Adjoint approach to optimization

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1 Adjoint approach to optimization Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore Health, Safety and Environment Group BARC 6-7 October, 2010 Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

2 Outline Minimum of a function Constrained minimization Finite difference approach Adjoint approach Automatic differentiation Example Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

3 Minimum of a function f(x) f'(x 0 )=0 x 0 x Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

4 Steepest descent method x 2 grad f(x 1,x 2 ) x 1 x n+1 = x n s n f(x n ) Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

5 Objectives and controls Objective function J(α) = J(α, u) mathematical representation of system performance Control variables α Parametric controls α R n Infinite dimensional controls α : X Y Shape α set of admissible shapes State variable u: solution of an ODE or PDE R(α, u) = 0 u = u(α) Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

6 Gradient-based minimization: blackbox/fd approach Initialize β 0, n = 0 For n = 0,..., N iter Solve R(β n, Q n ) = 0 For j = 1,..., N min I(β, Q(β)) β RN β n (j) = [β n 1,..., β n j + β j,..., β n N ] Solve R(β n (j), Q n (j)) = 0 di dβ j I(β n (j),qn (j) ) I(βn,Q n ) β j Steepest descent step Cost of FD-based steepest-descent β n+1 = β n s n di dβ (βn ) Cost = O(N + 1)N iter = O(N + 1)O(N) = O(N 2 ) Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

7 Accuracy of FD: Choice of step size d dx f(x 0) = f(x 0 + δ) f(x 0 ) + O(δ) δ In principle, if we choose a small δ, we reduce the error. But computers have finite precision. Instead of f(x 0 ) the computers gives f(x 0 ) + O(ɛ) where ɛ = machine precision. [f(x 0 + δ) + O(ɛ)] [f(x 0 ) + O(ɛ)] δ = d dx f(x ɛ 0) + C 2 δ + C 1 }{{ δ} Total error = f(x 0 + δ) f(x 0 ) δ, C 1, C 2 depend on f, x 0, precision + C 1 ɛ δ Total error is least when δ = δ opt = C 3 ɛ, C3 = C 1 /C 2 Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

8 Accuracy of FD: Choice of step size Error Precision ɛ δ opt Single Double C 1 δ C 2 ǫ/δ See: Brian J. McCartin, Seven Deadly Sins of Numerical Computation The American Mathematical Monthly, Vol. 105, No. 10 (Dec., 1998), pp δ opt δ Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

9 Drag gradient using FD (Samareh) Errors in Finite Difference Approximations 10 1 Mid chord % Error Round!off LE Sweep2 Tip chord LE sweep3 Twist (root) Twist (mid) Twist (tip) Truncation Twist (root) Twist (mid) Mid chord LE Sweep2 LE Sweep3 Tip chord Twist (tip) Scaled Step Size Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

10 Iterative problems I(β, Q), where R(β, Q) = 0 Q is implicitly defined, require an iterative solution method. Assume a Q 0 and iterate Q n Q n+1 until R(β, Q n ) TOL If TOL is too small, need too many iterations Many problems, we cannot reduce R(β, Q n ) to small values This means that numerical value of I will be noisy Finite difference will contain too much error, and is useless RAE5243 airfoil, Mach=0.68, Re=19 million, AOA=2.5 deg. iter Lift Drag E E E E E-002 Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

11 Complex variable method f(x 0 + iδ) = f(x 0 ) + iδf (x 0 ) + O(δ 2 ) + io(δ 3 ) f (x 0 ) = 1 δ imag[f(x 0 + iδ)] + O(δ 2 ) No roundoff error We can take δ to be very small, δ = Can be easily implemented fortran: redeclare real variables as complex matlab: no change Iterative problems: β β + i β Obtain Q = Q(β + i β) by solving R(β + i β, Q) = 0 Then gradient I (β) 1 imag[i(β + i β, Q(β + i β)] β Computational cost is O(N 2 ) or higher (due to complex arithmetic) Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

12 Objectives and controls Objective function J(α) = J(α, u) mathematical representation of system performance Control variables α Parametric controls α R n Infinite dimensional controls α : X Y Shape α set of admissible shapes State variable u: solution of an ODE or PDE R(α, u) = 0 Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

13 Mathematical formulation Constrained minimization problem Find δα such that δj < 0 min α J(α, u) subject to R(α, u) = 0 δj = J J δα + α u δu = J J u δα + = [ α J α + J u u α δα u α ] δα =: Gδα Steepest descent δα = ɛg δj = ɛgg = ɛ G 2 < 0 Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

