6.6. Using an external optimizer¶
Hazel comes with a preconfigured efficient Levenberg-Marquard algorithm, but it can also use any external optimizer. It can seamlessly integrate all SciPy optimizers but it also provides the tools to use other optimizers.
[1]:
%matplotlib inline
import numpy as np
import matplotlib.pyplot as pl
import hazel
import h5py
from scipy.optimize import minimize, differential_evolution
print(hazel.__version__)
label = ['I', 'Q', 'U', 'V']
2018.9.22
Let’s start by generating a model by reading the following configuration file:
[2]:
%cat conf_single.ini
# Hazel configuration File
[Working mode]
Output file = output.h5
Number of cycles = 1
# Topology
# Always photosphere and then chromosphere
# Photospheres are only allowed to be added with a filling factor
# Atmospheres share a filling factor if they are in parenthesis
# Atmospheres are one after the other with the -> operator
# Atmosphere 1 = ph2 -> ch1 -> ch2
[Spectral regions]
[[Region 1]]
Name = spec1
Wavelength = 10826, 10833, 150
Topology = ph1 -> ch1 -> te1
Stokes weights = 1.0, 1.0, 1.0, 1.0
LOS = 0.0, 0.0, 90.0
Boundary condition = 1.0, 0.0, 0.0, 0.0 # I/Ic(mu=1), Q/Ic(mu=1), U/Ic(mu=1), V/Ic(mu=1)
Wavelength file = 'observations/10830.wavelength'
Wavelength weight file = 'observations/10830.weights'
Observations file = 'observations/10830_stokes.1d'
Weights Stokes I = 1.0, 0.0, 0.0, 0.0
Weights Stokes Q = 0.0, 0.0, 0.0, 0.0
Weights Stokes U = 0.0, 0.0, 0.0, 0.0
Weights Stokes V = 0.0, 0.0, 0.0, 0.0
Mask file = None
[Atmospheres]
[[Photosphere 1]]
Name = ph1
Reference atmospheric model = 'photospheres/model_photosphere_200.1d'
Spectral region = spec1
Wavelength = 10826, 10833
Spectral lines = 300,
[[[Ranges]]]
T = 2500.0, 9000.0
vmic = 0.0, 3.0
v = -10.0, 10.0
Bx = -1000.0, 1000.0
By = -1000.0, 1000.0
Bz = -1000.0, 1000.0
ff = 0.0, 1.001
[[[Nodes]]]
T = 1, 1, 5, 5
vmic = 1, 0, 1, 1
v = 0, 0, 1, 1
Bx = 0, 0, 1, 1
By = 0, 0, 1, 1
Bz = 0, 0, 1, 1
ff = 0, 0, 0, 0
[[Chromosphere 1]]
Name = ch1 # Name of the atmosphere component
Spectral region = spec1 # Spectral region to be used for synthesis
Height = 3.0 # Height of the slab
Line = 10830 # 10830, 5876
Wavelength = 10828, 10833 # Wavelength range used for synthesis
Reference atmospheric model = 'chromospheres/model_chromosphere.1d' # File with model parameters
[[[Ranges]]]
Bx = -500, 500
By = -500, 500
Bz = -500, 500
tau = 0.1, 2.0
v = -10.0, 10.0
deltav = 3.0, 12.0
beta = 0.9, 2.0
a = 0.0, 1.0
ff = 0.0, 1.001
[[[Nodes]]]
Bx = 0, 0, 1, 1
By = 0, 0, 1, 1
Bz = 0, 0, 1, 1
tau = 0, 0, 0, 0
v = 0, 0, 0, 0
deltav = 0, 0, 0, 0
beta = 0, 0, 0, 0
a = 0, 0, 0, 0
ff = 0, 0, 0, 0
[[Parametric 1]]
Name = te1
Spectral region = spec1
Wavelength = 10828, 10833
Reference atmospheric model = 'telluric/model_telluric.1d' # File with model parameters
Type = Voigt # Voigt, MoGaussian, MoVoigt
[[[Ranges]]]
Lambda0 = 10832.0, 10834.0
Sigma = 0.1, 0.5
Depth = 0.2, 0.8
a = 0.0, 1.2
ff = 0.0, 1.001
[[[Nodes]]]
Lambda0 = 0, 0, 0, 0
Sigma = 0, 0, 0, 0
Depth = 0, 0, 0, 0
a = 0, 0, 0, 0
ff = 0, 0, 0, 0
This is a very simple inversion in which only two variables are inverted. This is surely not realistic but serves for the purpose of showing how more complicated things can be done. Let’s then read the configuration file and read the observation file defined in the configuration file.
[ ]:
# Test a single inversion in non-iterator mode
mod = hazel.Model('conf_spot.ini', working_mode='inversion', verbose=2)
mod.read_observation()
Next, let us invert it with the Levenberg-Marquartd algorithm provided by Hazel:
[4]:
mod.open_output()
mod.invert()
mod.write_output()
mod.close_output()
2018-10-05 13:15:04,949 - -------------
2018-10-05 13:15:04,950 - Cycle 0
2018-10-05 13:15:04,952 - Weights for region spec1 : SI=1.0 - SQ=0.0 - SU=0.0 - SV=0.0
2018-10-05 13:15:04,953 - -------------
2018-10-05 13:15:04,955 - Free parameters for ph1
2018-10-05 13:15:04,957 - - T with 1 node
2018-10-05 13:15:04,959 - - vmic with 1 node
2018-10-05 13:15:05,270 - It: 0 - chi2: 111.73491265894616 - lambda: 0.01827468471342611 - rel: -1.9999999553060352
2018-10-05 13:15:05,437 - It: 1 - chi2: 94.9879533212469 - lambda: 0.004287504963923828 - rel: -0.1620232890860157
2018-10-05 13:15:05,540 - It: 2 - chi2: 94.9844932019989 - lambda: 0.09602514964800882 - rel: -3.642759054093184e-05
[5]:
final = np.loadtxt('photospheres/model_photosphere.1d', skiprows=4)
start = np.loadtxt('photospheres/model_photosphere_200.1d', skiprows=4)
f = h5py.File('output.h5')
pl.plot(f['ph1']['T'][0,0,:])
pl.plot(final[:,1])
pl.plot(start[:,1], 'x')
f.close()
6.6.1. Using SciPy¶
Then we open the output file, call invert_external, write the output and close it. invert_external allows the user to pass an SciPy optimizer and the arguments for the optimizer. The advantage of the SciPy optimizers is that they share the arguments. In this example, we’ll use the Nelder-Mead algorithm, that does not use derivatives.
