[petsc-users] TAO: Finite Difference vs Continuous Adjoint gradient issues

[Emil Constantinescu]

On 11/22/17 3:48 AM, Julian Andrej wrote:
> Hello,
> 
> we prepared a small example which computes the gradient via the 
> continuous adjoint method of a heating problem with a cost functional.
> 
> We implemented the text book example and tested the gradient via a 
> Taylor Remainder (which works fine). Now we wanted to solve the
> optimization problem with TAO and checked the gradient vs. the finite 
> difference gradient and run into problems.
> 
> Testing hand-coded gradient (hc) against finite difference gradient 
> (fd), if the ratio ||fd - hc|| / ||hc|| is
> 0 (1.e-8), the hand-coded gradient is probably correct.
> Run with -tao_test_display to show difference
> between hand-coded and finite difference gradient.
> ||fd|| 0.000147076, ||hc|| = 0.00988136, angle cosine = 
> (fd'hc)/||fd||||hc|| = 0.99768
> 2-norm ||fd-hc||/max(||hc||,||fd||) = 0.985151, difference ||fd-hc|| = 
> 0.00973464
> max-norm ||fd-hc||/max(||hc||,||fd||) = 0.985149, difference ||fd-hc|| = 
> 0.00243363
> ||fd|| 0.000382547, ||hc|| = 0.0257001, angle cosine = 
> (fd'hc)/||fd||||hc|| = 0.997609
> 2-norm ||fd-hc||/max(||hc||,||fd||) = 0.985151, difference ||fd-hc|| = 
> 0.0253185
> max-norm ||fd-hc||/max(||hc||,||fd||) = 0.985117, difference ||fd-hc|| = 
> 0.00624562
> ||fd|| 8.84429e-05, ||hc|| = 0.00594196, angle cosine = 
> (fd'hc)/||fd||||hc|| = 0.997338
> 2-norm ||fd-hc||/max(||hc||,||fd||) = 0.985156, difference ||fd-hc|| = 
> 0.00585376
> max-norm ||fd-hc||/max(||hc||,||fd||) = 0.985006, difference ||fd-hc|| = 
> 0.00137836
> 
> Despite these differences we achieve convergence with our hand coded 
> gradient, but have to use -tao_ls_type unit.

Both give similar (assume descent) directions, but seem to be scaled 
differently. It could be a bad scaling by the mass matrix somewhere in 
the continuous adjoint. This could be seen if you plot them side by side 
as a quick diagnostic.

Emil

> $ python heat_adj.py -tao_type blmvm -tao_view -tao_monitor -tao_gatol 
> 1e-7 -tao_ls_type unit
> iter =   0, Function value: 0.000316722,  Residual: 0.00126285
> iter =   1, Function value: 3.82272e-05,  Residual: 0.000438094
> iter =   2, Function value: 1.26011e-07,  Residual: 8.4194e-08
> Tao Object: 1 MPI processes
>    type: blmvm
>        Gradient steps: 0
>    TaoLineSearch Object: 1 MPI processes
>      type: unit
>    Active Set subset type: subvec
>    convergence tolerances: gatol=1e-07,   steptol=0.,   gttol=0.
>    Residual in Function/Gradient:=8.4194e-08
>    Objective value=1.26011e-07
>    total number of iterations=2,                          (max: 2000)
>    total number of function/gradient evaluations=3,      (max: 4000)
>    Solution converged:    ||g(X)|| <= gatol
> 
> $ python heat_adj.py -tao_type blmvm -tao_view -tao_monitor 
> -tao_fd_gradient
> iter =   0, Function value: 0.000316722,  Residual: 4.87343e-06
> iter =   1, Function value: 0.000195676,  Residual: 3.83011e-06
> iter =   2, Function value: 1.26394e-07,  Residual: 1.60262e-09
> Tao Object: 1 MPI processes
>    type: blmvm
>        Gradient steps: 0
>    TaoLineSearch Object: 1 MPI processes
>      type: more-thuente
>    Active Set subset type: subvec
>    convergence tolerances: gatol=1e-08,   steptol=0.,   gttol=0.
>    Residual in Function/Gradient:=1.60262e-09
>    Objective value=1.26394e-07
>    total number of iterations=2,                          (max: 2000)
>    total number of function/gradient evaluations=3474,      (max: 4000)
>    Solution converged:    ||g(X)|| <= gatol
> 
> 
> We think, that the finite difference gradient should be in line with our 
> hand coded gradient for such a simple example.
> 
> We appreciate any hints on debugging this issue. It is implemented in 
> python (firedrake) and i can provide the code if this is needed.
> 
> Regards
> Julian