14 Mathematical formulation Constrained minimization problem Find δα such that δj < 0 min α J(α, u) subject to R(α, u) = 0 δj = J J δα + α u δu = J J u δα + = [ α J α + J u u α δα u α ] δα =: Gδα Steepest descent δα = ɛg δj = ɛgg = ɛ G 2 < 0 Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

15 Sensitivity approach Linearized state equation R(α, u) = 0 or R R δα + α u δu = 0 R u u α = R α Solve sensitivity equation iteratively, e.g., Gradient u t α + R u u α = R α dj dα = J α + J u u α Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

16 Sensitivity approach: Computational cost n design variables: α = (α 1,..., α n ) Solve primal problem R(α, u) = 0 to get u(α) For i = 1,..., n Solve sensitivity equation wrt αi R u u = R α i α i Compute derivative wrt αi dj = J + J dα i α i u u α i One primal equation, n sensitivity equations Computational cost = n + 1 Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

17 Adjoint approach We have δj = J J R R δα + δu and δα + α u α u δu = 0 Introduce a new unknown v δj = J ( ) J R R δα + δu + v δα + α u α u δu ( ) ( ) J R J R = + v δα + + v δu α α u u Adjoint equation Iterative solution ( ) R ( ) J v = u u v t + ( ) R v = u ( ) J u Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

18 Adjoint approach: Computational cost n design variables: α = (α 1,..., α n ) Solve primal problem R(α, u) = 0 to get u(α) Solve adjoint problem For i = 1,..., n Compute derivative wrt αi ( ) R ( ) J v = u u dj = J + v R dα i α i α i One primal equation, one adjoint equation Computational cost = 2, independent of n Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

19 Adjoint: Two approaches Continuous or differentiate-discretize PDE Discrete or discretize-differentiate PDE Adjoint PDE Discrete PDE Discrete adjoint Discrete adjoint Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

20 Techniques for computing gradients Hand differentiation Finite difference method Complex variable method Automatic Differentiation (AD) Computer code to compute J(α, u) and R(α, u) Chain rule of differentiation Generates a code to compute derivatives ADIFOR, ADOLC, ODYSEE, TAMC, TAF, TAPENADE see Code for J(α, u) α AD tool Code for J α Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

21 Derivatives Given a program P computing a function F F : R m R n X Y build a program that computes derivatives of F X : independent variables Y : dependent variables Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

22 Derivatives [ ] Yj Jacobian matrix: J = X i Directional or tangent derivative Ẏ = JẊ Adjoint mode X = J Ȳ Gradients (n = 1 output) [ ] Y J = = Y X i Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

23 Forward differentiation Program P is a sequence of computations F k T 0 = X, given k th line t k = F k (T k 1 ), T k = Function is a composition [ Tk 1 F = F p F p 1... F 2 F 1 Chain rule Ẏ = F (X)Ẋ = F p(t p 1 )F p 1(T p 2 )... F 2(T 1 )F 1(T 0 )Ẋ X, Ẋ Y, Ẏ cost(ẏ ) = 4 cost(y ) t k ] Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

24 Differentiation: Example A simple example f = (xy + sin x + 4)(3y 2 + 6) Computer code, f = t 10 t1 = x t 2 = y t 3 = t 1 t 2 t 4 = sin t 1 t 5 = t 3 + t 4 t 6 = t t 7 = t 2 2 t 8 = 3t 7 t 9 = t t 10 = t 6 t 9 Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

25 F77 code: costfunc.f s ubroutine c o s t f u n c ( x, y, f ) t1 = x t2 = y t3 = t1 t2 t4 = s i n ( t1 ) t5 = t3 + t4 t6 = t5 + 4 t7 = t2 2 t8 = 3.0 t7 t9 = t t10 = t6 t9 f = t10 end Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

26 Differentiation: Direct mode Apply chain rule of differentiation t 1 = x ṫ 1 = ẋ t 2 = y ṫ 2 = ẏ t 3 = t 1 t 2 ṫ 3 = ṫ 1 t 2 + t 1 ṫ 2 t 4 = sin(t 1 ) ṫ 4 = cos(t 1 )ṫ 1 t 5 = t 3 + t 4 ṫ 5 = ṫ 3 + ṫ 4 t 6 = t ṫ 6 = ṫ 5 t 7 = t 2 2 ṫ 7 = t 8 = 3t 7 ṫ 8 = 2t 2 ṫ 2 3ṫ 7 t 9 = t ṫ 9 = ṫ 8 t 10 = t 6 t 9 ṫ 10 = ṫ 6 t 9 + t 6 ṫ 9 ẋ = 1, ẏ = 0, ṫ 10 = f x and ẋ = 0, ẏ = 1, ṫ 10 = f y tapenade -d -vars "x y" -outvars f costfunc.f Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