[4]:
mod = hazel.Model('conf_spot.ini', working_mode='inversion', verbose=2)
mod.read_observation()
mod.open_output()
mod.invert_external(minimize, method='Nelder-Mead')
mod.write_output()
mod.close_output()
2019-02-04 08:19:45,202 - Using configuration from file : conf_spot.ini
2019-02-04 08:19:45,219 - Backtracking mode : brent
2019-02-04 08:19:45,220 - Adding spectral region spec1
2019-02-04 08:19:45,221 - - Reading wavelength axis from 10830_spot.wavelength
2019-02-04 08:19:45,237 - - Reading wavelength weights from 10830_spot.weights
2019-02-04 08:19:45,242 - - Using observations from 10830_spot.1d
2019-02-04 08:19:45,244 - - No mask for pixels
2019-02-04 08:19:45,245 - - No instrumental profile
2019-02-04 08:19:45,246 - - Using LOS ['0.0', '0.0', '90.0']
2019-02-04 08:19:45,247 - - Using boundary condition ['1.0', '0.0', '0.0', '0.0']
2019-02-04 08:19:45,249 - Using 2 cycles
2019-02-04 08:19:45,250 - Not using randomizations
2019-02-04 08:19:45,251 - Adding atmospheres
2019-02-04 08:19:45,251 - - New available chromosphere : ch1
2019-02-04 08:19:45,253 - * Adding line : 10830
2019-02-04 08:19:45,254 - * Magnetic field reference frame : vertical
2019-02-04 08:19:45,255 - * Reading 1D model chromospheres/init_spot.1d as reference
2019-02-04 08:19:45,265 - - New available photosphere : ph1
2019-02-04 08:19:45,267 - * Adding line : [300]
2019-02-04 08:19:45,268 - * Magnetic field reference frame : vertical
2019-02-04 08:19:45,269 - * Reading 1D model photospheres/init_spot.1d as reference
2019-02-04 08:19:45,274 - Adding topologies
2019-02-04 08:19:45,275 - - ph1 -> ch1
2019-02-04 08:19:45,276 - Removing unused atmospheres
2019-02-04 08:19:45,276 - Number of pixels to invert : 1
2019-02-04 08:19:45,279 - Total number of free parameters in all cycles : 13
2019-02-04 08:19:45,327 - -------------
2019-02-04 08:19:45,328 - Cycle 0
2019-02-04 08:19:45,329 - Weights for region spec1 : SI=1.0 - SQ=0.0 - SU=0.0 - SV=0.0
2019-02-04 08:19:45,330 - -------------
/scratch/Dropbox/GIT/hazel2/hazel/transforms.py:21: RuntimeWarning: divide by zero encountered in double_scalars
return np.log(x / (1.0 - x))
2019-02-04 08:19:46,292 - chi2: 3795.7470631716324
2019-02-04 08:19:46,422 - chi2: 3796.1194380671154
2019-02-04 08:19:46,549 - chi2: 3796.2258347793313
2019-02-04 08:19:46,680 - chi2: 3796.047389383589
2019-02-04 08:19:46,814 - chi2: 3795.7989963009313
2019-02-04 08:19:47,080 - chi2: 3794.437467178893
2019-02-04 08:19:47,218 - chi2: 3794.7686379109255
2019-02-04 08:19:47,490 - chi2: 3792.2741215269457
2019-02-04 08:19:47,625 - chi2: 3793.637377973455
2019-02-04 08:19:47,758 - chi2: 3793.9444598726545
2019-02-04 08:19:47,889 - chi2: 3793.2676842463034
2019-02-04 08:19:48,016 - chi2: 3792.754195633086
2019-02-04 08:19:48,278 - chi2: 3790.1629463683885
2019-02-04 08:19:48,416 - chi2: 3791.099772509932
2019-02-04 08:19:48,541 - chi2: 3790.1736856313914
2019-02-04 08:19:48,799 - chi2: 3787.8098061559413
2019-02-04 08:19:48,925 - chi2: 3788.9482285283016
2019-02-04 08:19:49,054 - chi2: 3788.481903112946
2019-02-04 08:19:49,311 - chi2: 3784.532841733785
2019-02-04 08:19:49,439 - chi2: 3786.0017537637823
2019-02-04 08:19:49,567 - chi2: 3785.0149022143673
2019-02-04 08:19:49,822 - chi2: 3779.7396744184375
2019-02-04 08:19:49,953 - chi2: 3782.5713100132284
2019-02-04 08:19:50,085 - chi2: 3781.740528116949
2019-02-04 08:19:50,220 - chi2: 3780.0608165916196
2019-02-04 08:19:50,499 - chi2: 3774.834428389877
2019-02-04 08:19:50,635 - chi2: 3776.487648250191
2019-02-04 08:19:50,763 - chi2: 3774.968724662549
2019-02-04 08:19:51,018 - chi2: 3768.6477274335466
2019-02-04 08:19:51,149 - chi2: 3771.495193122646
2019-02-04 08:19:51,281 - chi2: 3769.3854055743946
2019-02-04 08:19:51,536 - chi2: 3761.672732852133
2019-02-04 08:19:51,662 - chi2: 3764.7609407835707
2019-02-04 08:19:51,791 - chi2: 3762.8604648080877
2019-02-04 08:19:52,048 - chi2: 3752.9246379213
2019-02-04 08:19:52,177 - chi2: 3758.0050607902194
2019-02-04 08:19:52,305 - chi2: 3755.376981941634
2019-02-04 08:19:52,561 - chi2: 3741.9262799360645
2019-02-04 08:19:52,690 - chi2: 3748.5903017188916
2019-02-04 08:19:52,818 - chi2: 3745.92717241418
2019-02-04 08:19:52,947 - chi2: 3742.9055654011504
2019-02-04 08:19:53,205 - chi2: 3733.8475233082418
2019-02-04 08:19:53,334 - chi2: 3738.049604872633
2019-02-04 08:19:53,462 - chi2: 3735.3126750186534
2019-02-04 08:19:53,718 - chi2: 3725.0236142234457
2019-02-04 08:19:53,848 - chi2: 3729.586379429031
2019-02-04 08:19:53,976 - chi2: 3725.4905400248977
2019-02-04 08:19:54,106 - chi2: 3725.9483068754043
