.. 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 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 2.27e+06 3.68e+03 1.03e-09 3.68e+03 2.00e+01 0 1 1.13e+06 2.08e+03 2.58e-04 2.38e+03 1.92e+01 0 2 5.66e+05 1.52e+03 6.19e-04 1.87e+03 1.90e+01 0 Skip BFGS 3 2.83e+05 9.67e+02 1.32e-03 1.34e+03 1.83e+01 0 Skip BFGS 4 1.42e+05 5.30e+02 2.41e-03 8.71e+02 1.72e+01 0 Skip BFGS 5 7.08e+04 2.50e+02 3.79e-03 5.18e+02 1.41e+01 0 Skip BFGS 6 3.54e+04 1.03e+02 5.22e-03 2.88e+02 1.19e+01 0 Skip BFGS 7 1.77e+04 3.97e+01 6.45e-03 1.54e+02 9.97e+00 0 Skip BFGS Reached starting chifact with l2-norm regularization: Start IRLS steps... irls_threshold 1.2285608123285758 8 8.85e+03 1.64e+01 1.01e-02 1.06e+02 4.83e+00 0 Skip BFGS 9 1.54e+04 1.35e+01 1.17e-02 1.94e+02 1.74e+01 0 10 1.08e+04 3.05e+01 1.14e-02 1.54e+02 5.42e+00 0 11 7.99e+03 2.76e+01 1.25e-02 1.27e+02 5.51e+00 0 12 6.23e+03 2.47e+01 1.31e-02 1.06e+02 5.15e+00 0 Skip BFGS 13 6.23e+03 2.20e+01 1.32e-02 1.04e+02 8.91e+00 0 Skip BFGS 14 5.08e+03 2.27e+01 1.26e-02 8.68e+01 5.17e+00 0 15 5.08e+03 1.96e+01 1.20e-02 8.05e+01 7.36e+00 0 16 5.08e+03 1.87e+01 1.10e-02 7.44e+01 7.90e+00 0 17 7.92e+03 1.79e+01 9.90e-03 9.63e+01 1.53e+01 0 18 6.32e+03 2.36e+01 8.50e-03 7.74e+01 7.40e+00 0 19 6.32e+03 2.08e+01 7.65e-03 6.92e+01 1.05e+01 0 20 6.32e+03 1.93e+01 6.62e-03 6.12e+01 1.08e+01 0 21 9.93e+03 1.75e+01 5.61e-03 7.33e+01 1.47e+01 0 22 9.93e+03 2.04e+01 4.49e-03 6.50e+01 1.20e+01 0 23 9.93e+03 1.89e+01 3.85e-03 5.72e+01 1.18e+01 0 24 1.58e+04 1.69e+01 3.27e-03 6.86e+01 1.58e+01 0 25 1.58e+04 1.98e+01 2.59e-03 6.07e+01 1.19e+01 0 26 1.58e+04 1.90e+01 2.28e-03 5.51e+01 1.25e+01 0 Skip BFGS 27 1.58e+04 1.80e+01 2.03e-03 5.00e+01 1.29e+01 0 28 2.52e+04 1.68e+01 1.78e-03 6.16e+01 1.68e+01 0 29 2.52e+04 1.86e+01 1.40e-03 5.39e+01 1.18e+01 0 30 2.52e+04 1.81e+01 1.16e-03 4.73e+01 1.17e+01 0 Skip BFGS 31 3.98e+04 1.73e+01 9.65e-04 5.57e+01 1.69e+01 0 32 6.19e+04 1.80e+01 7.85e-04 6.65e+01 1.70e+01 0 33 6.19e+04 1.99e+01 5.82e-04 5.59e+01 1.10e+01 0 34 6.19e+04 1.88e+01 4.92e-04 4.93e+01 1.23e+01 0 35 9.73e+04 1.75e+01 4.28e-04 5.91e+01 1.69e+01 0 36 9.73e+04 1.82e+01 3.40e-04 5.13e+01 1.16e+01 0 37 1.54e+05 1.70e+01 2.80e-04 6.03e+01 1.72e+01 0 38 2.42e+05 1.77e+01 2.22e-04 7.13e+01 1.70e+01 0 39 2.42e+05 1.88e+01 1.83e-04 6.30e+01 1.16e+01 0 40 2.42e+05 1.81e+01 1.54e-04 5.52e+01 1.17e+01 0 41 3.84e+05 1.70e+01 1.30e-04 6.69e+01 1.74e+01 0 42 6.00e+05 1.78e+01 1.08e-04 8.24e+01 1.71e+01 0 43 6.00e+05 1.94e+01 8.86e-05 7.25e+01 1.18e+01 0 44 6.00e+05 1.88e+01 7.42e-05 6.33e+01 1.16e+01 0 45 9.38e+05 1.77e+01 6.26e-05 7.64e+01 1.69e+01 0 46 9.38e+05 1.88e+01 5.09e-05 6.65e+01 1.14e+01 0 47 9.38e+05 1.81e+01 4.19e-05 5.74e+01 1.16e+01 0 Reach maximum number of IRLS cycles: 40 ------------------------- STOP! ------------------------- 1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.6816e+02 0 : |xc-x_last| = 1.1286e-01 <= tolX*(1+|x0|) = 1.0010e-01 0 : |proj(x-g)-x| = 1.1565e+01 <= tolG = 1.0000e-01 0 : |proj(x-g)-x| = 1.1565e+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 21.778 seconds) **Estimated memory usage:** 8 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 `_