Adjoint Optimization

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1 Adjoint Optimization p. Adjoint Optimization On State Constraint and Second Order Adjoint Computation Eka Suwartadi Norwegian University of Science and Technology

2 Adjoint Optimization p. Why adjoint optimization? An efficient way to compute gradients for optimal control

3 Adjoint Optimization p. Why adjoint optimization? An efficient way to compute gradients for optimal control Requires only two simulations regardless of the number of decision variables(n)

4 Adjoint Optimization p. Why adjoint optimization? An efficient way to compute gradients for optimal control Requires only two simulations regardless of the number of decision variables(n) Much more efficient than finite differences or integrating sensitivity equations which requires N+1 simulations

5 Adjoint State Constraint Adjoint Optimization p.

6 Adjoint Optimization p. Big Picture Objective Function : NPV Constraints Adjoint Optimization control inputs Oil Reservoir states control inputs: BHP, injection/production rate states: pressure, water saturation

7 Adjoint Optimization p. Adjoint Optimization Algorithm 1. Define the Lagrangian max uɛu J(u) subject to : c(x,u) = 0 L(x,u,λ) = J(x,u) + λ T c(x,u)

8 Adjoint Optimization p. Adjoint Optimization Algorithm cont d 2. Take first order approximation as optimality condition L(x,u,λ) = J(x,u) + λ T c(x,u) x L(x,u,λ) x=x(u),λ=λ(u) = 0 adjoint equations : T c λ + J T x x = 0

9 Adjoint Optimization p. Adjoint Optimization Algorithm cont d 3. The gradient w.r.t u: J(u) = u L(x,u,λ) x=x(u),λ=λ(u) u L(x,u,λ) T = J u + λt c u

10 Adjoint Optimization p. Adjoint in the presence of state constraints max uɛu J(u) subject to : c(x,u) = 0 g(x, u) = 0 It is detrimental to adjoint optimization

11 Adjoint Optimization p. Adjoint in the presence of state constraints max uɛu J(u) subject to : c(x,u) = 0 g(x, u) = 0 It is detrimental to adjoint optimization Since the states are functions of the control inputs

12 Adjoint Optimization p. Adjoint in the presence of state constraints max uɛu J(u) subject to : c(x,u) = 0 g(x, u) = 0 It is detrimental to adjoint optimization Since the states are functions of the control inputs Lead to difficulties to compute the Jacobian of the constraints

13 Adjoint Optimization p. Why difficult to compute Jacobian? Objective function : J ( x 1,,x n 1,u 1,,u n 1) = n i=1 J i J : U R Adjoint u J : R R n u g : R n x n u R n g Jacobian x(u) g : R n g R n g n u

14 Adjoint Optimization p. 1 Mitigating the state constraint problem 1. KS (Kreisselmeier-Steinhauser) function - aggregating into one constraint KS(x,ρ) = 1 m ρ ln (ρg j (x)) j=1

15 Adjoint Optimization p. 1 Mitigating the state constraint problem 1. KS (Kreisselmeier-Steinhauser) function - aggregating into one constraint KS(x,ρ) = 1 m ρ ln (ρg j (x)) j=1 2. Smoothed penalty function, e.g Sarma s work (2006)

16 Adjoint Optimization p. 1 Mitigating the state constraint problem 1. KS (Kreisselmeier-Steinhauser) function - aggregating into one constraint KS(x,ρ) = 1 m ρ ln (ρg j (x)) j=1 2. Smoothed penalty function, e.g Sarma s work (2006) 3. Jacobian approximation,e.g TR1, TR2 algorithms (A.Griewank,2005)

17 Adjoint Optimization p. 1 Mitigating the state constraint problem 1. KS (Kreisselmeier-Steinhauser) function - aggregating into one constraint KS(x,ρ) = 1 m ρ ln (ρg j (x)) j=1 2. Smoothed penalty function, e.g Sarma s work (2006) 3. Jacobian approximation,e.g TR1, TR2 algorithms (A.Griewank,2005) 4. Barrier function or exact penalty function