2019-02-04 08:19:54,239 - chi2: 3725.9384478156358
2019-02-04 08:19:54,512 - chi2: 3726.256981982228
2019-02-04 08:19:54,646 - chi2: 3728.833590002748
2019-02-04 08:19:54,916 - chi2: 3728.343851834904
2019-02-04 08:19:55,177 - chi2: 3726.9398530782937
2019-02-04 08:19:55,305 - chi2: 3726.343330643527
2019-02-04 08:19:55,434 - chi2: 3724.9263704044192
2019-02-04 08:19:55,687 - chi2: 3717.3016862108925
2019-02-04 08:19:55,948 - chi2: 3723.2245413624796
2019-02-04 08:19:56,075 - chi2: 3720.5979319575185
2019-02-04 08:19:56,206 - chi2: 3717.9143986723816
2019-02-04 08:19:56,336 - chi2: 3724.1015604382437
2019-02-04 08:19:56,465 - chi2: 3724.4258123595355
2019-02-04 08:19:56,592 - chi2: 3718.374195032217
2019-02-04 08:19:56,848 - chi2: 3721.234309864587
2019-02-04 08:19:57,101 - chi2: 3715.7095658819558
2019-02-04 08:19:57,227 - chi2: 3718.5870168672714
2019-02-04 08:19:57,485 - chi2: 3707.3036330781556
2019-02-04 08:19:57,614 - chi2: 3709.824782246664
2019-02-04 08:19:57,743 - chi2: 3708.034816665729
2019-02-04 08:19:57,871 - chi2: 3710.289183182335
2019-02-04 08:19:58,126 - chi2: 3702.8427242651474
2019-02-04 08:19:58,378 - chi2: 3695.287347645473
2019-02-04 08:19:58,506 - chi2: 3696.770067982812
2019-02-04 08:19:58,774 - chi2: 3685.368545843614
2019-02-04 08:19:58,910 - chi2: 3689.419737180983
2019-02-04 08:19:59,045 - chi2: 3700.2284966565408
2019-02-04 08:19:59,307 - chi2: 3673.898917582945
2019-02-04 08:19:59,435 - chi2: 3677.901944920598
2019-02-04 08:19:59,707 - chi2: 3654.885230027137
2019-02-04 08:19:59,852 - chi2: 3663.038439669272
2019-02-04 08:19:59,988 - chi2: 3660.225634917521
2019-02-04 08:20:00,122 - chi2: 3668.745454773832
2019-02-04 08:20:00,385 - chi2: 3622.901158944818
2019-02-04 08:20:00,521 - chi2: 3635.8266276944714
2019-02-04 08:20:00,656 - chi2: 3632.689286599287
2019-02-04 08:20:00,795 - chi2: 3624.021680794471
2019-02-04 08:20:01,092 - chi2: 3589.8464841308655
2019-02-04 08:20:01,229 - chi2: 3600.6840500843805
2019-02-04 08:20:01,354 - chi2: 3609.769061071951
2019-02-04 08:20:01,622 - chi2: 3551.973339879864
2019-02-04 08:20:01,758 - chi2: 3564.1515021757778
2019-02-04 08:20:02,021 - chi2: 3498.670061762152
2019-02-04 08:20:02,152 - chi2: 3536.7170735698196
2019-02-04 08:20:02,284 - chi2: 3526.267380165079
2019-02-04 08:20:02,425 - chi2: 3505.3479670189117
2019-02-04 08:20:02,688 - chi2: 3409.8867319832566
2019-02-04 08:20:02,824 - chi2: 3483.4387055336038
2019-02-04 08:20:02,957 - chi2: 3439.8869670917798
2019-02-04 08:20:03,215 - chi2: 3323.349628689581
2019-02-04 08:20:03,347 - chi2: 3402.1425181887716
2019-02-04 08:20:03,473 - chi2: 3359.7693023438947
2019-02-04 08:20:03,604 - chi2: 3339.960627270747
2019-02-04 08:20:03,867 - chi2: 3202.1650754763837
2019-02-04 08:20:04,000 - chi2: 3281.828792129558
2019-02-04 08:20:04,136 - chi2: 3226.603709856249
2019-02-04 08:20:04,393 - chi2: 3054.203948052697
2019-02-04 08:20:04,527 - chi2: 3134.493612119521
2019-02-04 08:20:04,655 - chi2: 3113.6517904448588
2019-02-04 08:20:04,809 - chi2: 3057.281428930169
2019-02-04 08:20:05,080 - chi2: 2880.1100474994178
2019-02-04 08:20:05,215 - chi2: 2944.471433590623
2019-02-04 08:20:05,349 - chi2: 2900.4127085172217
2019-02-04 08:20:05,620 - chi2: 2698.8060149956973
2019-02-04 08:20:05,755 - chi2: 2786.153961733687
2019-02-04 08:20:05,915 - chi2: 2712.5931566574372
2019-02-04 08:20:06,187 - chi2: 2514.4988060587057
2019-02-04 08:20:06,321 - chi2: 2607.5528205760634
2019-02-04 08:20:06,451 - chi2: 2541.2797976481415
2019-02-04 08:20:06,707 - chi2: 2320.6113922855125
2019-02-04 08:20:06,838 - chi2: 2409.901444588529
2019-02-04 08:20:06,967 - chi2: 2396.928482455016
2019-02-04 08:20:07,099 - chi2: 2340.343709300031
2019-02-04 08:20:07,359 - chi2: 2251.9842366637818
2019-02-04 08:20:07,492 - chi2: 2278.686794671709
2019-02-04 08:20:07,623 - chi2: 2256.5604224838835
2019-02-04 08:20:07,754 - chi2: 2256.421231041984
2019-02-04 08:20:07,882 - chi2: 2288.2370961353063
2019-02-04 08:20:08,008 - chi2: 2348.744823424255
2019-02-04 08:20:08,138 - chi2: 2356.4340491217004
2019-02-04 08:20:08,393 - chi2: 2287.3121126777787
2019-02-04 08:20:08,649 - chi2: 2279.394620449786
2019-02-04 08:20:08,908 - chi2: 2277.218772609839
2019-02-04 08:20:09,164 - chi2: 2273.527730202047
2019-02-04 08:20:09,419 - chi2: 2262.919241044638
2019-02-04 08:20:09,547 - chi2: 2262.7418555533213
2019-02-04 08:20:09,806 - chi2: 2261.218896719727
2019-02-04 08:20:09,936 - chi2: 2273.75504163616
2019-02-04 08:20:10,064 - chi2: 2267.749157878059
2019-02-04 08:20:10,319 - chi2: 2258.589346607892
2019-02-04 08:20:10,572 - chi2: 2257.3466223028363
