a nonlinear trust region framework for pde-constrained...

MotivationPDE-Constrained OptimizationROM-Constrained Optimization

Numerical ExperimentsConclusionReferences

A Nonlinear Trust Region Framework forPDE-Constrained Optimization Using

Progressively-Constructed Reduced-Order Models

Matthew J. Zahr and Charbel Farhat

Institute for Computational and Mathematical EngineeringFarhat Research Group

Stanford University

SIAM Conference on Computational Science and EngineeringMS4: Adaptive Model Order Reduction

Salt Lake City, UTMarch 14, 2015

Zahr and Farhat Progressive ROM-Constrained Optimization



1 Motivation

2 PDE-Constrained Optimization

3 ROM-Constrained Optimization

4 Numerical ExperimentsAirfoil DesignRocket Nozzle Design

5 Conclusion




Reduced-Order Models (ROMs)

ROMs as Enabling Technology

Many-query analysesOptimization: design, control

Single objective, single-pointMultiobjective, multi-point

Uncertainty Quantification

Optimization under uncertainty

Real-time analysis

Model Predictive Control (MPC)

Flapping Bat Flight Simulation

Visualization of Mach number on isosurface of entropy

Unphysical separation around simplified animal “body”

Figure: Flapping Wing(Persson et al., 2012)

REDUCED ORDER MODEL (ROM)

o Perturbation problems (stability, trends, control, etc.)!

o Response problems (behavior, performance, etc.)!

- linearized !

- nonlinear !

!  Complex, time-dependent problems!




Application I: Compressible, Turbulent Flow over Vehicle

Benchmark in automotiveindustry

Mesh

2,890,434 vertices17,017,090 tetra17,342,604 DOF

CFD

CompressibleNavier-StokesDES + Wall func

Single forward simulation

≈ 0.5 day on 512 cores

Desired: shape optimization

unsteady effectsminimize average drag

and LES turbulence models, as well as a wall function. It performs a second-order semi-discretization of the convective fluxesusing a method based on the Roe, HLLE, or HLLC upwind scheme. It can also perform second- and fourth-order explicit andimplicit temporal discretizations using a variety of time integrators. The GNAT implementation in AERO-F is characterized bythe sample-mesh concept described in Section 5. All linear least-squares problems and singular value decompositions arecomputed in parallel using the ScaLAPACK library [50]. AERO-F is used here to demonstrate GNAT’s potential when appliedto a realistic, large-scale, nonlinear benchmark CFD problem: turbulent flow around the Ahmed body.

The Ahmed-body geometry [47] is a simplied car geometry. It can be described as a modified parallelepiped featuringround corners at the front end and a slanted face at the rear end (see Fig. 6). Depending on the inclination of this face, dif-ferent flow characteristics and wake structure may be observed. For a slant angle uP 30!, the flow features a large detach-ment. For smaller slant angles, the flow reattaches on the slant. Consequently, the drag coefficient suddenly decreases whenthe slant angle is increased beyond its critical value of u " 30!. Due to this phenomenon, predicting the flow past the Ahmedbody for varying slant angles has become a popular benchmark in the automotive industry.

This work considers the subcritical angleu " 20! and treats the drag coefficient CD " D12q1V2

15:6016#10$2 m2 around the body as

the output of interest. The free-stream velocity is set to V1 " 60 m/s, and the Reynolds number based on a reference lengthof 1.0 m is set to Re " 4:29# 106. The free-stream angle of attack is set to 0!.

6.2.1. High-dimensional CFD modelThe high-dimensional CFD model corresponds to an unsteady Navier–Stokes simulation using AERO-F’s DES turbulence

model and wall function. The fluid domain is discretized by a mesh with 2,890,434 nodes and 17,017,090 tetrahedra (Fig. 7).A symmetry plane is employed to exploit the symmetry of the body about the x–z plane. Due to the turbulence model andthree-dimensional domain, the number of conservation equations per node is m " 6, and therefore the dimension of the CFDmodel is N " 17;342;604. Roe’s scheme is employed to discretize the convective fluxes; a linear variation of the solution isassumed within each control volume, which leads to a second-order space-accurate scheme.

