.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "content/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        Click :ref:`here <sphx_glr_download_content_examples_10-pgi_plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_content_examples_10-pgi_plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:


Petrophysically guided inversion: Joint linear example with nonlinear relationships
===================================================================================

We do a comparison between the classic least-squares inversion
and our formulation of a petrophysically guided inversion.
We explore it through coupling two linear problems whose respective physical
properties are linked by polynomial relationships that change between rock units.

.. GENERATED FROM PYTHON SOURCE LINES 11-432



.. image-sg:: /content/examples/10-pgi/images/sphx_glr_plot_inv_1_PGI_Linear_1D_joint_WithRelationships_001.png
   :alt: Problem 1, Problem 2, Petrophysical Distribution, Problem 1, Problem 2, Petrophysical Distribution, Problem 1, Problem 2, Petro Distribution
   :srcset: /content/examples/10-pgi/images/sphx_glr_plot_inv_1_PGI_Linear_1D_joint_WithRelationships_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    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.***
                    
    Alpha scales: [3.456351510282569, 0.0, 3.4567008422694375e-06, 0.0]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.09369146 0.90630854]
    <class 'SimPEG.regularization.pgi.PGIsmallness'>
    Initial data misfit scales:  [0.09369146 0.90630854]
    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  1.89e+01  1.50e+05  0.00e+00  1.50e+05    1.41e+02      0              
    geophys. misfits: 535.4 (target 15.0 [False]); 33.8 (target 15.0 [False]) | smallness misfit: 1488.4 (target: 100.0 [False])
    Beta cooling evaluation: progress: [535.4  33.8] ; minimum progress targets: [120000. 120000.]
       1  1.89e+01  8.08e+01  2.05e+01  4.69e+02    7.45e+01      0              
    geophys. misfits: 242.5 (target 15.0 [False]); 7.5 (target 15.0 [True]) | smallness misfit: 680.5 (target: 100.0 [False])
    Beta cooling evaluation: progress: [242.5   7.5] ; minimum progress targets: [428.4  27. ]
    Updating scaling for data misfits by  2.010519744454068
    New scales: [0.1720768 0.8279232]
       2  1.89e+01  4.79e+01  1.99e+01  4.25e+02    8.26e+01      0   Skip BFGS  
    geophys. misfits: 104.7 (target 15.0 [False]); 7.3 (target 15.0 [True]) | smallness misfit: 612.1 (target: 100.0 [False])
    Beta cooling evaluation: progress: [104.7   7.3] ; minimum progress targets: [194.  15.]
    Updating scaling for data misfits by  2.0462720853966854
    New scales: [0.29839345 0.70160655]
       3  1.89e+01  3.64e+01  2.08e+01  4.30e+02    7.18e+01      0              
    geophys. misfits: 50.8 (target 15.0 [False]); 7.5 (target 15.0 [True]) | smallness misfit: 571.3 (target: 100.0 [False])
    Beta cooling evaluation: progress: [50.8  7.5] ; minimum progress targets: [83.7 15. ]
    Updating scaling for data misfits by  1.9972672672546747
    New scales: [0.45929528 0.54070472]
       4  1.89e+01  2.74e+01  2.14e+01  4.33e+02    7.08e+01      0   Skip BFGS  
    geophys. misfits: 30.7 (target 15.0 [False]); 8.3 (target 15.0 [True]) | smallness misfit: 540.4 (target: 100.0 [False])
    Beta cooling evaluation: progress: [30.7  8.3] ; minimum progress targets: [40.7 15. ]
    Updating scaling for data misfits by  1.8067317121786268
    New scales: [0.60547709 0.39452291]
       5  1.89e+01  2.18e+01  2.18e+01  4.34e+02    6.48e+01      0   Skip BFGS  
    geophys. misfits: 23.1 (target 15.0 [False]); 9.9 (target 15.0 [True]) | smallness misfit: 513.2 (target: 100.0 [False])
    Beta cooling evaluation: progress: [23.1  9.9] ; minimum progress targets: [24.5 15. ]
    Updating scaling for data misfits by  1.5127839790264341
    New scales: [0.69894754 0.30105246]
       6  1.89e+01  1.91e+01  2.20e+01  4.35e+02    4.92e+01      0   Skip BFGS  
    geophys. misfits: 20.2 (target 15.0 [False]); 12.0 (target 15.0 [True]) | smallness misfit: 491.9 (target: 100.0 [False])
    Beta cooling evaluation: progress: [20.2 12. ] ; minimum progress targets: [18.5 15. ]
    Decreasing beta to counter data misfit decrase plateau.
    Updating scaling for data misfits by  1.2485129783447346
    New scales: [0.74350082 0.25649918]
       7  9.45e+00  1.81e+01  2.20e+01  2.26e+02    7.15e+01      0   Skip BFGS  
    geophys. misfits: 11.7 (target 15.0 [True]); 8.2 (target 15.0 [True]) | smallness misfit: 517.5 (target: 100.0 [False])
    Beta cooling evaluation: progress: [11.7  8.2] ; minimum progress targets: [16.2 15. ]
    Warming alpha_pgi to favor clustering:  1.5588936356655474
       8  9.45e+00  1.08e+01  2.32e+01  2.30e+02    2.78e+01      0              
    geophys. misfits: 11.6 (target 15.0 [True]); 10.6 (target 15.0 [True]) | smallness misfit: 459.1 (target: 100.0 [False])
    Beta cooling evaluation: progress: [11.6 10.6] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  2.1127903907176244
       9  9.45e+00  1.13e+01  2.37e+01  2.35e+02    2.67e+01      0              
    geophys. misfits: 11.5 (target 15.0 [True]); 13.1 (target 15.0 [True]) | smallness misfit: 416.6 (target: 100.0 [False])
    Beta cooling evaluation: progress: [11.5 13.1] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  2.588137855814081
      10  9.45e+00  1.19e+01  2.40e+01  2.39e+02    3.71e+01      0   Skip BFGS  