2019-02-04 08:20:10,831 - chi2: 2256.952372793815
2019-02-04 08:20:11,088 - chi2: 2254.2155138923135
2019-02-04 08:20:11,216 - chi2: 2256.194468166094
2019-02-04 08:20:11,469 - chi2: 2253.9256852259273
2019-02-04 08:20:11,721 - chi2: 2254.606139744805
2019-02-04 08:20:11,848 - chi2: 2253.4346247550025
2019-02-04 08:20:11,975 - chi2: 2253.4398674893746
2019-02-04 08:20:12,233 - chi2: 2252.371687056321
2019-02-04 08:20:12,486 - chi2: 2239.44338665025
2019-02-04 08:20:12,622 - chi2: 2251.0993234609155
2019-02-04 08:20:12,760 - chi2: 2253.996979120541
2019-02-04 08:20:12,896 - chi2: 2244.3365802325675
2019-02-04 08:20:13,033 - chi2: 2250.637924023315
2019-02-04 08:20:13,169 - chi2: 2251.4602225602016
2019-02-04 08:20:13,301 - chi2: 2244.7599633741606
2019-02-04 08:20:13,431 - chi2: 2247.242068256648
2019-02-04 08:20:13,559 - chi2: 2248.865059755322
2019-02-04 08:20:13,685 - chi2: 2241.664174475986
2019-02-04 08:20:13,813 - chi2: 2239.4821495842143
2019-02-04 08:20:13,940 - chi2: 2241.395885161215
2019-02-04 08:20:14,202 - chi2: 2230.845324249745
2019-02-04 08:20:14,331 - chi2: 2235.9759999761513
2019-02-04 08:20:14,461 - chi2: 2235.2857309088067
2019-02-04 08:20:14,598 - chi2: 2237.6124710197328
2019-02-04 08:20:14,870 - chi2: 2223.0186018395175
2019-02-04 08:20:15,000 - chi2: 2224.2625719999874
2019-02-04 08:20:15,131 - chi2: 2225.4856335622535
2019-02-04 08:20:15,260 - chi2: 2231.1908877673695
2019-02-04 08:20:15,387 - chi2: 2228.3422743337496
2019-02-04 08:20:15,643 - chi2: 2204.2631501142037
2019-02-04 08:20:15,770 - chi2: 2214.0913622743305
2019-02-04 08:20:15,899 - chi2: 2207.7154689844183
2019-02-04 08:20:16,028 - chi2: 2206.8012974764556
2019-02-04 08:20:16,157 - chi2: 2207.5518133942205
2019-02-04 08:20:16,408 - chi2: 2202.4525999272037
2019-02-04 08:20:16,535 - chi2: 2210.846103632488
2019-02-04 08:20:16,792 - chi2: 2173.679986965395
2019-02-04 08:20:16,919 - chi2: 2193.0424692654824
2019-02-04 08:20:17,047 - chi2: 2176.020063230032
2019-02-04 08:20:17,303 - chi2: 2153.9191227524375
2019-02-04 08:20:17,429 - chi2: 2192.6043514867506
2019-02-04 08:20:17,557 - chi2: 2166.8556758792565
2019-02-04 08:20:17,684 - chi2: 2170.948089169805
2019-02-04 08:20:17,812 - chi2: 2156.0133327862363
2019-02-04 08:20:18,065 - chi2: 2119.0084819390877
2019-02-04 08:20:18,193 - chi2: 2134.984179818933
2019-02-04 08:20:18,321 - chi2: 2143.9152354175335
2019-02-04 08:20:18,448 - chi2: 2120.645668721253
2019-02-04 08:20:18,702 - chi2: 2081.303490540617
2019-02-04 08:20:18,830 - chi2: 2102.4739455658646
2019-02-04 08:20:18,959 - chi2: 2095.103723467974
2019-02-04 08:20:19,218 - chi2: 2030.0673988692663
2019-02-04 08:20:19,346 - chi2: 2064.402093388032
2019-02-04 08:20:19,474 - chi2: 2034.5862368581772
2019-02-04 08:20:19,602 - chi2: 2050.2494471853197
2019-02-04 08:20:19,731 - chi2: 2034.2234848039204
2019-02-04 08:20:19,986 - chi2: 1955.4833649947848
2019-02-04 08:20:20,118 - chi2: 1980.40737554571
2019-02-04 08:20:20,246 - chi2: 1987.438146189222
2019-02-04 08:20:20,498 - chi2: 1881.8666952909437
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2019-02-04 08:23:18,953 - chi2: 55.73973416072686
2019-02-04 08:23:19,211 - chi2: 55.73972709684265
2019-02-04 08:23:19,464 - chi2: 55.73973585467345
2019-02-04 08:23:20,851 - chi2: 55.7396515873017
2019-02-04 08:23:20,977 - chi2: 55.73974317730569
2019-02-04 08:23:21,104 - chi2: 55.73971014665481
2019-02-04 08:23:21,361 - chi2: 55.73976612133344
2019-02-04 08:23:22,754 - chi2: 55.73975206910569
2019-02-04 08:23:22,880 - chi2: 55.73973176564177
2019-02-04 08:23:23,132 - chi2: 55.73972718627426
2019-02-04 08:23:23,260 - chi2: 55.73963950416585
2019-02-04 08:23:23,516 - chi2: 55.73979429942284
2019-02-04 08:23:23,789 - chi2: 55.73958012096499
2019-02-04 08:23:24,047 - chi2: 55.73966916626964
2019-02-04 08:23:24,177 - chi2: 55.739667084662265
2019-02-04 08:23:24,432 - chi2: 55.739580463258335
2019-02-04 08:23:25,828 - chi2: 55.73967742153505
2019-02-04 08:23:26,082 - chi2: 55.73977636215099
2019-02-04 08:23:26,212 - chi2: 55.73966988204012
2019-02-04 08:23:26,352 - chi2: 55.73960659835409
2019-02-04 08:23:26,612 - chi2: 55.73959497740292
2019-02-04 08:23:28,013 - chi2: 55.73967048882171
2019-02-04 08:23:28,267 - chi2: 55.73962518923347
2019-02-04 08:23:28,867 - chi2: 55.73952778023619
2019-02-04 08:23:28,869 - -------------
2019-02-04 08:23:28,870 - Cycle 1
2019-02-04 08:23:28,871 - Weights for region spec1 : SI=0.0 - SQ=10.0 - SU=10.0 - SV=10.0
2019-02-04 08:23:28,872 - -------------
2019-02-04 08:23:29,722 - chi2: 30.15786115286162