Flow simulations are performed within a time interval t 2 0 s;0:1 s% &, the second-order accurate implicit three-pointbackward difference scheme is used for time integration, and the computational time-step size is fixed to Dt " 8# 10$5 s.For the chosen CFD mesh, this time-step size corresponds to a maximum CFL number of roughly 2000. The nonlinear systemof algebraic equations arising at each time step is solved by Newton’s method. Convergence is declared at the kth iterationfor the nth time step when the residual satisfies kRn'k(k 6 0:001kRn'0(k. All flow computations are performed in a non-dimen-sional setting.

A steady-state simulation computes the initial condition for the unsteady simulation. This steady-state calculation ischaracterized by the same parameters as above, except that it employs local time stepping with a maximum CFL numberof 50, it uses the first-order implicit backward Euler time integration scheme, and it employs only one Newton iterationper (pseudo) time step.

All computations are performed in double-precision arithmetic on a parallel Linux cluster5 using a variable number ofcores.

6.2.2. Comparison with experimentRef. [47] reports an experimental drag coefficient of 0.250 around the Ahmed body for a slant angle of u " 20!. Fig. 8

reports the time history of the drag coefficient computed using the high-dimensional CFD model described in the previoussection. Indeed, the time-averaged value of the computed drag coefficient obtained using the trapezoidal rule is CD " 0:2524.

Fig. 6. Geometry of the Ahmed body (from Ref. [51].)

5 The cluster contains compute nodes with 16 GB of memory. Each node consists of two quad-core Intel Xeon E5345 processors running at 2.33 GHz inside aDELL Poweredge 1950. The interconnect is Cisco DDR InfiniBand.

K. Carlberg et al. / Journal of Computational Physics 242 (2013) 623–647 637

(a) Ahmed Body: Geometry (Ahmed et al, 1984)

Hence, it is within less than 1% of the reported experimental value. This asserts the quality of the constructed CFD model andAERO-F’s computations. For reference, this high-dimensional CFD simulation consumed 13.28 h on 512 cores.

6.2.3. ROM performance metricsThe following metrics will be used to assess GNAT’s performance. The relative discrepancy in the drag coefficient, which

assesses the accuracy of a GNAT simulation, is measured as follows:

RD !1nt

Xnt

n!1jCn

DI " CnDIII

jmax

nCnDI "min

nCnDI

; #31$

where CnDI denotes the drag coefficient computed at the nth time step using the high-dimensional CFD model (tier I model),

and CnDIII denotes the corresponding value computed using the GNAT ROM (tier III model).

The improvement in CPU performance delivered by GNAT as measured in wall time is defined as

WT ! T I

T III; #32$

where T I denotes the wall time consumed by a flow simulation associated with the high-dimensional CFD model, and T III

denotes the wall time consumed online by its counterpart based on a GNAT ROM. For the high-dimensional model, thereported wall time includes the solution of the governing equations and the output of the state vector; for the GNATreduced-order model, it includes the execution of Algorithm 2. After the completion of Algorithms 1 and 2 is executed to

Fig. 7. CFD mesh with 2,890,434 grid points and 17,017,090 tetrahedra (partial view, u ! 20%). Darker areas indicate a more refined area of the mesh.

Fig. 8. Time history of the drag coefficient predicted for u ! 20% using DES and a CFD mesh with N ! 17;342;604 unknowns. Oscillatory behavior due tovortex shedding is apparent.

638 K. Carlberg et al. / Journal of Computational Physics 242 (2013) 623–647

(b) Ahmed Body: Mesh (Carlberg et al, 2011)




Application II: Turbulent Flow over Flapping Wing

Biologically-inspired flight

Micro aerial vehicles

Mesh

43,000 vertices231,000 tetra (p = 3)2,310,000 DOF

CFD

Compressible Navier-StokesDiscontinuous Galerkin

Desired: shape optimization +control

unsteady effectsmaximize thrust

Flapping Bat Flight Simulation

Visualization of Mach number on isosurface of entropy

Unphysical separation around simplified animal “body”

Figure: Flapping Wing (Persson et al., 2012)




Problem Formulation

Goal: Rapidly solve PDE-constrained optimization problems of the form

minimizew∈RN , µ∈Rp

f(w,µ)

subject to R(w,µ) = 0Discretize-then-optimize

where R : RN × Rp → RN is the discretized (steady, nonlinear) PDE, w is thePDE state vector, µ is the vector of parameters, and N is assumed to be verylarge.

