Semi-automatic transition from simulation to one-shot optimization with equality constraints

Transcription

1 Semi-automatic transition from simulation to one-shot optimization with equality constraints Lisa Kusch, Tim Albring, Andrea Walther, Nicolas Gauger Chair for Scientific Computing, TU Kaiserslautern, Institute of Mathematics, Universität Paderborn September 15, 2016 AD 2016

2 Outline 1 The One-Shot Approach with Additional Constraints 2 AD-Based Discrete Adjoint in SU2 3 Multi-Objective Optimization 4 Application and Results Application to Airfoil Design Multi-Objective Optimization with Single Constraint Multi-Objective Optimization with Multiple Constraints Second-Order Adjoints 5 Summary and Outlook Kusch,Albring,Walther,Gauger From simulation to one-shot optimization with constraints 2/ 18

3 Problem Definition PDE-constrained optimization problem min u,y f (y, u) s.t. c(y, u) = 0, h(y, u) = 0 fixed point iteration G y ρ < 1 min u,y f (y, u) s.t. G(y, u) = y, h(y, u) = 0 objective function f R, state y Y R n, design u U R m state equation c : Y U Y, constraints h : Y U V R p (notation: Ĝ = (G(y, u) y, h(y, u)) T ) one-shot approach using the AD-based discrete adjoint [Hamdi, Griewank(2010)],[Walther, Gauger, Kusch, Richert(2016)] L(ȳ, y, u) = f (y, u) + (G(y, u) y) T ȳ + h T µ = N(y, ȳ, µ, u) y T ȳ Kusch,Albring,Walther,Gauger From simulation to one-shot optimization with constraints 3/ 18

4 The One-Shot Approach with Additional Constraints KKT point (discrete) y = Nȳ (y, ȳ, µ, u ) T = G(y, u) ȳ = N y (y, ȳ, µ, u ) T = f y (y, u ) T + G y (y, u ) T ȳ 0 = N u (y, ȳ, µ, u ) T = f u (y, u ) T + G u (y, u ) T ȳ 0 = N µ (y, ȳ, µ, u ) T = h(y, u ) One-shot approach with equality constraints for k = 1,... y k+1 = G(y k, u k ) ȳ k+1 = N y (y k, ȳ k, µ k, u k ) u k+1 = u k B 1 k N u (y k, ȳ k, µ k, u k ) 1 µ k+1 = µ k ˇB k h(y k, u k ) Kusch,Albring,Walther,Gauger From simulation to one-shot optimization with constraints 4/ 18

5 Convergence Properties doubly augmented Lagrangian L a (y, ȳ, µ, u) = α 2 Ĝ(y, u) 2 + β 2 N y (y, ȳ, µ, u) T ȳ 2 + N ȳ T y correspondence and descent condition αβ(1 ρ) > 1 + β 2 N yy (1) and α 2(Ĝ u ) 1...p (Ĝ y ) + (N yu ) 1...p (N uu ) 1...p 1...p / (Ĝu) 1...p 2 exact penalty function, descent for all large symmetric pos. def. B, ˇB choice of B B 2 uul a (y, ȳ, u) B = αĝ u T Ĝu + βnyun T yu + N uu Kusch,Albring,Walther,Gauger From simulation to one-shot optimization with constraints 5/ 18

6 Implementation Details BFGS Update: B u u L a (y, ȳ, µ, u + u) u L a (y, ȳ, µ, u) =: r calculate u L a = α yk T G u + αhu T h + β ȳk T N yu + N u N u and yk T G u + hu T h can be obtained with reverse mode of AD from ū y = G(y, u) v = f (y, u) ū = f u (y, u) T v + G u (y, u) T ȳ + h u (y, u) T w w = h(y, u) finite difference over reverse / tangent over reverse ȳk T N yu 1 h (N u(y k + h ȳ k, ȳ k, µ k, u k ) N u (y k, ȳ k, µ k, u k )) (2) curvature condition: set B k = I if r T k u k 0 Kusch,Albring,Walther,Gauger From simulation to one-shot optimization with constraints 6/ 18

7 Open-source multiphysics suite SU2 Under active development (Stanford University, TU Kaiserslautern, TU Delft,...) FV discretization of (U)RANS equations and turbulence models using highly modular C++ code-structure, additional PDE solvers Multi-grid and local time-stepping methods Several space and time discretizations, MPI, python scripts optimization environment based on continuous or discrete adjoint, integrated AD support [Albring, Sagebaum, Gauger(2015)] Kusch,Albring,Walther,Gauger From simulation to one-shot optimization with constraints 7/ 18