27 Automatic Differentiation: Direct mode SUBROUTINE COSTFUNC_D(x, xd, y, yd, f, fd) t1d = xd t1 = x t2d = yd t2 = y t3d = t1d*t2 + t1*t2d t3 = t1*t2 t4d = t1d*cos(t1) t4 = SIN(t1) t5d = t3d + t4d t5 = t3 + t4 t6d = t5d t6 = t5 + 4 t7d = 2*t2*t2d t7 = t2**2 t8d = 3.0*t7d t8 = 3.0*t7 t9d = t8d t9 = t t10d = t6d*t9 + t6*t9d t10 = t6*t9 fd = t10d f = t10 END Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

28 Backward differentiation Program P is a sequence of computations F k T 0 = X, given k th line t k = F k (T k 1 ), T k = Function is a composition [ Tk 1 F = F p F p 1... F 2 F 1 Chain rule X = [F (X)] Ȳ = [F 1(T 0 )] [F 2(T 1 )]... [F p 1(T p 2 )] [F p(t p 1 )] Ȳ X, Ȳ X cost( X) = 4 cost(y ) t k ] Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

29 Differentiation: Reverse mode Apply chain rule of differentiation in reverse t 1 = x t 10 = 1 t 2 = y t 9 = t 10 t 10,9 = t 6 t 3 = t 1 t 2 t 8 = t 9 t 9,8 = t 6 t 4 = sin(t 1 ) t 7 = t 8 t 8,7 = 3t 6 t 5 = t 3 + t 4 t 6 = t 10 t 10,6 = t 9 t 6 = t t 5 = t 6 t 6,5 = t 9 t 7 = t 2 2 t 4 = t 5 t 5,4 = t 9 t 8 = 3t 7 t 3 = t 5 t 5,3 = t 9 t 9 = t t 2 = t 7 t 7,2 + t 3 t 3,2 = 6t 2 t 6 + t 1 t 9 t 10 = t 6 t 9 t 1 = t 4 t 4,1 + t 3 t 3,1 = t 9 cos(t 1 ) + t 9 t 2 t 1 = f x, t 2 = f y tapenade -b -vars "x y" -outvars f costfunc.f Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

30 Automatic Differentiation: Reverse mode SUBROUTINE COSTFUNC_B(x, xb, y, yb, f, fb) t1 = x t2 = y t3 = t1*t2 t4 = SIN(t1) t5 = t3 + t4 t6 = t5 + 4 t7 = t2**2 t8 = 3.0*t7 t9 = t t10b = fb t6b = t9*t10b t9b = t6*t10b t8b = t9b t7b = 3.0*t8b t5b = t6b t3b = t5b t2b = t1*t3b + 2*t2*t7b t4b = t5b t1b = t2*t3b + COS(t1)*t4b yb = t2b xb = t1b fb = 0.0 END Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

31 Direct versus reverse AD F : R m R n Direct mode Reverse mode cost(j) = m 4 cost(p ) cost(j) = n 4 cost(p ) Scalar output F R, n = 1 Direct mode gives F Ẋ for given vector Ẋ Reverse mode gives F, hence preferred Vector output F R n Direct mode gives F Ẋ for given vector Ẋ use for sensitivity equation approach Reverse mode gives ( F ) Ȳ use for adjoint approach Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

32 Black-box AD cost depends on alpha call ComputeCost(alpha, cost) Subroutine for cost function subroutine ComputeCost(alpha, cost) call SolveState(alpha, u) call CostFun(alpha, u, cost) end Reverse differentiation using AD tapenade -backward \ -head ComputeCost \ -vars alpha \ -outvars cost \ ComputeCost.f SolveState.f CostFun.f Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

33 Black-box AD Diffentiated subroutines: ComputeCost b.f, SolveState b.f, CostFun b.f subroutine ComputeCost_b(alpha,alphab,cost,costb) call SolveState(alpha, u) call CostFun_b(alpha, alphab, u, ub, cost, costb) call SolveState_b(alpha, alphab, u, ub) end Compute gradient using costb = 1.0 call ComputeCost_b(alpha, alphab, cost, costb) Gradient given by alphab alphab = (cost) (alpha) Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

34 AD for iterative problems Solve state equation R(α, u) = 0 as steady state of du dt + R(α, u) = 0, u(0) = u o State solver subroutine SolveState(alpha, u) u = 0.0 do call Residue(alpha, u, res) u = u - dt * res while( abs(res) > TOL) end subroutine Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