    geophys. misfits: 11.5 (target 15.0 [True]); 15.3 (target 15.0 [False]) | smallness misfit: 386.6 (target: 100.0 [False])
    Beta cooling evaluation: progress: [11.5 15.3] ; minimum progress targets: [15. 15.]
    Decreasing beta to counter data misfit increase.
    Updating scaling for data misfits by  1.3061828151514434
    New scales: [0.68936138 0.31063862]
      11  4.73e+00  1.27e+01  2.39e+01  1.26e+02    6.83e+01      0   Skip BFGS  
    geophys. misfits: 9.2 (target 15.0 [True]); 8.0 (target 15.0 [True]) | smallness misfit: 436.4 (target: 100.0 [False])
    Beta cooling evaluation: progress: [9.2 8. ] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  4.5495688280985105
      12  4.73e+00  8.80e+00  2.63e+01  1.33e+02    3.92e+01      0              
    geophys. misfits: 9.3 (target 15.0 [True]); 11.0 (target 15.0 [True]) | smallness misfit: 351.8 (target: 100.0 [False])
    Beta cooling evaluation: progress: [ 9.3 11. ] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  6.776672564636674
      13  4.73e+00  9.82e+00  2.77e+01  1.41e+02    4.32e+01      0              
    geophys. misfits: 9.3 (target 15.0 [True]); 13.9 (target 15.0 [True]) | smallness misfit: 290.2 (target: 100.0 [False])
    Beta cooling evaluation: progress: [ 9.3 13.9] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  9.11746526965902
      14  4.73e+00  1.07e+01  2.88e+01  1.47e+02    4.04e+01      0   Skip BFGS  
    geophys. misfits: 9.5 (target 15.0 [True]); 17.5 (target 15.0 [False]) | smallness misfit: 246.6 (target: 100.0 [False])
    Beta cooling evaluation: progress: [ 9.5 17.5] ; minimum progress targets: [15. 15.]
    Decreasing beta to counter data misfit increase.
    Updating scaling for data misfits by  1.5788233229219961
    New scales: [0.58430118 0.41569882]
      15  2.36e+00  1.28e+01  2.84e+01  8.00e+01    5.76e+01      0   Skip BFGS  
    geophys. misfits: 8.6 (target 15.0 [True]); 8.5 (target 15.0 [True]) | smallness misfit: 292.0 (target: 100.0 [False])
    Beta cooling evaluation: progress: [8.6 8.5] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  16.052059946206228
      16  2.36e+00  8.53e+00  3.39e+01  8.87e+01    6.37e+01      0              
    geophys. misfits: 8.9 (target 15.0 [True]); 11.1 (target 15.0 [True]) | smallness misfit: 213.3 (target: 100.0 [False])
    Beta cooling evaluation: progress: [ 8.9 11.1] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  24.46565941156895
      17  2.36e+00  9.78e+00  3.67e+01  9.66e+01    7.84e+01      0              
    geophys. misfits: 9.3 (target 15.0 [True]); 12.2 (target 15.0 [True]) | smallness misfit: 171.5 (target: 100.0 [False])
    Beta cooling evaluation: progress: [ 9.3 12.2] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  34.81630831322642
      18  2.36e+00  1.05e+01  3.98e+01  1.05e+02    7.37e+01      0              
    geophys. misfits: 9.6 (target 15.0 [True]); 14.0 (target 15.0 [True]) | smallness misfit: 139.4 (target: 100.0 [False])
    Beta cooling evaluation: progress: [ 9.6 14. ] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  45.85270548053809
      19  2.36e+00  1.14e+01  4.24e+01  1.12e+02    8.05e+01      0              
    geophys. misfits: 10.2 (target 15.0 [True]); 17.2 (target 15.0 [False]) | smallness misfit: 114.2 (target: 100.0 [False])
    Beta cooling evaluation: progress: [10.2 17.2] ; minimum progress targets: [15. 15.]
    Decreasing beta to counter data misfit increase.
    Updating scaling for data misfits by  1.471322693112527
    New scales: [0.48857543 0.51142457]
      20  1.18e+00  1.38e+01  4.11e+01  6.24e+01    8.24e+01      0              
    geophys. misfits: 9.5 (target 15.0 [True]); 12.0 (target 15.0 [True]) | smallness misfit: 131.4 (target: 100.0 [False])
    Beta cooling evaluation: progress: [ 9.5 12. ] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  64.97532712199992
      21  1.18e+00  1.08e+01  4.86e+01  6.82e+01    8.50e+01      0              
    geophys. misfits: 9.1 (target 15.0 [True]); 8.7 (target 15.0 [True]) | smallness misfit: 124.0 (target: 100.0 [False])
    Beta cooling evaluation: progress: [9.1 8.7] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  109.31457730535797
      22  1.18e+00  8.92e+00  6.14e+01  8.15e+01    9.55e+01      0              
    geophys. misfits: 10.2 (target 15.0 [True]); 11.8 (target 15.0 [True]) | smallness misfit: 104.9 (target: 100.0 [False])
    Beta cooling evaluation: progress: [10.2 11.8] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  149.6925382535705
      23  1.18e+00  1.10e+01  6.89e+01  9.24e+01    1.04e+02      0              
    geophys. misfits: 10.5 (target 15.0 [True]); 11.5 (target 15.0 [True]) | smallness misfit: 96.5 (target: 100.0 [True])
    All targets have been reached
    Beta cooling evaluation: progress: [10.5 11.5] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  204.71934366895056
    ------------------------- STOP! -------------------------
    1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 1.5000e+04
    0 : |xc-x_last| = 5.4168e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 1.0394e+02 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 1.0394e+02 <= 1e3*eps       = 1.0000e-02
    0 : maxIter   =      50    <= iter          =     24
    ------------------------- DONE! -------------------------
    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.***
                    