2019-02-04 08:23:30,027 - chi2: 29.82684235462965
2019-02-04 08:23:30,181 - chi2: 30.046654397040943
2019-02-04 08:23:30,483 - chi2: 29.42486845544968
2019-02-04 08:23:30,637 - chi2: 29.86955143153041
2019-02-04 08:23:30,942 - chi2: 29.24415921179823
2019-02-04 08:23:31,093 - chi2: 29.417780284469607
2019-02-04 08:23:31,249 - chi2: 29.73259502577965
2019-02-04 08:23:31,558 - chi2: 29.027193350098727
2019-02-04 08:23:31,711 - chi2: 29.30529263379414
2019-02-04 08:23:32,012 - chi2: 28.831429729964366
2019-02-04 08:23:32,166 - chi2: 28.899444670035233
2019-02-04 08:23:32,470 - chi2: 28.707293158473902
2019-02-04 08:23:32,773 - chi2: 28.387338034312876
2019-02-04 08:23:33,079 - chi2: 28.114861446059095
2019-02-04 08:23:33,233 - chi2: 28.153796632525097
2019-02-04 08:23:33,543 - chi2: 27.561915378379492
2019-02-04 08:23:33,846 - chi2: 26.946213810765947
2019-02-04 08:23:34,001 - chi2: 27.005904560917365
2019-02-04 08:23:34,324 - chi2: 25.996915583649177
2019-02-04 08:23:34,630 - chi2: 24.551946550221533
2019-02-04 08:23:34,790 - chi2: 24.728385634162702
2019-02-04 08:23:35,095 - chi2: 22.83900400239949
2019-02-04 08:23:35,402 - chi2: 20.321717015751428
2019-02-04 08:23:35,561 - chi2: 20.435142999680785
2019-02-04 08:23:35,872 - chi2: 17.520673110441027
2019-02-04 08:23:36,190 - chi2: 14.476508532125646
2019-02-04 08:23:36,344 - chi2: 14.556135255571284
2019-02-04 08:23:36,652 - chi2: 12.913213908605094
2019-02-04 08:23:36,805 - chi2: 13.028849361832417
2019-02-04 08:23:36,960 - chi2: 13.918048379214559
2019-02-04 08:23:37,271 - chi2: 13.317056806755444
2019-02-04 08:23:37,570 - chi2: 13.207677265413524
2019-02-04 08:23:37,869 - chi2: 13.06139580971638
2019-02-04 08:23:38,172 - chi2: 13.003506617668947
2019-02-04 08:23:38,479 - chi2: 12.962858923143
2019-02-04 08:23:38,780 - chi2: 12.936484349065289
2019-02-04 08:23:39,085 - chi2: 12.924829661327701
2019-02-04 08:23:39,402 - chi2: 12.93013477939974
2019-02-04 08:23:39,702 - chi2: 12.915454965066834
2019-02-04 08:23:39,858 - chi2: 12.92052630902305
2019-02-04 08:23:40,161 - chi2: 12.912771884113535
2019-02-04 08:23:40,464 - chi2: 12.911642149633957
2019-02-04 08:23:40,768 - chi2: 12.91077900750835
2019-02-04 08:23:41,072 - chi2: 12.910790030053843
2019-02-04 08:23:41,381 - chi2: 12.909613213133587
2019-02-04 08:23:41,693 - chi2: 12.91031900277726
2019-02-04 08:23:41,997 - chi2: 12.910138810951427
2019-02-04 08:23:42,150 - chi2: 12.90998301413335
2019-02-04 08:23:42,458 - chi2: 12.909863318569476
2019-02-04 08:23:42,786 - chi2: 12.91033494367943
2019-02-04 08:23:43,087 - chi2: 12.907824149092587
2019-02-04 08:23:43,395 - chi2: 12.909075879650107
2019-02-04 08:23:43,546 - chi2: 12.90785803961736
2019-02-04 08:23:43,700 - chi2: 12.908781389451082
2019-02-04 08:23:44,009 - chi2: 12.906115400722424
2019-02-04 08:23:44,317 - chi2: 12.903721909035953
2019-02-04 08:23:44,467 - chi2: 12.907388984356846
2019-02-04 08:23:44,619 - chi2: 12.906730185092737
2019-02-04 08:23:44,944 - chi2: 12.902082447638834
2019-02-04 08:23:45,247 - chi2: 12.89860803147282
2019-02-04 08:23:45,397 - chi2: 12.900115378334746
2019-02-04 08:23:45,702 - chi2: 12.88915358721836
2019-02-04 08:23:45,860 - chi2: 12.890122480102228
2019-02-04 08:23:46,014 - chi2: 12.891377829119683
2019-02-04 08:23:46,316 - chi2: 12.877089442483864
2019-02-04 08:23:46,619 - chi2: 12.862289214219588
2019-02-04 08:23:46,771 - chi2: 12.876726435848974
2019-02-04 08:23:47,071 - chi2: 12.849711944439118
2019-02-04 08:23:47,372 - chi2: 12.835946775633605
2019-02-04 08:23:47,521 - chi2: 12.83727526225312
2019-02-04 08:23:47,824 - chi2: 12.797540024001908
2019-02-04 08:23:47,974 - chi2: 12.82343102214567
2019-02-04 08:23:48,126 - chi2: 12.819920414387655
2019-02-04 08:23:48,280 - chi2: 12.81341846742597
2019-02-04 08:23:48,596 - chi2: 12.812814585236184
2019-02-04 08:23:48,747 - chi2: 12.797852961527662
2019-02-04 08:23:49,052 - chi2: 12.782299358923359
2019-02-04 08:23:49,357 - chi2: 12.783075801857446
2019-02-04 08:23:49,508 - chi2: 12.785160925407872
2019-02-04 08:23:49,659 - chi2: 12.797417253011048
2019-02-04 08:23:49,810 - chi2: 12.793056213992983
2019-02-04 08:23:49,962 - chi2: 12.790385077125636
2019-02-04 08:23:50,269 - chi2: 12.782958099148576
2019-02-04 08:23:50,580 - chi2: 12.78211703797221
2019-02-04 08:23:50,884 - chi2: 12.781103225269254
2019-02-04 08:23:51,185 - chi2: 12.78078878130472
2019-02-04 08:23:51,336 - chi2: 12.780997829454677
2019-02-04 08:23:51,637 - chi2: 12.780399463078384
2019-02-04 08:23:51,942 - chi2: 12.780344012631192
2019-02-04 08:23:52,099 - chi2: 12.780382680355988
