GammaLib

Here the output of the optimizer when fitting the diffuse model in an unbinned analysis:

>Iteration   0: -logL=32632.073, Lambda=1.0e-03
>Iteration   1: -logL=32632.024, Lambda=1.0e-03, delta=0.049, max(|grad|)=-70.118144 [Index:7]
 Iteration   2: -logL=32632.024, Lambda=1.0e-04, delta=-0.519, max(|grad|)=-30.237027 [Index:2] (stalled)
 Iteration   3: -logL=32632.024, Lambda=1.0e-03, delta=-0.389, max(|grad|)=-29.206366 [Index:2] (stalled)
>Iteration   4: -logL=32631.434, Lambda=1.0e-02, delta=0.590, max(|grad|)=-21.309557 [Index:2]
 Iteration   5: -logL=32631.434, Lambda=1.0e-03, delta=-0.893, max(|grad|)=-54.630362 [Index:7] (stalled)
 Iteration   6: -logL=32631.434, Lambda=1.0e-02, delta=-0.570, max(|grad|)=-41.063943 [Index:7] (stalled)
>Iteration   7: -logL=32629.450, Lambda=1.0e-01, delta=1.984, max(|grad|)=4.124423 [Sigma:5]
>Iteration   8: -logL=32629.412, Lambda=1.0e-02, delta=0.038, max(|grad|)=1.424688 [Index:7]
 Iteration   9: -logL=32629.412, Lambda=1.0e-03, delta=-0.004, max(|grad|)=-1.126451 [Index:7] (stalled)
 Iteration  10: -logL=32629.412, Lambda=1.0e-02, delta=-0.002, max(|grad|)=-0.771307 [Index:7] (stalled)
>Iteration  11: -logL=32629.409, Lambda=1.0e-01, delta=0.003, max(|grad|)=0.751852 [Index:7]
>Iteration  12: -logL=32629.409, Lambda=1.0e-02, delta=0.000, max(|grad|)=-0.163063 [Index:2]
 Iteration  13: -logL=32629.409, Lambda=1.0e-03, delta=-0.000, max(|grad|)=-0.334147 [Index:7] (stalled)
 Iteration  14: -logL=32629.409, Lambda=1.0e-02, delta=-0.000, max(|grad|)=-0.227450 [Index:7] (stalled)
>Iteration  15: -logL=32629.409, Lambda=1.0e-01, delta=0.000, max(|grad|)=0.237633 [Index:7]
>Iteration  16: -logL=32629.409, Lambda=1.0e-02, delta=0.000, max(|grad|)=-0.064887 [Index:2]
 Iteration  17: -logL=32629.409, Lambda=1.0e-03, delta=-0.000, max(|grad|)=-0.130886 [Index:7] (stalled)
 Iteration  18: -logL=32629.409, Lambda=1.0e-02, delta=-0.000, max(|grad|)=-0.088992 [Index:7] (stalled)
>Iteration  19: -logL=32629.408, Lambda=1.0e-01, delta=0.000, max(|grad|)=0.096239 [Index:7]
>Iteration  20: -logL=32629.408, Lambda=1.0e-02, delta=0.000, max(|grad|)=-0.026185 [Index:2]

I played around with the various integration precision in the Irf and Npred computation without any notable change in the number of iterations required for the fit or in the log-likelihood profile of the spectral index.

Maybe this is just a question of fit precision. Increasing the requirement on the optimizer precision increases the number of iterations but leads to a better minimum in the spectral index, as indicated in the plots below that show the log-likelihood profile for a precision of 1e-6 (left), 1e-8 (centre) and 1e-10 (right).

The table below shows the fit results as function of requested precision:

So the results do not depend sensitively on the precision, but the gradients do. Note that the default GOptimizeLM precision is 1.0-6, which is also the value used in ctlike.

The question remains why the fit needs so many iterations? Possibly do to the diffuse nature of the source, which leads to a pretty strong correlation of source and background model. The biggest gradients are observed in the index of the background and the source model, and the fit alters between improvements and stalls. Maybe this could be changed if the full Hessian would be used in the fitting, but with the present scheme it looks like we have to live with this.