    Alpha scales: [0.0003501319300388227, 0.0, 3.482371221944663e-06, 0.0]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.09369146 0.90630854]
    <class 'SimPEG.regularization.pgi.PGIsmallness'>
    Initial data misfit scales:  [0.09369146 0.90630854]
    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  1.90e+03  1.50e+05  0.00e+00  1.50e+05    1.41e+02      0              
    geophys. misfits: 44336.6 (target 15.0 [False]); 31031.5 (target 15.0 [False]) | smallness misfit: 137.5 (target: 100.0 [False])
    Beta cooling evaluation: progress: [44336.6 31031.5] ; minimum progress targets: [120000. 120000.]
       1  1.90e+03  3.23e+04  3.32e-01  3.29e+04    9.20e+01      0              
    geophys. misfits: 319.5 (target 15.0 [False]); 8.2 (target 15.0 [True]) | smallness misfit: 49.6 (target: 100.0 [True])
    Beta cooling evaluation: progress: [319.5   8.2] ; minimum progress targets: [35469.3 24825.2]
    Updating scaling for data misfits by  1.8359491168853845
    New scales: [0.15951901 0.84048099]
       2  1.90e+03  5.78e+01  1.39e-01  3.21e+02    9.46e+01      0   Skip BFGS  
    geophys. misfits: 36.0 (target 15.0 [False]); 8.6 (target 15.0 [True]) | smallness misfit: 25.3 (target: 100.0 [True])
    Beta cooling evaluation: progress: [36.   8.6] ; minimum progress targets: [255.6  15. ]
    Updating scaling for data misfits by  1.7390449378966708
    New scales: [0.24815528 0.75184472]
       3  1.90e+03  1.54e+01  6.88e-02  1.46e+02    6.84e+01      0   Skip BFGS  
    geophys. misfits: 22.3 (target 15.0 [False]); 7.8 (target 15.0 [True]) | smallness misfit: 34.3 (target: 100.0 [True])
    Beta cooling evaluation: progress: [22.3  7.8] ; minimum progress targets: [28.8 15. ]
    Updating scaling for data misfits by  1.9260734111057487
    New scales: [0.3886497 0.6113503]
       4  1.90e+03  1.34e+01  8.75e-02  1.80e+02    6.85e+01      0              
    geophys. misfits: 16.1 (target 15.0 [False]); 8.5 (target 15.0 [True]) | smallness misfit: 30.6 (target: 100.0 [True])
    Beta cooling evaluation: progress: [16.1  8.5] ; minimum progress targets: [17.9 15. ]
    Updating scaling for data misfits by  1.7605247883552084
    New scales: [0.52812534 0.47187466]
       5  1.90e+03  1.25e+01  7.61e-02  1.57e+02    6.72e+01      0              
    geophys. misfits: 13.1 (target 15.0 [True]); 10.8 (target 15.0 [True]) | smallness misfit: 24.5 (target: 100.0 [True])
    All targets have been reached
    Beta cooling evaluation: progress: [13.1 10.8] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  1.2642239917598124
    ------------------------- STOP! -------------------------
    1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 1.5000e+04
    0 : |xc-x_last| = 1.3963e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 6.7141e+01 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 6.7141e+01 <= 1e3*eps       = 1.0000e-02
    0 : maxIter   =      50    <= iter          =      6
    ------------------------- DONE! -------------------------
    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.***
                    