2019-02-04 08:23:52,409 - chi2: 12.78027156085927
2019-02-04 08:23:52,709 - chi2: 12.780552398564563
2019-02-04 08:23:52,862 - chi2: 12.780085477934943
2019-02-04 08:23:53,168 - chi2: 12.779940191384028
2019-02-04 08:23:53,475 - chi2: 12.779711870103956
2019-02-04 08:23:53,629 - chi2: 12.77989376276615
2019-02-04 08:23:53,934 - chi2: 12.779886378933709
2019-02-04 08:23:54,084 - chi2: 12.779661594622592
2019-02-04 08:23:54,385 - chi2: 12.779208143071184
2019-02-04 08:23:54,686 - chi2: 12.77968711624677
2019-02-04 08:23:54,838 - chi2: 12.779220597754904
2019-02-04 08:23:54,992 - chi2: 12.779368372675663
2019-02-04 08:23:55,302 - chi2: 12.778480823774974
2019-02-04 08:23:55,603 - chi2: 12.777853380681993
2019-02-04 08:23:55,756 - chi2: 12.778551461282525
2019-02-04 08:23:55,906 - chi2: 12.778386948776536
2019-02-04 08:23:56,206 - chi2: 12.776625758266245
2019-02-04 08:23:56,356 - chi2: 12.77704294956321
2019-02-04 08:23:56,506 - chi2: 12.777022533716867
2019-02-04 08:23:56,807 - chi2: 12.774285789321915
2019-02-04 08:23:56,956 - chi2: 12.774751773280805
2019-02-04 08:23:57,106 - chi2: 12.774567765340482
2019-02-04 08:23:57,413 - chi2: 12.771073984133327
2019-02-04 08:23:57,713 - chi2: 12.767699308006502
2019-02-04 08:23:57,865 - chi2: 12.768862955091306
2019-02-04 08:23:58,017 - chi2: 12.768028980499803
2019-02-04 08:23:58,318 - chi2: 12.758079393747787
2019-02-04 08:23:58,468 - chi2: 12.760258681229493
2019-02-04 08:23:58,618 - chi2: 12.758405720151254
2019-02-04 08:23:58,918 - chi2: 12.750313213128376
2019-02-04 08:23:59,219 - chi2: 12.737403125757707
2019-02-04 08:23:59,370 - chi2: 12.741135368218927
2019-02-04 08:23:59,673 - chi2: 12.72628428812559
2019-02-04 08:23:59,981 - chi2: 12.7223886536483
2019-02-04 08:24:00,283 - chi2: 12.708772497261116
2019-02-04 08:24:00,436 - chi2: 12.711844194169956
2019-02-04 08:24:00,737 - chi2: 12.717540042228842
2019-02-04 08:24:01,039 - chi2: 12.715222692679347
2019-02-04 08:24:01,341 - chi2: 12.70194844777076
2019-02-04 08:24:01,490 - chi2: 12.703482746875661
2019-02-04 08:24:01,790 - chi2: 12.706077225787544
2019-02-04 08:24:01,941 - chi2: 12.703967234131598
2019-02-04 08:24:02,091 - chi2: 12.70241590623188
2019-02-04 08:24:02,394 - chi2: 12.699838226174199
2019-02-04 08:24:02,693 - chi2: 12.693286865333524
2019-02-04 08:24:02,844 - chi2: 12.701441436197191
2019-02-04 08:24:03,145 - chi2: 12.69550410683864
2019-02-04 08:24:03,449 - chi2: 12.685164842033764
2019-02-04 08:24:03,747 - chi2: 12.692369679855439
2019-02-04 08:24:03,897 - chi2: 12.686455635946269
2019-02-04 08:24:04,199 - chi2: 12.679671087868805
2019-02-04 08:24:04,353 - chi2: 12.686362581944236
2019-02-04 08:24:04,651 - chi2: 12.671543219190996
2019-02-04 08:24:04,952 - chi2: 12.679043062508086
2019-02-04 08:24:05,102 - chi2: 12.673016901983535
2019-02-04 08:24:05,404 - chi2: 12.663379248157579
2019-02-04 08:24:05,706 - chi2: 12.663193932436853
2019-02-04 08:24:05,856 - chi2: 12.669563610893107
2019-02-04 08:24:06,156 - chi2: 12.665377089532841
2019-02-04 08:24:06,310 - chi2: 12.662629236086849
2019-02-04 08:24:06,609 - chi2: 12.662566888927916
2019-02-04 08:24:06,908 - chi2: 12.66149897575618
2019-02-04 08:24:07,211 - chi2: 12.66117288954166
2019-02-04 08:24:07,509 - chi2: 12.66104448009875
2019-02-04 08:24:07,808 - chi2: 12.66115298843556
2019-02-04 08:24:08,106 - chi2: 12.660803670370393
2019-02-04 08:24:08,409 - chi2: 12.66086058016407
2019-02-04 08:24:08,706 - chi2: 12.66086610587556
2019-02-04 08:24:09,004 - chi2: 12.660823319491321
2019-02-04 08:24:09,305 - chi2: 12.660751233905517
2019-02-04 08:24:09,604 - chi2: 12.660779973597979
2019-02-04 08:24:09,903 - chi2: 12.660824043643744
2019-02-04 08:24:10,054 - chi2: 12.660809565509775
2019-02-04 08:24:10,358 - chi2: 12.660782043416722
2019-02-04 08:24:10,508 - chi2: 12.660791751571072
2019-02-04 08:24:10,807 - chi2: 12.660715352472677
2019-02-04 08:24:10,957 - chi2: 12.660770911373216
2019-02-04 08:24:11,257 - chi2: 12.660766948336683
2019-02-04 08:24:11,559 - chi2: 12.660769296489576
2019-02-04 08:24:11,858 - chi2: 12.66078023501115
2019-02-04 08:24:12,008 - chi2: 12.660691043786723
2019-02-04 08:24:12,311 - chi2: 12.66065959973836
2019-02-04 08:24:13,213 - chi2: 12.660682870572243
2019-02-04 08:24:13,515 - chi2: 12.660678229388962
2019-02-04 08:24:13,835 - chi2: 12.660669855518012
2019-02-04 08:24:14,135 - chi2: 12.660655010859308
2019-02-04 08:24:15,041 - chi2: 12.66065407305971
2019-02-04 08:24:15,193 - chi2: 12.660662717025312
2019-02-04 08:24:16,098 - chi2: 12.660692267505796
2019-02-04 08:24:16,732 - chi2: 12.66065407305971
Now we plot the results.