18 Adjoint Optimization p. 1 Barrier function A barrier function(logarithmic) ensures feasibility max J(u) + µ u n log(g n (u)) approach optimal solution with µ 0

19 A Simple Example Optimization Adjoint Optimization p. 1

20 Adjoint Optimization p. 1 A Simple Example - cont d J(v n,s n,u n,y n ) = N J n s.t : B n (s n 1 ) C D C T 0 0 D T 0 0 v n p n π n = n=1 Fu n 0 Hu n Saturation equation which is numerically integrated using Newton method. v 0 q inj u 1 x 0 = p 0 q prd1 u 2 π 0, un = q prd2 = u 3 s 0 q prd3 u 4 q prd4 u 5

21 Adjoint Optimization p. 1 A Simple Example - cont d yn := u n s n prd1 s n prd2 s n prd3 s n prd

22 Objective function evolution Adjoint Optimization p. 1

23 Constraint Satisfaction Adjoint Optimization p. 1

24 Adjoint Optimization p. 1 Conclusion Efficient adjoint method honouring state constraints has been tested successfully

25 Second Order Adjoint Computation Adjoint Optimization p. 1

26 Adjoint Optimization p. 1 Overview Adjoint Optimization End up with BFGS/L-BFGS method

27 Adjoint Optimization p. 1 Overview Adjoint Optimization End up with BFGS/L-BFGS method BFGS gives superlinear convergence rate

28 Adjoint Optimization p. 1 Overview Adjoint Optimization End up with BFGS/L-BFGS method BFGS gives superlinear convergence rate Why not use second order adjoint / Newton method?

29 Adjoint Optimization p. 1 Overview Adjoint Optimization End up with BFGS/L-BFGS method BFGS gives superlinear convergence rate Why not use second order adjoint / Newton method? Newton method gives quadratic convergence rate

30 Adjoint Optimization p. 1 Overview Adjoint Optimization End up with BFGS/L-BFGS method BFGS gives superlinear convergence rate Why not use second order adjoint / Newton method? Newton method gives quadratic convergence rate Second order information increases convergence rate in optimization

31 Adjoint Optimization p. 2 A motivating example of the Adjoint Newton Navon et.al (1997,2007)

32 Adjoint Optimization p. 2 References of Adjoint Newton Method 1. M. Heinkenschloss, Rice University(2008), Numerical Solution of Implicitly Constrained Optimization Problems

33 Adjoint Optimization p. 2 References of Adjoint Newton Method 1. M. Heinkenschloss, Rice University(2008), Numerical Solution of Implicitly Constrained Optimization Problems 2. M. Heinkenschloss, ACM(1999), An interface between Optimization and Application for the Numerical Solution of Optimal Control Problems

34 Adjoint Optimization p. 2 References of Adjoint Newton Method 1. M. Heinkenschloss, Rice University(2008), Numerical Solution of Implicitly Constrained Optimization Problems 2. M. Heinkenschloss, ACM(1999), An interface between Optimization and Application for the Numerical Solution of Optimal Control Problems 3. Chapter 5 of Ito and Kunisch book, SIAM(2008): Langrange Multiplier Approach to Variational Problems and Applications

35 Adjoint Optimization p. 2 Optimization Formulation U is a convex set. U R n u f : U R c : R n x n u R n x minĵ(u) u U subject to : c(x,u) = 0

36 Adjoint Optimization p. 2 Hessian Computation Procedure 1. Define the Lagrangian L(x,u,λ) = J(x,u) + λ T c(x,u)

37 Adjoint Optimization p. 2 Hessian Computation Procedure cont d 2. Take first order approximation as optimality condition L(x,u,λ) = J(x,u) + λ T c(x,u) x L(x,u,λ) x=x(u),λ=λ(u) = 0 adjoint equations : T c λ + J T x x = 0

38 Adjoint Optimization p. 2 Hessian Computation Procedure cont d 3. The gradient w.r.t u: J(u) = u L(x,u,λ) x=x(u),λ=λ(u) u L(x,u,λ) T = J u + λt c u

39 Adjoint Optimization p. 2 Hessian Computation Procedure cont d 4. Solve Newton equation using conjugate-gradient to get δu or v