    Alpha scales: [3.482672955213921e-05, 0.0, 3.5050139896186864e-05, 0.0]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.09369146 0.90630854]
    /home/vsts/work/1/s/SimPEG/directives/directives.py:1058: UserWarning:

    There is no PGI regularization. Smallness target is turned off (TriggerSmall flag)

    Initial data misfit scales:  [0.09369146 0.90630854]
    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  1.02e+06  1.50e+05  0.00e+00  1.50e+05    1.41e+02      0              
    geophys. misfits: 28267.2 (target 15.0 [False]); 17613.7 (target 15.0 [False])
       1  2.04e+05  1.86e+04  2.17e-02  2.30e+04    1.37e+02      0              
    geophys. misfits: 4212.9 (target 15.0 [False]); 1890.7 (target 15.0 [False])
       2  4.08e+04  2.11e+03  5.39e-02  4.30e+03    1.24e+02      0   Skip BFGS  
    geophys. misfits: 284.8 (target 15.0 [False]); 118.9 (target 15.0 [False])
       3  8.15e+03  1.34e+02  7.10e-02  7.13e+02    9.86e+01      0   Skip BFGS  
    geophys. misfits: 23.0 (target 15.0 [False]); 11.9 (target 15.0 [True])
    Updating scaling for data misfits by  1.2572187806737076
    New scales: [0.1150188 0.8849812]
       4  1.63e+03  1.32e+01  7.62e-02  1.37e+02    6.39e+01      0   Skip BFGS  
    geophys. misfits: 8.5 (target 15.0 [True]); 5.6 (target 15.0 [True])
    All targets have been reached
    ------------------------- STOP! -------------------------
    1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 1.5000e+04
    0 : |xc-x_last| = 3.2547e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 6.3884e+01 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 6.3884e+01 <= 1e3*eps       = 1.0000e-02
    0 : maxIter   =      50    <= iter          =      5
    ------------------------- DONE! -------------------------
    /home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:301: UserWarning:

    marker is redundantly defined by the 'marker' keyword argument and the fmt string "b.-" (-> marker='.'). The keyword argument will take precedence.