[16]:
final = np.loadtxt('photospheres/model_photosphere.1d', skiprows=4)
start = np.loadtxt('photospheres/model_photosphere_200.1d', skiprows=4)
f = h5py.File('output.h5')
pl.plot(f['ph1']['T'][0,0,:])
pl.plot(final[:,1])
pl.plot(start[:,1], 'x')
f.close()
/scratch/miniconda3/envs/py36/lib/python3.6/site-packages/matplotlib/figure.py:2267: UserWarning: This figure includes Axes that are not compatible with tight_layout, so results might be incorrect.
warnings.warn("This figure includes Axes that are not compatible "
Now let’s use an optimizer that uses first-order derivatives. Let’s select the Newton-CG method:
[17]:
mod = hazel.Model('conf_single.ini', working_mode='inversion', verbose=2)
mod.read_observation()
mod.open_output()
mod.invert_external(minimize, use_jacobian=True, method='Newton-CG')
mod.write_output()
mod.close_output()
2018-07-10 16:50:45,019 - Adding spectral region spec1
2018-07-10 16:50:45,021 - - Reading wavelength axis from observations/10830.wavelength
2018-07-10 16:50:45,027 - - Reading wavelength weights from observations/10830.weights
2018-07-10 16:50:45,034 - - Using observations from observations/10830_stokes.1d
2018-07-10 16:50:45,036 - - No mask for pixels
2018-07-10 16:50:45,038 - - Using LOS ['0.0', '0.0', '90.0']
2018-07-10 16:50:45,039 - - Using boundary condition ['1.0', '0.0', '0.0', '0.0']
2018-07-10 16:50:45,041 - Using 1 cycles
2018-07-10 16:50:45,043 - Adding atmospheres
2018-07-10 16:50:45,045 - - New available photosphere : ph1
2018-07-10 16:50:45,046 - * Adding line : [300]
2018-07-10 16:50:45,048 - * Magnetic field reference frame : vertical
2018-07-10 16:50:45,050 - * Reading 1D model photospheres/model_photosphere_200.1d as reference
2018-07-10 16:50:45,057 - - New available chromosphere : ch1
2018-07-10 16:50:45,059 - * Adding line : 10830
2018-07-10 16:50:45,060 - * Magnetic field reference frame : vertical
2018-07-10 16:50:45,062 - * Reading 1D model chromospheres/model_chromosphere.1d as reference
/scratch/Dropbox/GIT/hazel2/hazel/transforms.py:19: RuntimeWarning: divide by zero encountered in log
return np.log(x / (1.0 - x))
2018-07-10 16:50:45,066 - - New available parametric : te1
2018-07-10 16:50:45,068 - * Reading 1D model telluric/model_telluric.1d as reference
2018-07-10 16:50:45,070 - Adding topologies
2018-07-10 16:50:45,071 - - ph1 -> ch1 -> te1
2018-07-10 16:50:45,073 - Removing unused atmospheres
2018-07-10 16:50:45,074 - Number of pixels to invert : 1
2018-07-10 16:50:45,098 - -------------
2018-07-10 16:50:45,099 - Cycle 0
2018-07-10 16:50:45,101 - Weights for region spec1 : SI=1.0 - SQ=0.0 - SU=0.0 - SV=0.0
2018-07-10 16:50:45,102 - -------------
2018-07-10 16:50:45,930 - chi2: 60596.95373734312
2018-07-10 16:50:46,542 - chi2: 243.02449839565205
2018-07-10 16:50:54,937 - chi2: 243.02449839565205
[5]:
final = np.loadtxt('photospheres/model_photosphere.1d', skiprows=4)
start = np.loadtxt('photospheres/model_photosphere_200.1d', skiprows=4)
f = h5py.File('output.h5')
pl.plot(f['ph1']['T'][0,0,:])
pl.plot(final[:,1])
pl.plot(start[:,1], 'x')
f.close()
---------------------------------------------------------------------------
OSError Traceback (most recent call last)
<ipython-input-5-224f8c68ccab> in <module>
----> 1 final = np.loadtxt('photospheres/model_photosphere.1d', skiprows=4)
2 start = np.loadtxt('photospheres/model_photosphere_200.1d', skiprows=4)
3 f = h5py.File('output.h5')
4 pl.plot(f['ph1']['T'][0,0,:])
5 pl.plot(final[:,1])
/scratch/miniconda3/envs/py36/lib/python3.6/site-packages/numpy/lib/npyio.py in loadtxt(fname, dtype, comments, delimiter, converters, skiprows, usecols, unpack, ndmin, encoding)
924 fname = str(fname)
925 if _is_string_like(fname):
--> 926 fh = np.lib._datasource.open(fname, 'rt', encoding=encoding)
927 fencoding = getattr(fh, 'encoding', 'latin1')
928 fh = iter(fh)
/scratch/miniconda3/envs/py36/lib/python3.6/site-packages/numpy/lib/_datasource.py in open(path, mode, destpath, encoding, newline)
260
261 ds = DataSource(destpath)
--> 262 return ds.open(path, mode, encoding=encoding, newline=newline)
263
264
/scratch/miniconda3/envs/py36/lib/python3.6/site-packages/numpy/lib/_datasource.py in open(self, path, mode, encoding, newline)
616 encoding=encoding, newline=newline)
617 else:
--> 618 raise IOError("%s not found." % path)
619
620
OSError: photospheres/model_photosphere.1d not found.
