1. Adjoint based gradient calculation
- advantantages and challenges
Bjarne Foss, Ruben Ringset
The Norwegian University of Science & Technology – NTNU
IO center
Outline
Motivation
A simple example to illustrate the potential of adjoints
Where are the hurdles?
Conclusions

2. Motivation
Model
Norne field
StatoilHydro
Eni, Petoro
Data

3. Motivation + now now time history Parameter estimation well schedule
model
simulator
forecast
for k=1 to N simulate(k) end +
Optimize
Let us take a look at some important workflows
Uncertainty

4. Optimization requires a large number of gradient calculations
Motivation
Inlet
separator
Pipelines/tankers
Market
Reservoir
Wells Pipelines
Process Utilities
Reservoir and
well models
(Eclipse)
Network model
(GAP, MaxPro,
OLGA)
Process model
(HYSIS)
Application
Value chain optimization
Optimization requires a large number of gradient calculations
Efficient gradient computations are important

5. A simple example

6. A simple example

7. A simple example

8. The explicit Jacobian exists
A simple example
The explicit Jacobian exists

9. Adjoint gradient computation

10. Adjoint gradient computation
Forward
simulation

11. Adjoint gradient computation
One forward
simulation
One reverse
simulation

12. Forward method
N forward
simulations
(nested loops)

13. The output constraint challenge – possible remedies
Reducing the number of constraints
Enforcing them on parts of a prediction horizon
Lumping output constraints together
One interesting application of this is found in the Standford GPRS reservoir simulator (Sarma et al, 2006)

14. The output constraint challenge
The feasible cone for the lumped constraints will be at least as large as the true feasible cone given by the null space of the active constraints
In GPRS they found a way to recover from such a situation by exploiting information about the structure of the model, in particular u->z

15. The output constraint challenge – possible remedies
Reducing the number of constraints
Enforcing them on parts of a prediction horizon
Lumping output constraints together
One interesting application of this is found in the Standford GPRS reservoir simulator (Sarma et al, 2006)
Taking advantage of barrier or interior point optimization methods
Removing output constraints without introducing slack variables
Model constraints (i.e. equality constraints) can be removed by a single shooting method (in eg. MPC)

16. Adjoint based gradient calculation
- advantantages and challenges
Conclusions
Adjoint based gradient calculation may give huge improvements in run-time
Output constraints is a challenge

18. Once again - A very simple example
Let
Lagrangian function
and assume that is the independent variable, i.e.
Compute the gradient wrt
Choose
(”reverse simulation”)