    /home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:308: UserWarning:

    marker is redundantly defined by the 'marker' keyword argument and the fmt string "r.-" (-> marker='.'). The keyword argument will take precedence.

    /home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:346: UserWarning:

    marker is redundantly defined by the 'marker' keyword argument and the fmt string "b.-" (-> marker='.'). The keyword argument will take precedence.

    /home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:353: UserWarning:

    marker is redundantly defined by the 'marker' keyword argument and the fmt string "r.-" (-> marker='.'). The keyword argument will take precedence.

    /home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:366: UserWarning:

    The following kwargs were not used by contour: 'label'

    /home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:375: UserWarning:

    The following kwargs were not used by contour: 'label'

    /home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:402: UserWarning:

    marker is redundantly defined by the 'marker' keyword argument and the fmt string "b.-" (-> marker='.'). The keyword argument will take precedence.

    /home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:409: UserWarning:

    marker is redundantly defined by the 'marker' keyword argument and the fmt string "r.-" (-> marker='.'). The keyword argument will take precedence.







|

.. code-block:: default


    import discretize as Mesh
    import matplotlib.pyplot as plt
    import numpy as np
    from SimPEG import (
        data_misfit,
        directives,
        inverse_problem,
        inversion,
        maps,
        optimization,
        regularization,
        simulation,
        utils,
    )

    # Random seed for reproductibility
    np.random.seed(1)
    # Mesh
    N = 100
    mesh = Mesh.TensorMesh([N])

    # Survey design parameters
    nk = 30
    jk = np.linspace(1.0, 59.0, nk)
    p = -0.25
    q = 0.25


    # Physics
    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((nk, mesh.nC))

    for i in range(nk):
        G[i, :] = g(i)

    m0 = np.zeros(mesh.nC)
    m0[20:41] = np.linspace(0.0, 1.0, 21)
    m0[41:57] = np.linspace(-1, 0.0, 16)

    poly0 = maps.PolynomialPetroClusterMap(coeffyx=np.r_[0.0, -4.0, 4.0])
    poly1 = maps.PolynomialPetroClusterMap(coeffyx=np.r_[-0.0, 3.0, 6.0, 6.0])
    poly0_inverse = maps.PolynomialPetroClusterMap(coeffyx=-np.r_[0.0, -4.0, 4.0])
    poly1_inverse = maps.PolynomialPetroClusterMap(coeffyx=-np.r_[0.0, 3.0, 6.0, 6.0])
    cluster_mapping = [maps.IdentityMap(), poly0_inverse, poly1_inverse]

    m1 = np.zeros(100)
    m1[20:41] = 1.0 + (poly0 * np.vstack([m0[20:41], m1[20:41]]).T)[:, 1]
    m1[41:57] = -1.0 + (poly1 * np.vstack([m0[41:57], m1[41:57]]).T)[:, 1]

    model2d = np.vstack([m0, m1]).T
    m = utils.mkvc(model2d)

    clfmapping = utils.GaussianMixtureWithNonlinearRelationships(
        mesh=mesh,
        n_components=3,
        covariance_type="full",
        tol=1e-8,
        reg_covar=1e-3,
        max_iter=1000,
        n_init=100,
        init_params="kmeans",
        random_state=None,
        warm_start=False,
        means_init=np.array(
            [
                [0, 0],
                [m0[20:41].mean(), m1[20:41].mean()],
                [m0[41:57].mean(), m1[41:57].mean()],
            ]
        ),
        verbose=0,
        verbose_interval=10,
        cluster_mapping=cluster_mapping,
    )
    clfmapping = clfmapping.fit(model2d)

    clfnomapping = utils.WeightedGaussianMixture(
        mesh=mesh,
        n_components=3,
        covariance_type="full",
        tol=1e-8,
        reg_covar=1e-3,
        max_iter=1000,
        n_init=100,
        init_params="kmeans",
        random_state=None,
        warm_start=False,
        verbose=0,
        verbose_interval=10,
    )
    clfnomapping = clfnomapping.fit(model2d)

    wires = maps.Wires(("m1", mesh.nC), ("m2", mesh.nC))