6.6.2. Using other external optimizers (pro-mode)¶
Using non-SciPy optimizers is slightly more complicated. For this purpose Hazel provides the necessary tools to use them. Here we show an example using the well-known Adam optimizer, broadly used for training deep neural networks. It is a first-order optimizer which uses derivative information. Let us first define a class that implements the Adam optimizer:
[8]:
class AdamOptimizer:
def __init__(self, fun, x0, alpha=0.001, beta1=0.9, beta2=0.999, epsilon=1e-8):
self.alpha = alpha
self.beta1 = beta1
self.beta2 = beta2
self.epsilon = epsilon
self.m = 0
self.v = 0
self.t = 0
self.theta = x0
self.fun = fun
def optimize(self, n_iter):
for i in range(n_iter):
f, gradient = self.fun(self.theta)
self.t = self.t + 1
self.m = self.beta1*self.m + (1 - self.beta1)*gradient
self.v = self.beta2*self.v + (1 - self.beta2)*(gradient**2)
m_hat = self.m/(1 - self.beta1**self.t)
v_hat = self.v/(1 - self.beta2**self.t)
self.theta = self.theta - self.alpha*(m_hat/(np.sqrt(v_hat) - self.epsilon))
print("Iter: {0} - chi2: {1}".format(self.t, f))
return self.theta
Now we instantiate the model, read the observations and open the output. Next, we need to follow what invert_external does. First, compute some factors used internally by Hazel. Loop over all cycles with the following steps: i) find the active parameters, ii) instantiate the optimizer with the _func_grad function provided by Hazel, iii) call the optimizer, iv) set the final model and v) propagate the new parameters to the final ones. Finally, we write the output.
[9]:
mod = hazel.Model('conf_single.ini', working_mode='inversion', verbose=2)
mod.read_observation()
mod.open_output()
for k, v in mod.spectrum.items():
v.factor_chi2 = 1.0 / (v.noise**2 * v.dof)
for mod.cycle in range(mod.n_cycles):
mod.find_active_parameters(mod.cycle)
n_pars = len(mod.nodes)
optimizer = AdamOptimizer(mod._func_grad, mod.nodes)
tmp = optimizer.optimize(300)
mod.set_new_model(tmp)
mod.flatten_parameters_to_reference(mod.cycle)
mod.write_output()
mod.close_output()
2018-10-05 13:15:27,266 - Adding spectral region spec1
2018-10-05 13:15:27,268 - - Reading wavelength axis from observations/10830.wavelength
2018-10-05 13:15:27,271 - - Reading wavelength weights from observations/10830.weights
2018-10-05 13:15:27,275 - - Using observations from observations/10830_stokes.1d
2018-10-05 13:15:27,277 - - No mask for pixels
2018-10-05 13:15:27,279 - - No instrumental profile
2018-10-05 13:15:27,280 - - Using LOS ['0.0', '0.0', '90.0']
2018-10-05 13:15:27,282 - - Using boundary condition ['1.0', '0.0', '0.0', '0.0']
2018-10-05 13:15:27,283 - Using 1 cycles
2018-10-05 13:15:27,285 - Not using randomizations
2018-10-05 13:15:27,290 - Adding atmospheres
2018-10-05 13:15:27,292 - - New available photosphere : ph1
2018-10-05 13:15:27,294 - * Adding line : [300]
2018-10-05 13:15:27,296 - * Magnetic field reference frame : vertical
2018-10-05 13:15:27,301 - * Reading 1D model photospheres/model_photosphere_200.1d as reference
2018-10-05 13:15:27,305 - - New available chromosphere : ch1
2018-10-05 13:15:27,307 - * Adding line : 10830
2018-10-05 13:15:27,308 - * Magnetic field reference frame : vertical
2018-10-05 13:15:27,310 - * Reading 1D model chromospheres/model_chromosphere.1d as reference
2018-10-05 13:15:27,315 - - New available parametric : te1
2018-10-05 13:15:27,317 - * Reading 1D model telluric/model_telluric.1d as reference
2018-10-05 13:15:27,320 - Adding topologies
2018-10-05 13:15:27,321 - - ph1 -> ch1 -> te1
2018-10-05 13:15:27,323 - Removing unused atmospheres
2018-10-05 13:15:27,324 - Number of pixels to invert : 1
2018-10-05 13:15:27,329 - Total number of free parameters in all cycles : 2
---------------------------------------------------------------------------
OSError Traceback (most recent call last)
<ipython-input-9-a3e31dbce02b> in <module>()
1 mod = hazel.Model('conf_single.ini', working_mode='inversion', verbose=2)
2 mod.read_observation()
----> 3 mod.open_output()
4
5 for k, v in mod.spectrum.items():
/scratch/Dropbox/GIT/hazel2/hazel/model.py in open_output(self)
311 def open_output(self):
312 self.output_handler = Generic_output_file(self.output_file)
--> 313 self.output_handler.open(self)
314
315 def close_output(self):
/scratch/Dropbox/GIT/hazel2/hazel/io.py in open(self, model)
31 # Open the file
32 if (self.extension == 'h5'):
---> 33 self.handler = h5py.File(self.filename, 'w')
34
35 if (self.extension == 'zarr'):
~/miniconda3/envs/py36/lib/python3.6/site-packages/h5py/_hl/files.py in __init__(self, name, mode, driver, libver, userblock_size, swmr, **kwds)
267 with phil:
268 fapl = make_fapl(driver, libver, **kwds)
--> 269 fid = make_fid(name, mode, userblock_size, fapl, swmr=swmr)
270
271 if swmr_support:
~/miniconda3/envs/py36/lib/python3.6/site-packages/h5py/_hl/files.py in make_fid(name, mode, userblock_size, fapl, fcpl, swmr)
103 fid = h5f.create(name, h5f.ACC_EXCL, fapl=fapl, fcpl=fcpl)
104 elif mode == 'w':
--> 105 fid = h5f.create(name, h5f.ACC_TRUNC, fapl=fapl, fcpl=fcpl)
106 elif mode == 'a':
107 # Open in append mode (read/write).
h5py/_objects.pyx in h5py._objects.with_phil.wrapper()
h5py/_objects.pyx in h5py._objects.with_phil.wrapper()
h5py/h5f.pyx in h5py.h5f.create()
OSError: Unable to create file (unable to truncate a file which is already open)
[4]:
final = np.loadtxt('photospheres/model_photosphere.1d', skiprows=4)
start = np.loadtxt('photospheres/model_photosphere_200.1d', skiprows=4)
f = h5py.File('output.h5')
pl.plot(f['ph1']['T'][0,0,:])
pl.plot(final[:,1])
pl.plot(start[:,1], 'x')
f.close()
/scratch/miniconda3/envs/py36/lib/python3.6/site-packages/matplotlib/figure.py:2267: UserWarning: This figure includes Axes that are not compatible with tight_layout, so results might be incorrect.
warnings.warn("This figure includes Axes that are not compatible "
[ ]: