# Inversion: Linear Problem¶

Here we go over the basics of creating a linear problem and inversion.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 from __future__ import print_function import numpy as np from SimPEG import Mesh from SimPEG import Problem from SimPEG import Survey from SimPEG import DataMisfit from SimPEG import Directives from SimPEG import Optimization from SimPEG import Regularization from SimPEG import InvProblem from SimPEG import Inversion import matplotlib.pyplot as plt def run(N=100, plotIt=True): """ Inversion: Linear Problem ========================= Here we go over the basics of creating a linear problem and inversion. """ np.random.seed(1) mesh = Mesh.TensorMesh([N]) nk = 20 jk = np.linspace(1., 60., nk) p = -0.25 q = 0.25 def g(k): return ( np.exp(p*jk[k]*mesh.vectorCCx) * np.cos(np.pi*q*jk[k]*mesh.vectorCCx) ) G = np.empty((nk, mesh.nC)) for i in range(nk): G[i, :] = g(i) mtrue = np.zeros(mesh.nC) mtrue[mesh.vectorCCx > 0.3] = 1. mtrue[mesh.vectorCCx > 0.45] = -0.5 mtrue[mesh.vectorCCx > 0.6] = 0 prob = Problem.LinearProblem(mesh, G=G) survey = Survey.LinearSurvey() survey.pair(prob) survey.makeSyntheticData(mtrue, std=0.01) M = prob.mesh reg = Regularization.Tikhonov(mesh, alpha_s=1., alpha_x=1.) dmis = DataMisfit.l2_DataMisfit(survey) opt = Optimization.InexactGaussNewton(maxIter=60) invProb = InvProblem.BaseInvProblem(dmis, reg, opt) directives = [ Directives.BetaEstimate_ByEig(beta0_ratio=1e-2), Directives.TargetMisfit() ] inv = Inversion.BaseInversion(invProb, directiveList=directives) m0 = np.zeros_like(survey.mtrue) mrec = inv.run(m0) if plotIt: fig, axes = plt.subplots(1, 2, figsize=(12*1.2, 4*1.2)) for i in range(prob.G.shape[0]): axes[0].plot(prob.G[i, :]) axes[0].set_title('Columns of matrix G') axes[1].plot(M.vectorCCx, survey.mtrue, 'b-') axes[1].plot(M.vectorCCx, mrec, 'r-') axes[1].legend(('True Model', 'Recovered Model')) axes[1].set_ylim([-2, 2]) return prob, survey, mesh, mrec if __name__ == '__main__': run() plt.show()