    relatrive_error = 0.01
    noise_floor = 0.0

    prob1 = simulation.LinearSimulation(mesh, G=G, model_map=wires.m1)
    survey1 = prob1.make_synthetic_data(
        m, relative_error=relatrive_error, noise_floor=noise_floor, add_noise=True
    )

    prob2 = simulation.LinearSimulation(mesh, G=G, model_map=wires.m2)
    survey2 = prob2.make_synthetic_data(
        m, relative_error=relatrive_error, noise_floor=noise_floor, add_noise=True
    )


    dmis1 = data_misfit.L2DataMisfit(simulation=prob1, data=survey1)
    dmis2 = data_misfit.L2DataMisfit(simulation=prob2, data=survey2)
    dmis = dmis1 + dmis2
    minit = np.zeros_like(m)

    # Distance weighting
    wr1 = np.sum(prob1.G ** 2.0, axis=0) ** 0.5 / mesh.cell_volumes
    wr1 = wr1 / np.max(wr1)
    wr2 = np.sum(prob2.G ** 2.0, axis=0) ** 0.5 / mesh.cell_volumes
    wr2 = wr2 / np.max(wr2)

    reg_simple = regularization.PGI(
        mesh=mesh,
        gmmref=clfmapping,
        gmm=clfmapping,
        approx_gradient=True,
        wiresmap=wires,
        non_linear_relationships=True,
        weights_list=[wr1, wr2],
    )

    opt = optimization.ProjectedGNCG(
        maxIter=50,
        tolX=1e-6,
        maxIterCG=100,
        tolCG=1e-3,
        lower=-10,
        upper=10,
    )

    invProb = inverse_problem.BaseInvProblem(dmis, reg_simple, opt)

    # directives
    scales = directives.ScalingMultipleDataMisfits_ByEig(
        chi0_ratio=np.r_[1.0, 1.0], verbose=True, n_pw_iter=10
    )
    scaling_schedule = directives.JointScalingSchedule(verbose=True)
    alpha0_ratio = np.r_[1e6, 1e4, 1, 1]
    alphas = directives.AlphasSmoothEstimate_ByEig(
        alpha0_ratio=alpha0_ratio, n_pw_iter=10, verbose=True
    )
    beta = directives.BetaEstimate_ByEig(beta0_ratio=1e-5, n_pw_iter=10)
    betaIt = directives.PGI_BetaAlphaSchedule(
        verbose=True,
        coolingFactor=2.0,
        progress=0.2,
    )
    targets = directives.MultiTargetMisfits(verbose=True)
    petrodir = directives.PGI_UpdateParameters(update_gmm=False)

    # Setup Inversion
    inv = inversion.BaseInversion(
        invProb,
        directiveList=[alphas, scales, beta, petrodir, targets, betaIt, scaling_schedule],
    )

    mcluster_map = inv.run(minit)

    # Inversion with no nonlinear mapping
    reg_simple_no_map = regularization.PGI(
        mesh=mesh,
        gmmref=clfnomapping,
        gmm=clfnomapping,
        approx_gradient=True,
        wiresmap=wires,
        non_linear_relationships=False,
        weights_list=[wr1, wr2],
    )

    opt = optimization.ProjectedGNCG(
        maxIter=50,
        tolX=1e-6,
        maxIterCG=100,
        tolCG=1e-3,
        lower=-10,
        upper=10,
    )

    invProb = inverse_problem.BaseInvProblem(dmis, reg_simple_no_map, opt)

    # directives
    scales = directives.ScalingMultipleDataMisfits_ByEig(
        chi0_ratio=np.r_[1.0, 1.0], verbose=True, n_pw_iter=10
    )
    scaling_schedule = directives.JointScalingSchedule(verbose=True)
    alpha0_ratio = np.r_[100.0 * np.ones(2), 1, 1]
    alphas = directives.AlphasSmoothEstimate_ByEig(
        alpha0_ratio=alpha0_ratio, n_pw_iter=10, verbose=True
    )
    beta = directives.BetaEstimate_ByEig(beta0_ratio=1e-5, n_pw_iter=10)
    betaIt = directives.PGI_BetaAlphaSchedule(
        verbose=True,
        coolingFactor=2.0,
        progress=0.2,
    )
    targets = directives.MultiTargetMisfits(
        chiSmall=1.0, TriggerSmall=True, TriggerTheta=False, verbose=True
    )
    petrodir = directives.PGI_UpdateParameters(update_gmm=False)

