1

Adjoint based gradient calculation - advantantages and challenges

Bjarne Foss, Ruben RingsetThe Norwegian University of Science & Technology – NTNUIO center

Outline1. Motivation

2. A simple example to illustrate the potential of adjoints

3. Where are the hurdles?

4. Conclusions

2

Motivation

Norne fieldStatoilHydroEni, Petoro

Model

Data

3

for k=1 to N ...simulate(k) end

model well schedule

+

simulator forecast

timenow

Optimize

now

history

Parameter estimation

Uncertainty

Motivation

4

Reservoir Wells Pipelines Process Utilities Pipelines/tankers

Market

Inletseparator

Reservoir and well models

(Eclipse)

Network model(GAP, MaxPro,

OLGA)

Process model(HYSIS)

X 1

N G

L N G

C W 1

C 2

C W2 A / B

E 1 A

E 1 B

E 2

E 3

P r e - c o o l i n gP r e - c o o l i n gS e c t i o nS e c t i o n

L i q u e f a c t i o nL i q u e f a c t i o n S e c t i o nS e c t i o n

S u b - c o o l i n gS u b - c o o l i n gS e c t i o nS e c t i o n

C W3 A / B

C 1

C 3G

Application Value chain optimization

Application Value chain optimization

Motivation

Optimization requires a large number of gradient calculations

Efficient gradient computations are important

5

A simple example

kkkkk

kkkk

kkk

k

uuxxz

ukxkxx

uxk

xx

,2,1,2,1

,2,2,11,2

,12,2

,11,1

2.01.0

Output

)sin(

systemNonlinear

]2.01.0[ ],11[

10

01 ,

)sin(

21

Jacobians

,2

kk

kk

k

DC

Bkk

xkA

kk

kk

xku

xu

,1,2

2,2,1

law control leStabilizab

N

TNk

Tk

N

kk

Tk QxxRuuQzzJ

2

1][

2

1

function Objective1

0

6

A simple example

7

A simple example

8

length simulation theis where

with increases Runtime 2

N

NNwith

linearly increases Runtime

A simple example

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)

0),0(min(),0(max(

,,,1,

,min1 1

max,,

maxmin

ik

N

k

N

iiik

Nkk

zzzzy

RzNkzzzz

z

14

The output constraint challenge

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)

0),0(min(),0(max(

,,,1,

,min1 1

max,,

maxmin

ik

N

k

N

iiik

Nkk

zzzzy

RzNkzzzz

z

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

17

18

Once again - A very simple example

ufuxf u )(2

1 22

u

Let

and assume that is the independent variable, i.e. Compute the gradient wrt

01),( uxuxg

)(),( uux

)1()(2

1 22 uxuxL Lagrangian function

du

dL

du

dx

x

L

u

LLu

du

duxxuLu

)1()1)(()(

u

Choose x

12))1(( uuuLu 0),( when 12 uxgufu

(”reverse simulation”)