40 Adjoint Optimization p. 2 Hessian Computation Procedure cont d 4. Solve Newton equation using conjugate-gradient to get δu or v 5. Solve w :c x (x(u),u) w = c u (x(u),u) v

41 Adjoint Optimization p. 2 Hessian Computation Procedure cont d 4. Solve Newton equation using conjugate-gradient to get δu or v 5. Solve w :c x (x(u),u) w = c u (x(u),u) v 6. Solve p :c x (x(u),u) T p = xx L(x,u,λ) w xu L(x,u,λ) v

42 Adjoint Optimization p. 2 Hessian Computation Procedure cont d 4. Solve Newton equation using conjugate-gradient to get δu or v 5. Solve w :c x (x(u),u) w = c u (x(u),u) v 6. Solve p :c x (x(u),u) T p = xx L(x,u,λ) w xu L(x,u,λ) v 7. Then the Hessian-vector is 2 Ĵ(u)v = c u (x(u),u) T p ux L(x,u,λ)w+ uu L(x,u,λ)v

43 Adjoint Optimization p. 2 Application to Oil Reservoir Model max u J (u k,x k ) = N k=1 J k s.t : B k ( s k 1) C D C T 0 0 D T 0 0 v k p k π k = Fu k 0 Hu k (1) s k = s k 1 + t i k ( v k,s k,u k) (2) with initial input u 0 and state x 0 = ( p 0 s 0 ) T

44 Adjoint Optimization p. 2 Application to Oil Reservoir Model cont d Apply the Hessian-vector procedure : 1. Compute the pressure and saturation solution using IMPES method 2. Compute the Lagrangian multipliers by solving J = N k=1 J k + λ kt v +λ kt s ( B k v k Cp k + Dπ k Fu k f k) +λ kt p C T v k +λ kt ( ( π Dv k Hu k h k) s k s k 1 + ti k ( v k,s k,u k))

45 Adjoint Optimization p. 2 Application to Oil Reservoir Model cont d J v k = Jk v k + λkt v B k + λ kt p C T + λ kt π D + tλ kt s J p k = λ kt v C J π k = λ kt v D J s k = Jk s k + λ(k+1)t v for k = N,...,1 with s n(bk+1 v k+1 ) + λ kt s i k v k ) (I t ik s k J u k = Jk u k λkt v F λ kt π H

46 Adjoint Optimization p. 3 Application to Oil Reservoir Model cont d 3. Using conjugate gradient method to compute δ u 4. Solve w or derivative of the states { ( s k B k+1 v k+1) } I ws k+1 = for k = 0,...,N 5. Solve p or derivative of the Lagrangian multipliers ( ( ) I t i k T s )p k k s = ( ) t ik s I w k k s ( B k + C k + D k + t ik v )w k k v Cwp k Dwπ k Fδ k u Hδ k u p k+1 s { 2 J k s k w k s } T { ( s B k+1 v k+1)} T p k+1 k v and

47 Adjoint Optimization p. 3 Application to Oil Reservoir Model cont d C D C k 0 0 D k 0 0 B k for k = N,...,1 p k v p k p p k π = 6. The second order adjoint: t ( i k v k ) T p k s J u 2 = pkt v F p kt π H

48 Adjoint Optimization p. 3 Hessian Validation Compare the Hessian from finite difference Check property of self-adjointness of the Hessian

49 Adjoint Optimization p. 3 Current Status Theoretical derivation has been completed Implementing Adjoint-Hessian Collaboration with Stein Krogstad (SINTEF) A paper will be presented at SPE-RCSC

50 Thank you! Adjoint Optimization p. 3

AM 205: lecture 19. Last time: Conditions for optimality, Newton s method for optimization Today: survey of optimization methods

AM 205: lecture 19. Last time: Conditions for optimality, Newton s method for optimization Today: survey of optimization methods AM 205: lecture 19 Last time: Conditions for optimality, Newton s method for optimization Today: survey of optimization methods Quasi-Newton Methods General form of quasi-newton methods: x k+1 = x k α