    # Setup Inversion
    inv = inversion.BaseInversion(
        invProb,
        directiveList=[alphas, scales, beta, petrodir, targets, betaIt, scaling_schedule],
    )

    mcluster_no_map = inv.run(minit)

    # WeightedLeastSquares Inversion

    reg1 = regularization.WeightedLeastSquares(
        mesh, alpha_s=1.0, alpha_x=1.0, mapping=wires.m1
    )
    reg1.cell_weights = wr1
    reg2 = regularization.WeightedLeastSquares(
        mesh, alpha_s=1.0, alpha_x=1.0, mapping=wires.m2
    )
    reg2.cell_weights = wr2
    reg = reg1 + reg2

    opt = optimization.ProjectedGNCG(
        maxIter=50,
        tolX=1e-6,
        maxIterCG=100,
        tolCG=1e-3,
        lower=-10,
        upper=10,
    )

    invProb = inverse_problem.BaseInvProblem(dmis, reg, opt)

    # directives
    alpha0_ratio = np.r_[1, 1, 1, 1]
    alphas = directives.AlphasSmoothEstimate_ByEig(
        alpha0_ratio=alpha0_ratio, n_pw_iter=10, verbose=True
    )
    scales = directives.ScalingMultipleDataMisfits_ByEig(
        chi0_ratio=np.r_[1.0, 1.0], verbose=True, n_pw_iter=10
    )
    scaling_schedule = directives.JointScalingSchedule(verbose=True)
    beta = directives.BetaEstimate_ByEig(beta0_ratio=1e-5, n_pw_iter=10)
    beta_schedule = directives.BetaSchedule(coolingFactor=5.0, coolingRate=1)
    targets = directives.MultiTargetMisfits(
        TriggerSmall=False,
        verbose=True,
    )

    # Setup Inversion
    inv = inversion.BaseInversion(
        invProb,
        directiveList=[alphas, scales, beta, targets, beta_schedule, scaling_schedule],
    )

    mtik = inv.run(minit)


    # Final Plot
    fig, axes = plt.subplots(3, 4, figsize=(25, 15))
    axes = axes.reshape(12)
    left, width = 0.25, 0.5
    bottom, height = 0.25, 0.5
    right = left + width
    top = bottom + height

    axes[0].set_axis_off()
    axes[0].text(
        0.5 * (left + right),
        0.5 * (bottom + top),
        ("Using true nonlinear\npetrophysical relationships"),
        horizontalalignment="center",
        verticalalignment="center",
        fontsize=20,
        color="black",
        transform=axes[0].transAxes,
    )

    axes[1].plot(mesh.cell_centers_x, wires.m1 * mcluster_map, "b.-", ms=5, marker="v")
    axes[1].plot(mesh.cell_centers_x, wires.m1 * m, "k--")
    axes[1].set_title("Problem 1")
    axes[1].legend(["Recovered Model", "True Model"], loc=1)
    axes[1].set_xlabel("X")
    axes[1].set_ylabel("Property 1")

    axes[2].plot(mesh.cell_centers_x, wires.m2 * mcluster_map, "r.-", ms=5, marker="v")
    axes[2].plot(mesh.cell_centers_x, wires.m2 * m, "k--")
    axes[2].set_title("Problem 2")
    axes[2].legend(["Recovered Model", "True Model"], loc=1)
    axes[2].set_xlabel("X")
    axes[2].set_ylabel("Property 2")

    x, y = np.mgrid[-1:1:0.01, -4:2:0.01]
    pos = np.empty(x.shape + (2,))
    pos[:, :, 0] = x
    pos[:, :, 1] = y
    CS = axes[3].contour(
        x,
        y,
        np.exp(clfmapping.score_samples(pos.reshape(-1, 2)).reshape(x.shape)),
        100,
        alpha=0.25,
        cmap="viridis",
    )
    axes[3].scatter(wires.m1 * mcluster_map, wires.m2 * mcluster_map, marker="v")
    axes[3].set_title("Petrophysical Distribution")
    CS.collections[0].set_label("")
    axes[3].legend(["True Petrophysical Distribution", "Recovered model crossplot"])
    axes[3].set_xlabel("Property 1")
    axes[3].set_ylabel("Property 2")