35 AD for iterative problems Adjoint solver, hand written subroutine SolveState_b(alpha,alphab,u,ub) resb = 0.0 do ub1 = 0.0 call Residue_bu(alpha,u,ub1,res,resb) resb = resb - (ub + ub1) while( abs(ub+ub1).gt. 1.0e-5) call Residue_ba(alpha,alphab,u,res,resb) end subroutine resb = Adjoint variable ub = J u, ub1 = [ ] R v u Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

36 Adjoint iterative scheme Forward iterations linearly stable Adjoint iteration u n+1 = u n t R(α, u n ), t < S(σ(R )) v n+1 = v n t { [R (α, u )] v n + J } u [R ] has same eigenvalues as R = adjoint iterations stable under same condition on t Preconditioner for adjoint = (preconditioner for primal problem) Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

37 AD tools Source transformation (ST) and operator overloading (OO) See Tool Modes Method Lang ADIFOR F ST Fortran ADOLC F/B OO C TAPENADE F/B ST Fortran/C TAF/TAMC F/B ST Fortran/C MAD F OO Matlab Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

38 Quasi 1-D flow Inflow x h(x) Outflow Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

39 Quasi 1-D flow Quasi 1-D flow in a duct t (hu) + dh (hf) = P, x dx x (a, b) t > 0 ρ ρu 0 U = ρu, f = p + ρu 2, P = p E (E + p)u 0 h(x) = cross-section height of duct Inverse design: find shape h to get pressure distribution p Optimization problem: find the shape h which minimizes J = b a (p p ) 2 dx Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

40 Quasi 1-D flow Inflow face 1 i N i 1/2 i + 1/2 Outflow face Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

41 Quasi 1-D flow Finite volume scheme for cell C i = [x i 1/2, x i+1/2 ] du i h i dt + h i+1/2f i+1/2 h i 1/2 F i 1/2 x Discrete cost function = (h i+1/2 h i 1/2 ) P i x J = N (p i p i ) 2 i=1 Control variables N = 100 h 1/2, h 1+1/2,..., h i+1/2,..., h N+1/2 Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

42 Duct shape Duct shape Target shape Current shape 1.6 h x Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

43 Target pressure distribution p Target pressure AUSMDV KFVS LF Pressure x Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

44 Current pressure distribution Starting pressure AUSMDV KFVS LF Pressure x Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

45 Adjoint density 5 0 AUSMDV KFVS LF 5 Adjoint density x Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

46 Convergence history: Explicit Euler Residue Convergence history with AUSMDV flux Flow Adjoint No of iterations Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

47 Shape gradient Gradient AUSMDV KFVS LF 25 Gradient x Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

48 Shape gradient Gradient Gradient AUSMDV KFVS LF x Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

49 Validation of Shape gradient 35 Gradient with AUSMDV flux B Gradient A C x Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

50 Validation of shape gradient J h J(h + h) J(h h) 2 h h A B C AD Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

51 Gradient smoothing Non-smooth gradients G especially in the presence of shocks Smooth using an elliptic equation ) (1 ɛ(x) d2 dx 2 Ḡ(x) = G(x) L i = ɛ i = { G i+1 G i + G i G i 1 }L i G i+1 2G i + G i 1 max( G i+1 G i + G i G i 1, TOL) Finite difference with Jacobi iterations Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

52 Gradient smoothing Gradient using AUSMDV flux Original gradient Smoothed gradient 25 Gradient x Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

53 Quasi 1-D optimization: Shape Target Initial h x Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

54 Quasi 1-D optimization: Final shape h Target Initial Final x Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

55 Quasi 1-D optimization: Pressure Pressure 1.5 Target Initial Final x Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

56 Quasi 1-D optimization: Convergence 10 0 Cost Gradient Cost/Gradient No. of iterations Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

57 Quasi 1-D optimization: Adjoint density 0 5 Adjoint density Initial Final x Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

58 Example codes 1-D example: nozzle flow (TAPENADE) 1-D example: nozzle flow (ADOLC) 2-D example: unstructured grid Euler solver (TAPENADE) 2/3-D example: structured grid Euler solver (TAPENADE) Praveen. C (TIFR-CAM) Optimization BARC, 6 Oct / 58

Adjoint code development and optimization using automatic differentiation (AD)

Adjoint code development and optimization using automatic differentiation (AD) Praveen. C Computational and Theoretical Fluid Dynamics Division National Aerospace Laboratories Bangalore - 560 037 CTFD