Semi-automatic transition from simulation to one-shot ... · Semi-automatic transition from simulation to one-shot optimization with equality constraints Lisa Kusch, Tim Albring,

Semi-automatic transition from simulation to one-shotoptimization with equality constraints

Lisa Kusch, Tim Albring, Andrea Walther, Nicolas Gauger

Chair for Scientific Computing, TU Kaiserslautern, www.scicomp.uni-kl.deInstitute of Mathematics, Universitat Paderborn

September 15, 2016

AD 2016

Outline

1 The One-Shot Approach with Additional Constraints

2 AD-Based Discrete Adjoint in SU2

3 Multi-Objective Optimization

4 Application and ResultsApplication to Airfoil DesignMulti-Objective Optimization with Single ConstraintMulti-Objective Optimization with Multiple ConstraintsSecond-Order Adjoints

5 Summary and Outlook

Kusch,Albring,Walther,Gauger From simulation to one-shot optimization with constraints 2/ 18

Problem Definition

PDE-constrained optimization problem

minu,y

f (y , u)

s.t. c(y , u) = 0,

h(y , u) = 0

→fixed point iteration||Gy || ≤ ρ < 1

minu,y

f (y , u)

s.t. G (y , u) = y ,

h(y , u) = 0

objective function f ∈ R, state y ∈ Y ⊂ Rn, design u ∈ U ⊂ Rm

state equation c : Y × U → Y , constraints h : Y × U → V ⊂ Rp

(notation: G = (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(y , y , u) = f (y , u) + (G (y , u)− y)T y + hTµ = N(y , y , µ, u)− yT y


The One-Shot Approach with Additional Constraints

KKT point (discrete)

y∗ = Ny (y∗, y∗, µ, u∗)T = G (y∗, u)

y∗ = Ny (y∗, y∗, µ, u∗)T = fy (y∗, u∗)T + Gy (y∗, u∗)T y∗

0 = Nu(y∗, y∗, µ, u∗)T = fu(y∗, u∗)T + Gu(y∗, u∗)T y∗

0 = Nµ(y∗, y∗, µ, u∗)T = h(y∗, u∗)

One-shot approach with equality constraints

for k = 1, ...

yk+1 = G (yk , uk)

yk+1 = Ny (yk , yk , µk , uk)

uk+1 = uk − B−1k Nu(yk , yk , µk , uk)

µk+1 = µk − B−1k h(yk , uk)


Convergence Propertiesdoubly augmented Lagrangian

La(y , y , µ, u) =α

2||G (y , u)||2 +

β

2||Ny (y , y , µ, u)T − y ||2 + N − yT y

correspondence and descent condition√αβ(1− ρ) > 1 +

β

2||Nyy || (1)

and α ≥ ‖2(G>u )1...p(G>y )+(Nyu)1...p − (Nuu)1...p1...p‖/‖(Gu)1...p‖2

⇒ exact penalty function, descent for all large symmetric pos. def. B, B

choice of B

B ≈ ∇2uuL

a(y , y , u) ⇒ B = αGTu Gu + βNT

yuNyu + Nuu


Implementation Details

BFGS Update: B∆u ≈ ∇uLa(y , y , µ, u + ∆u)−∇uL

a(y , y , µ, u) =: r

calculate ∇uLa = α∆yT

k Gu + αhTu h + β∆yTk Nyu + Nu

Nu and ∆yTk Gu + hTu h can be obtained with reverse mode of AD from u

y = G (y , u)

v = f (y , u)

w = h(y , u)

u = fu(y , u)T v + Gu(y , u)T y + hu(y , u)T w

finite difference over reverse / tangent over reverse

∆yTk Nyu ≈

1

h(Nu(yk + h∆yk , yk , µk , uk)− Nu(yk , yk , µk , uk)) (2)

curvature condition: set Bk = I if rTk ∆uk ≤ 0


Open-source multiphysics suite SU2Under active development (Stanford University, TU Kaiserslautern, TUDelft,...)

FV discretization of (U)RANS equations and turbulence models usinghighly 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)]


Discrete-Adjoint Solver in SU2

operator overloading (frequent extensions/improvements)

shape optimization → mesh deformation as additional constraint in theoptimization problem M(u) = x

fixed-point form yk+1 = Ny (yk , yk , µk , uk)T

CoDiPack

open-source (https://github.com/SciCompKL/CoDiPack)

use of expression templates

different tape implementations

Performance SU2 (NASA Common Research Model)

Runtime Factor: 1.2

Memory Factor: 7.2


Multi-Objective Optimization Problem

miny ,u

F(y , u)

s.t. c(y , u) = 0, (3)

h(y , u) ≥ 0

(4)

Pareto dominance:

x dominates x if

∀i ∈ {1, ..., k} : fi (x) ≤ fi (x)

∃j ∈ {1, ..., k} : fj(x) < fj(x)


Methods for Multi-Objective Optimization

common: weighted sum,evolutionary algorithms

ε-constraint method[Marglin(1967)]

miny ,u

fsj (y , u)

s.t. c(y , u) = 0, (5)

h(y , u) = 0,

fi (x) ≤ f(j)i

∀ i ∈ {1, ..., k} i 6= sj

connected: equality constraints


Application (Airfoil Design)

aerodynamic shape optimizationof NACA 0012 airfoil

objectives: drag coefficient cd , liftcoefficient cl

bounded design variables:38 Hicks-Henne bump functions(bounds: [-0.005,0.005])

constraints: moment, thickness

steady Euler equations, transonicflow (Mach 0.8 and AOA 1.25)


Result: Lift Constraintpreconditioner B = 20, µ0 = 0, target lift: 0.4backtracking line search, trial step size: 1, α = 20, β = 2

0 400 800 1,200 1,600 2,000

0

0.4

Iteration

10cdcl

10||Nu||

0 500 1,000 1,500 2,000

−8

−6

−4

−2

Iteration

primal residualadjoint residual


Resulting Pareto-optimal Front

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

1

2

3·10−3

lift coefficient

dra

gco

effici

ent


Result: Lift, Moment, Thickness Constrainttarget lift: 0.3, moment: 0.0, thickness: 0.12preconditioner diag(20, 20, 100), µi,0 = 0

0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000

0

0.3

Iteration

10cdcl

10||Nu||cm

thickn.


Resulting Pareto-optimal Front

0 0.2 0.4 0.60

1

2

3

4·10−3

lift coefficient

dra

gco

effici

ent


Using Second-Order Adjoints

Tangent over Reverse: ∆yTk Nyu

˙uT = (yTGuy + vT fyu + wThyu)y + other terms (6)

⇒ Set v = 1, w = µ, y = y , y = ∆y , Read from ˙u (Runtime Factor: 1.6)

DataType: ActiveReal<RealForward, ChunkTape<RealForward, int>>

GetMixedDerivative: tape.getGradient(x[adjPos++]).getGradient()

0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000−0.5

0

0.5

1·10−2

Iteration

objective second orderobjective finite differences


Summary and Outlook

SummarySemi-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!


References

Hamdi, A. and Griewank, A.

Properties of an augmented lagrangian for design optimization.Optimization Methods and Software 25(4), 645–664 (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 2015-3240 (2015).

Marglin, S. A.

Public investment criteria.Allen & Unwin London, (1967).


Documents

Semi-automatic transition from simulation to one-shot ... · Semi-automatic transition from simulation to one-shot optimization with equality constraints Lisa Kusch, Tim Albring,