    axes[4].set_axis_off()
    axes[4].text(
        0.5 * (left + right),
        0.5 * (bottom + top),
        ("Using a pure\nGaussian distribution"),
        horizontalalignment="center",
        verticalalignment="center",
        fontsize=20,
        color="black",
        transform=axes[4].transAxes,
    )

    axes[5].plot(mesh.cell_centers_x, wires.m1 * mcluster_no_map, "b.-", ms=5, marker="v")
    axes[5].plot(mesh.cell_centers_x, wires.m1 * m, "k--")
    axes[5].set_title("Problem 1")
    axes[5].legend(["Recovered Model", "True Model"], loc=1)
    axes[5].set_xlabel("X")
    axes[5].set_ylabel("Property 1")

    axes[6].plot(mesh.cell_centers_x, wires.m2 * mcluster_no_map, "r.-", ms=5, marker="v")
    axes[6].plot(mesh.cell_centers_x, wires.m2 * m, "k--")
    axes[6].set_title("Problem 2")
    axes[6].legend(["Recovered Model", "True Model"], loc=1)
    axes[6].set_xlabel("X")
    axes[6].set_ylabel("Property 2")

    CSF = axes[7].contour(
        x,
        y,
        np.exp(clfmapping.score_samples(pos.reshape(-1, 2)).reshape(x.shape)),
        100,
        alpha=0.5,
        label="True Petro. Distribution",
    )
    CS = axes[7].contour(
        x,
        y,
        np.exp(clfnomapping.score_samples(pos.reshape(-1, 2)).reshape(x.shape)),
        500,
        cmap="viridis",
        linestyles="--",
        label="Modeled Petro. Distribution",
    )
    axes[7].scatter(
        wires.m1 * mcluster_no_map,
        wires.m2 * mcluster_no_map,
        marker="v",
        label="Recovered model crossplot",
    )
    axes[7].set_title("Petrophysical Distribution")
    axes[7].legend()
    axes[7].set_xlabel("Property 1")
    axes[7].set_ylabel("Property 2")

    # Tikonov

    axes[8].set_axis_off()
    axes[8].text(
        0.5 * (left + right),
        0.5 * (bottom + top),
        ("Least-Squares\n~Using a single cluster"),
        horizontalalignment="center",
        verticalalignment="center",
        fontsize=20,
        color="black",
        transform=axes[8].transAxes,
    )

    axes[9].plot(mesh.cell_centers_x, wires.m1 * mtik, "b.-", ms=5, marker="v")
    axes[9].plot(mesh.cell_centers_x, wires.m1 * m, "k--")
    axes[9].set_title("Problem 1")
    axes[9].legend(["Recovered Model", "True Model"], loc=1)
    axes[9].set_xlabel("X")
    axes[9].set_ylabel("Property 1")

    axes[10].plot(mesh.cell_centers_x, wires.m2 * mtik, "r.-", ms=5, marker="v")
    axes[10].plot(mesh.cell_centers_x, wires.m2 * m, "k--")
    axes[10].set_title("Problem 2")
    axes[10].legend(["Recovered Model", "True Model"], loc=1)
    axes[10].set_xlabel("X")
    axes[10].set_ylabel("Property 2")

    CS = axes[11].contour(
        x,
        y,
        np.exp(clfmapping.score_samples(pos.reshape(-1, 2)).reshape(x.shape)),
        100,
        alpha=0.25,
        cmap="viridis",
    )
    axes[11].scatter(wires.m1 * mtik, wires.m2 * mtik, marker="v")
    axes[11].set_title("Petro Distribution")
    CS.collections[0].set_label("")
    axes[11].legend(["True Petro Distribution", "Recovered model crossplot"])
    axes[11].set_xlabel("Property 1")
    axes[11].set_ylabel("Property 2")
    plt.subplots_adjust(wspace=0.3, hspace=0.3, top=0.85)
    plt.show()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** ( 0 minutes  34.074 seconds)

**Estimated memory usage:**  18 MB


.. _sphx_glr_download_content_examples_10-pgi_plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example


    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py <plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py>`

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_inv_1_PGI_Linear_1D_joint_WithRelationships.ipynb <plot_inv_1_PGI_Linear_1D_joint_WithRelationships.ipynb>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_