Infinite loop in inversion

Potential Fields
1

What might be the reason for repeating iteration #20?

opt = Optimization.ProjectedGNCG(maxIter=30, maxIterLS = 20, maxIterCG=20, tolCG = 1e-3)
betaest = Directives.BetaEstimate_ByEig() 
IRLS = Directives.Update_IRLS(f_min_change = 1e-2, maxIRLSiter = 20, minGNiter = 5) 
update_Jacobi = Directives.UpdatePreconditioner()



=============================== Projected GNCG ===============================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS    Comment   
-----------------------------------------------------------------------------
x0 has any nan: 0
   0  2.32e+01  1.30e+06  1.87e+01  1.30e+06    2.03e+02      0              
   1  1.16e+01  1.24e+06  1.19e+03  1.26e+06    1.90e+02      0              
   2  5.81e+00  1.20e+06  4.26e+03  1.22e+06    1.87e+02      0   Skip BFGS  
   3  2.91e+00  1.12e+06  1.45e+04  1.16e+06    1.82e+02      0   Skip BFGS  
   4  1.45e+00  9.89e+05  4.66e+04  1.06e+06    1.76e+02      0   Skip BFGS  
   5  7.26e-01  8.08e+05  1.38e+05  9.08e+05    1.68e+02      0   Skip BFGS  
   6  3.63e-01  5.89e+05  3.57e+05  7.19e+05    1.60e+02      0   Skip BFGS  
   7  1.82e-01  3.77e+05  7.74e+05  5.18e+05    1.53e+02      0   Skip BFGS  
   8  9.08e-02  2.17e+05  1.40e+06  3.44e+05    1.48e+02      0   Skip BFGS  
   9  4.54e-02  1.17e+05  2.17e+06  2.15e+05    1.45e+02      0   Skip BFGS  
  10  2.27e-02  6.02e+04  3.03e+06  1.29e+05    1.42e+02      0   Skip BFGS  
  11  1.13e-02  3.05e+04  3.94e+06  7.53e+04    1.37e+02      0   Skip BFGS  
  12  5.67e-03  1.57e+04  4.84e+06  4.32e+04    1.30e+02      0   Skip BFGS  
  13  2.84e-03  8.33e+03  5.75e+06  2.46e+04    1.20e+02      0   Skip BFGS  
  14  1.42e-03  4.43e+03  6.71e+06  1.39e+04    1.06e+02      0   Skip BFGS  
  15  7.09e-04  2.27e+03  7.77e+06  7.78e+03    9.14e+01      0   Skip BFGS  
  16  3.55e-04  1.09e+03  8.93e+06  4.26e+03    7.49e+01      0   Skip BFGS  
  17  1.77e-04  4.82e+02  1.01e+07  2.28e+03    6.23e+01      0   Skip BFGS  
  18  8.87e-05  1.96e+02  1.12e+07  1.19e+03    5.07e+01      0   Skip BFGS  
Reached starting chifact with l2-norm regularization: Start IRLS steps...
eps_p: 0.4993707616846184 eps_q: 0.4993707616846184
delta phim: 1.304e+06
  19  4.43e-05  7.53e+01  2.18e+07  1.04e+03    3.83e+01      0   Skip BFGS  
Beta search step
  19  9.36e-05  7.53e+01  2.18e+07  2.11e+03    6.46e+01      0   Skip BFGS  
Beta search step
  19  6.28e-05  7.53e+01  2.18e+07  1.44e+03    4.79e+01      0              
  20  6.28e-05  9.53e+01  2.12e+07  1.43e+03    4.95e+01      0   Skip BFGS  
Beta search step
  20  1.03e-04  9.53e+01  2.12e+07  2.29e+03    6.16e+01      0              
Beta search step
  20  6.56e-05  9.53e+01  2.12e+07  1.49e+03    4.98e+01      0              
Beta search step
  20  1.06e-04  9.53e+01  2.12e+07  2.34e+03    6.24e+01      0              
Beta search step
  20  6.62e-05  9.53e+01  2.12e+07  1.50e+03    4.99e+01      0              
Beta search step
  20  1.06e-04  9.53e+01  2.12e+07  2.35e+03    6.26e+01      0              
Beta search step
  20  6.64e-05  9.53e+01  2.12e+07  1.50e+03    4.99e+01      0              
Beta search step
  20  1.06e-04  9.53e+01  2.12e+07  2.35e+03    6.26e+01      0              
Beta search step
  20  6.64e-05  9.53e+01  2.12e+07  1.50e+03    4.99e+01      0              
Beta search step
  20  1.06e-04  9.53e+01  2.12e+07  2.35e+03    6.26e+01      0              
Beta search step
  20  6.64e-05  9.53e+01  2.12e+07  1.50e+03    4.99e+01      0              
Beta search step
  20  1.06e-04  9.53e+01  2.12e+07  2.35e+03    6.26e+01      0              
Beta search step
  20  6.64e-05  9.53e+01  2.12e+07  1.50e+03    4.99e+01      0
2

That can happen with the beta_search=True switched on.
It’s not a super smart algorithm at this point, as it tries to re-adjust beta by multiplying it by fix constants to get to the target misfit.

if phi_d > target:
    beta *= mean([2, phi_d/target])
else:
    beta *= mean([0.75, phi_d/target])

So you can easily end up in an oscillatory pattern, like you have here, where you are just bouncing around the target but never take a step. I personally rarely use the beta_search, and just let it progress even if it moves away from the target for a little while. The important part is that the Directive.IRLS WILL re-adjust beta in subsequent iterations to bring it back to target even without beta_search=True.

Feel free to think of a better approach to estimate the appropriate beta and share with us!