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

[Smith, Barry F.]

> On Nov 22, 2017, at 3:48 AM, Julian Andrej <juan at tf.uni-kiel.de> wrote:
> 
> Hello,
> 
> we prepared a small example which computes the gradient via the continuous adjoint method of a heating problem with a cost functional.

   Julian,

     The first thing to note is that the continuous adjoint is not exactly the same as the adjoint for the actual algebraic system you are solving. (It is only, as I understand it possibly the same in the limit with very fine mesh and time step). Thus you would not actually expect these to match with PETSc fd. Now as your refine space/time do the numbers get closer to each other?

  Note the angle cosine is very close to one which means that they are producing the same search direction, just different lengths.

   How is the convergence of the solver if you use -tao_fd_gradient do you still need unit.

> but have to use -tao_ls_type unit.

   This is slightly odd, because this line search always just takes the full step, the other ones would normally be better since they are more sophisticated in picking the step size. Please run without the -tao_ls_type unit.  and send the output

   Also does your problem have bound constraints? If not use -tao_type lmvm  and send the output. 

   Just saw Emil's email, yes there could easily be a scaling issue with your continuous adjoint.

  Barry



> 
> 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