCG=1e-4 ) # 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 172-193 .. 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.Update_IRLS(max_irls_iterations=40, minGNiter=1, 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] .. GENERATED FROM PYTHON SOURCE LINES 194-200 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 200-210 .. 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 v0.22.1 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 =============================== Projected GNCG =============================== # beta phi_d phi_m f |proj(x-g)-x| LS Comment ----------------------------------------------------------------------------- x0 has any nan: 0 0 1.70e+06 3.72e+03 1.01e-09 3.72e+03 1.99e+01 0 1 8.49e+05 1.87e+03 3.86e-04 2.20e+03 1.93e+01 0 2 4.25e+05 1.28e+03 8.90e-04 1.66e+03 1.88e+01 0 Skip BFGS 3 2.12e+05 7.48e+02 1.78e-03 1.13e+03 1.80e+01 0 Skip BFGS 4 1.06e+05 3.74e+02 3.02e-03 6.94e+02 1.65e+01 0 Skip BFGS 5 5.31e+04 1.63e+02 4.39e-03 3.96e+02 1.42e+01 0 Skip BFGS 6 2.65e+04 6.65e+01 5.64e-03 2.16e+02 1.27e+01 0 Skip BFGS 7 1.33e+04 2.90e+01 6.61e-03 1.17e+02 1.01e+01 0 Skip BFGS Reached starting chifact with l2-norm regularization: Start IRLS steps... irls_threshold 1.2669107150802716 8 6.63e+03 1.60e+01 1.00e-02 8.25e+01 4.05e+00 0 Skip BFGS 9 1.12e+04 1.46e+01 1.15e-02 1.43e+02 1.69e+01 0 10 8.79e+03 2.43e+01 1.15e-02 1.25e+02 2.91e+00 0 11 6.86e+03 2.46e+01 1.23e-02 1.09e+02 3.25e+00 0 Skip BFGS 12 5.45e+03 2.38e+01 1.28e-02 9.35e+01 4.40e+00 0 Skip BFGS 13 4.47e+03 2.24e+01 1.29e-02 8.01e+01 4.58e+00 0 Skip BFGS 14 4.47e+03 2.07e+01 1.26e-02 7.69e+01 6.10e+00 0 15 4.47e+03 2.07e+01 1.17e-02 7.30e+01 6.39e+00 0 16 4.47e+03 2.06e+01 1.07e-02 6.84e+01 6.50e+00 0 17 4.47e+03 2.02e+01 9.64e-03 6.33e+01 6.84e+00 0 18 4.47e+03 1.97e+01 8.56e-03 5.79e+01 7.31e+00 0 19 4.47e+03 1.89e+01 7.41e-03 5.21e+01 7.79e+00 0 20 6.97e+03 1.79e+01 6.47e-03 6.30e+01 1.50e+01 0 21 6.97e+03 2.03e+01 5.31e-03 5.73e+01 1.04e+01 0 22 6.97e+03 1.98e+01 4.57e-03 5.17e+01 1.03e+01 0 Skip BFGS 23 6.97e+03 1.90e+01 4.11e-03 4.76e+01 1.13e+01 0 24 6.97e+03 1.84e+01 3.75e-03 4.45e+01 1.19e+01 0 25 6.97e+03 1.87e+01 3.20e-03 4.10e+01 1.77e+01 0 26 1.09e+04 1.77e+01 2.71e-03 4.72e+01 1.70e+01 0 27 1.09e+04 1.87e+01 2.17e-03 4.24e+01 1.35e+01 0 28 1.09e+04 1.82e+01 1.80e-03 3.79e+01 1.26e+01 0 29 1.71e+04 1.77e+01 1.50e-03 4.34e+01 1.87e+01 0 30 1.71e+04 1.85e+01 1.19e-03 3.88e+01 1.41e+01 0 31 1.71e+04 1.85e+01 1.02e-03 3.59e+01 1.64e+01 1 Skip BFGS 32 1.71e+04 1.89e+01 8.52e-04 3.34e+01 1.35e+01 0 33 1.71e+04 1.92e+01 7.23e-04 3.15e+01 1.32e+01 0 Skip BFGS 34 1.71e+04 1.95e+01 6.14e-04 3.00e+01 1.37e+01 0 35 1.71e+04 1.98e+01 5.23e-04 2.87e+01 1.39e+01 0 36 1.71e+04 2.01e+01 4.46e-04 2.77e+01 1.41e+01 0 37 1.71e+04 2.03e+01 3.80e-04 2.68e+01 1.42e+01 0 38 1.71e+04 2.05e+01 3.17e-04 2.59e+01 1.40e+01 0 39 1.71e+04 2.05e+01 2.66e-04 2.51e+01 1.40e+01 0 40 1.71e+04 2.06e+01 2.26e-04 2.44e+01 1.38e+01 0 41 1.71e+04 2.06e+01 1.89e-04 2.38e+01 1.37e+01 0 42 1.71e+04 2.06e+01 1.70e-04 2.35e+01 1.68e+01 2 43 1.71e+04 2.07e+01 1.53e-04 2.33e+01 1.76e+01 2 Skip BFGS 44 1.71e+04 2.11e+01 1.17e-04 2.31e+01 1.72e+01 0 45 1.71e+04 2.11e+01 9.65e-05 2.27e+01 1.36e+01 0 46 1.71e+04 2.11e+01 8.97e-05 2.26e+01 1.64e+01 8 Skip BFGS 47 1.71e+04 2.11e+01 8.29e-05 2.25e+01 1.78e+01 3 Reach maximum number of IRLS cycles: 40 ------------------------- STOP! ------------------------- 1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.7168e+02 1 : |xc-x_last| = 4.0540e-02 <= tolX*(1+|x0|) = 1.0010e-01 0 : |proj(x-g)-x| = 1.7789e+01 <= tolG = 1.0000e-01 0 : |proj(x-g)-x| = 1.7789e+01 <= 1e3*eps = 1.0000e-02 0 : maxIter = 100 <= iter = 48 ------------------------- DONE! ------------------------- .. GENERATED FROM PYTHON SOURCE LINES 211-214 Plotting Results ---------------- .. GENERATED FROM PYTHON SOURCE LINES 214-251 .. 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.iterStart, IRLS.iterStart], np.r_[0, np.max(saveDict.phi_d)], "k:") ax.text( IRLS.iterStart, 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/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_2_inversion_irls_003.png :alt: plot inv 2 inversion irls :srcset: /content/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_2_inversion_irls_003.png :class: sphx-glr-multi-img * .. image-sg:: /content/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_2_inversion_irls_004.png :alt: plot inv 2 inversion irls :srcset: /content/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 27.241 seconds) **Estimated memory usage:** 9 MB .. _sphx_glr_download_content_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 ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_