.. 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. self.save_txt = save_txt /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. on_disk = self.save_txt .. 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 v0.25.2 ================================================= Projected GNCG ================================================= # beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment ----------------------------------------------------------------------------------------------------------------- 0 1.70e+06 3.70e+03 1.02e-09 3.70e+03 0 inf inf 1 1.70e+06 1.89e+03 3.80e-04 2.53e+03 1.96e+01 0 8 3.89e-04 1.78e+00 2 8.48e+05 1.30e+03 8.83e-04 2.05e+03 1.90e+01 0 9 2.58e-04 2.06e-01 3 4.24e+05 7.64e+02 1.78e-03 1.52e+03 1.86e+01 0 10 5.04e-04 2.98e-01 4 2.12e+05 3.81e+02 3.05e-03 1.03e+03 1.76e+01 0 11 7.81e-04 3.25e-01 5 1.06e+05 1.62e+02 4.48e-03 6.37e+02 1.51e+01 0 14 4.58e-04 1.24e-01 6 5.30e+04 6.17e+01 5.79e-03 3.68e+02 1.39e+01 0 16 8.88e-04 1.44e-01 7 2.65e+04 2.29e+01 6.78e-03 2.03e+02 1.30e+01 0 20 8.81e-04 8.08e-02 8 1.32e+04 9.70e+00 7.46e-03 1.08e+02 1.07e+01 0 29 7.21e-04 3.58e-02 Reached starting chifact with l2-norm regularization: Start IRLS steps... irls_threshold 1.2492122951221412 9 1.32e+04 1.70e+01 9.22e-03 1.39e+02 1.51e+01 0 28 9.49e-04 2.98e-02 10 2.10e+04 3.90e+01 9.17e-03 2.32e+02 1.83e+01 0 20 8.85e-04 6.24e-02 11 1.33e+04 3.12e+01 1.07e-02 1.74e+02 9.92e+00 0 21 9.60e-04 2.92e-02 12 9.23e+03 2.71e+01 1.19e-02 1.37e+02 7.94e+00 0 23 9.19e-04 1.69e-02 13 6.87e+03 2.40e+01 1.28e-02 1.12e+02 6.97e+00 0 24 7.49e-04 1.00e-02 14 5.44e+03 2.13e+01 1.32e-02 9.29e+01 6.33e+00 0 24 7.07e-04 7.65e-03 15 5.44e+03 2.18e+01 1.24e-02 8.94e+01 7.46e+00 0 24 5.63e-04 4.55e-03 16 5.44e+03 2.20e+01 1.16e-02 8.51e+01 7.87e+00 0 23 8.38e-04 7.24e-03 17 5.44e+03 2.17e+01 1.06e-02 7.95e+01 8.03e+00 0 22 3.97e-04 3.59e-03 18 5.44e+03 2.10e+01 9.59e-03 7.31e+01 8.25e+00 0 21 9.08e-04 8.50e-03 19 5.44e+03 1.99e+01 8.52e-03 6.63e+01 8.62e+00 0 18 5.45e-04 5.42e-03 20 5.44e+03 1.83e+01 7.36e-03 5.83e+01 8.73e+00 0 19 9.93e-04 1.06e-02 21 5.44e+03 1.67e+01 6.40e-03 5.15e+01 9.45e+00 0 21 6.34e-04 7.32e-03 22 8.70e+03 1.94e+01 4.87e-03 6.18e+01 1.62e+01 0 18 9.86e-04 3.68e-02 23 8.70e+03 1.90e+01 4.23e-03 5.58e+01 9.50e+00 0 20 8.72e-04 1.25e-02 24 8.70e+03 1.85e+01 3.81e-03 5.17e+01 1.10e+01 0 20 4.24e-04 7.72e-03 25 8.70e+03 1.80e+01 3.44e-03 4.79e+01 1.17e+01 0 20 7.23e-04 1.58e-02 26 8.70e+03 1.76e+01 3.01e-03 4.38e+01 1.70e+01 0 18 4.42e-04 1.84e-02 27 1.36e+04 1.88e+01 2.37e-03 5.11e+01 1.67e+01 0 16 4.77e-04 3.01e-02 28 1.36e+04 1.82e+01 2.01e-03 4.56e+01 1.23e+01 0 16 9.36e-04 3.13e-02 29 1.36e+04 1.73e+01 1.68e-03 4.02e+01 1.19e+01 0 18 7.57e-04 2.36e-02 30 2.15e+04 1.77e+01 1.32e-03 4.60e+01 1.72e+01 0 17 6.51e-04 6.39e-02 31 3.37e+04 1.84e+01 1.06e-03 5.42e+01 1.78e+01 0 15 9.18e-04 1.33e-01 32 3.37e+04 1.80e+01 9.01e-04 4.84e+01 1.36e+01 0 22 3.39e-04 1.65e-02 33 3.37e+04 1.75e+01 7.53e-04 4.29e+01 1.61e+01 0 20 5.42e-04 3.27e-02 34 5.29e+04 1.81e+01 6.04e-04 5.01e+01 1.76e+01 0 19 7.16e-04 1.80e-01 35 5.29e+04 1.79e+01 5.09e-04 4.48e+01 1.38e+01 0 23 6.38e-04 4.70e-02 36 8.26e+04 1.85e+01 4.08e-04 5.21e+01 1.84e+01 0 19 7.73e-04 2.64e-01 37 8.26e+04 1.83e+01 3.40e-04 4.64e+01 1.39e+01 0 21 2.56e-04 2.41e-02 38 8.26e+04 1.79e+01 2.80e-04 4.10e+01 1.38e+01 0 19 7.43e-04 8.92e-02 39 1.29e+05 1.81e+01 2.23e-04 4.68e+01 1.83e+01 0 21 4.32e-04 2.43e-01 40 1.29e+05 1.79e+01 1.88e-04 4.21e+01 1.37e+01 0 24 2.92e-04 5.95e-02 41 2.01e+05 1.79e+01 1.53e-04 4.86e+01 1.94e+01 0 19 8.97e-04 4.05e-01 42 3.13e+05 1.83e+01 1.27e-04 5.80e+01 1.81e+01 0 21 3.72e-04 1.67e-01 43 3.13e+05 1.82e+01 1.07e-04 5.17e+01 1.36e+01 0 24 3.23e-04 3.35e-02 44 3.13e+05 1.78e+01 8.93e-05 4.58e+01 1.33e+01 0 25 3.14e-04 3.12e-02 45 4.88e+05 1.80e+01 7.35e-05 5.39e+01 1.88e+01 0 22 3.15e-04 1.52e-01 46 4.88e+05 1.78e+01 6.21e-05 4.81e+01 1.35e+01 0 23 2.39e-04 2.54e-02 47 7.63e+05 1.81e+01 5.09e-05 5.69e+01 1.79e+01 0 22 3.91e-04 1.84e-01 48 7.63e+05 1.79e+01 4.29e-05 5.07e+01 1.37e+01 0 25 6.42e-04 6.90e-02 Reach maximum number of IRLS cycles: 40 ------------------------- STOP! ------------------------- 1 : |fc-fOld| = 1.4670e-01 <= tolF*(1+|f0|) = 3.6979e+02 0 : |xc-x_last| = 1.1429e-01 <= tolX*(1+|x0|) = 1.0010e-01 0 : |proj(x-g)-x| = 1.3666e+01 <= tolG = 1.0000e-01 0 : |proj(x-g)-x| = 1.3666e+01 <= 1e3*eps = 1.0000e-02 0 : maxIter = 100 <= iter = 48 ------------------------- 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 28.405 seconds) **Estimated memory usage:** 325 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 `_