8 Discrete-Adjoint Solver in SU2 operator overloading (frequent extensions/improvements) shape optimization mesh deformation as additional constraint in the optimization problem M(u) = x fixed-point form ȳ k+1 = N y (y k, ȳ k, µ k, u k ) T CoDiPack open-source ( use of expression templates different tape implementations Performance SU2 (NASA Common Research Model) Runtime Factor: 1.2 Memory Factor: 7.2 Kusch,Albring,Walther,Gauger From simulation to one-shot optimization with constraints 8/ 18

9 Multi-Objective Optimization Problem min y,u F(y, u) s.t. c(y, u) = 0, (3) Pareto dominance: x dominates x if h(y, u) 0 (4) i {1,..., k} : j {1,..., k} : f i (x) f i (x) f j (x) < f j (x) Kusch,Albring,Walther,Gauger From simulation to one-shot optimization with constraints 9/ 18

10 Methods for Multi-Objective Optimization common: weighted sum, evolutionary algorithms ε-constraint method [Marglin(1967)] min y,u f sj (y, u) s.t. c(y, u) = 0, (5) h(y, u) = 0, f i (x) f (j) i i {1,..., k} i s j connected: equality constraints Kusch,Albring,Walther,Gauger From simulation to one-shot optimization with constraints 10/ 18

11 Application (Airfoil Design) aerodynamic shape optimization of NACA 0012 airfoil objectives: drag coefficient c d, lift coefficient c l bounded design variables: 38 Hicks-Henne bump functions (bounds: [-0.005,0.005]) constraints: moment, thickness steady Euler equations, transonic flow (Mach 0.8 and AOA 1.25) Kusch,Albring,Walther,Gauger From simulation to one-shot optimization with constraints 11/ 18

12 Result: Lift Constraint preconditioner ˇB = 20, µ 0 = 0, target lift: 0.4 backtracking line search, trial step size: 1, α = 20, β = c d c l 10 N u 2 4 primal residual adjoint residual ,200 1,600 2,000 Iteration ,000 1,500 2,000 Iteration Kusch,Albring,Walther,Gauger From simulation to one-shot optimization with constraints 12/ 18

13 Resulting Pareto-optimal Front drag coefficient lift coefficient Kusch,Albring,Walther,Gauger From simulation to one-shot optimization with constraints 13/ 18

14 Result: Lift, Moment, Thickness Constraint target lift: 0.3, moment: 0.0, thickness: 0.12 preconditioner diag(20, 20, 100), µ i,0 = c d c l 10 N u c m thickn ,000 1,500 2,000 2,500 3,000 3,500 4,000 Iteration Kusch,Albring,Walther,Gauger From simulation to one-shot optimization with constraints 14/ 18

15 Resulting Pareto-optimal Front drag coefficient lift coefficient Kusch,Albring,Walther,Gauger From simulation to one-shot optimization with constraints 15/ 18

16 Using Second-Order Adjoints Tangent over Reverse: ȳ T k N yu ū T = (ȳ T G uy + v T f yu + w T h yu )ẏ + other terms (6) Set v = 1, w = µ, ȳ = ȳ, ẏ = ȳ, Read from ū (Runtime Factor: 1.6) DataType: ActiveReal<RealForward, ChunkTape<RealForward, int>> GetMixedDerivative: tape.getgradient(x[adjpos++]).getgradient() objective second order objective finite differences ,000 1,200 1,400 1,600 1,800 2,000 Iteration Kusch,Albring,Walther,Gauger From simulation to one-shot optimization with constraints 16/ 18

17 Summary and Outlook Summary Semi-automatic transition from simulation to optimization involving AD-based discrete adjoint in SU2 constrained one-shot method and deterministic multi-objective optimization Outlook different application in SU2 (multi-disciplinary) investigations on preconditioner for constraints Thank you for your attention! Kusch,Albring,Walther,Gauger From simulation to one-shot optimization with constraints 17/ 18

18 References Hamdi, A. and Griewank, A. Properties of an augmented lagrangian for design optimization. Optimization Methods and Software 25(4), (2010). Walther, A., Gauger, N., Kusch, L., and Richert, N. On an extension of one-shot methods to incorporate additional constraints. Optimization Methods and Software (2016). Albring, T., Sagebaum, M., and Gauger, N. Development of a consistent discrete adjoint solver in an evolving aerodynamic design framework. AIAA (2015). Marglin, S. A. Public investment criteria. Allen & Unwin London, (1967). Kusch,Albring,Walther,Gauger From simulation to one-shot optimization with constraints 18/ 18