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

[Julian Andrej]

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.

$ 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