REDUCED ORDER MODEL (ROM)

o Perturbation problems (stability, trends, control, etc.)!

o Response problems (behavior, performance, etc.)!

- linearized !

- nonlinear !

!  Complex, time-dependent problems!




Definition of Φ: Proper Orthogonal Decomposition

MOR assumption

w − w ≈ Φy =⇒ ∂w

∂µ≈ Φ

∂y

∂µ

State-Sensitivity1 POD

Collect state and sensitivity snapshots by sampling HDM

X =[w(µ1)− w w(µ2)− w · · · w(µn)− w

]Y =

[∂w∂µ (µ1) ∂w

∂µ (µ2) · · · ∂w∂µ (µn)

]Use Proper Orthogonal Decomposition to generate reduced bases from eachindividually

ΦX = POD(X)

ΦY = POD(Y)

Concatenate to get ROBΦ =

[ΦX ΦY

]1(Washabaugh and Farhat, 2013),(Zahr and Farhat, 2014)




ROM-Constrained Optimization

ROM-constrained optimization:

minimizey∈Rn, µ∈Rp

f(w + Φy,µ)

subject to ΨTR(w + Φy,µ) = 0

whereRr(y,µ) = ΨTR(w + Φy,µ) = 0

is the reduced-order model




Progressive/Adaptive Approach

Progressive Approach to ROM-Constrained Optimization

Collect snapshots from HDM at sparse sampling of the parameter space

Initial condition for optimization problem

Build ROB Φ from sparse training

Solve optimization problem


f(w + Φy,µ)


1

2||R(w + Φy,µ)||22 ≤ ε

Use solution of above problem to enrich training and repeat untilconvergence

(Arian et al., 2000), (Fahl, 2001), (Afanasiev and Hinze, 2001), (Kunisch andVolkwein, 2008), (Hinze and Matthes, 2013), (Yue and Meerbergen, 2013), (Zahrand Farhat, 2014)




Progressive Approach

HDM

HDM

ROBΦ,ΨCompress

ROM

OptimizerHDM

Figure: Schematic of Algorithm





(a) Idealized Optimization Trajectory: Parameter Space

HDM

RO

M

RO

M

RO

M

RO

M

HDM

RO

M

RO

M

RO

M

RO

M

HDM

RO

M

RO

M

RO

M

RO

M

(b) Breakdown of Computational Effort





Ingredients of Proposed Approach (Zahr and Farhat, 2014)

Minimum-residual ROM (LSPG) and minimum-error sensitivities

fr(µ) = f(µ),dfrdµ

(µ) =df

dµ(µ) for training parameters µ

Reduced optimization (sub)problem


f(w + Φy,µ)


1

2||R(w + Φy,µ)||22 ≤ ε

Efficiently update ROB with additional snapshots or new translation vector

Without re-computing SVD of entire snapshot matrix

Adaptive selection of ε→ trust-region approach




Adaptive Selection of Trust-Region Radius

Let

µ∗−1 = µ(0)0 = initial condition for PDE-constrained optimization

µ∗j = solution of jth reduced optimization problem

Define

ρj =f(w(µ∗j ),µ

∗j )− f(w(µ∗j−1),µ∗j−1)

f(wr(µ∗j ),µ∗j )− f(wr(µ∗j−1),µ∗j−1)

Trust-Region Radius

ε′ =

1τ ε ρk ∈ [0.5, 2]

ε ρk ∈ [0.25, 0.5) ∪ (2, 4]

τε otherwise




Fast Updates to Reduced-Order Basis

Two situations where snapshot matrix modified (Zahr and Farhat, 2014)

Additional snapshots to be incorporated

Φ′ = POD([X Y

]) given Φ = POD(X)

Offset vector modified

Φ′ = POD(X− w1T ) given Φ = POD(X− w1T )

Compute new basis using singular factors of existing basis complete withoutcomplete recomputation

Fast, Low-Rank Updates to ROB

Compute (Brand, 2006)

Φ′ = POD(X + ABT ) given Φ = POD(X)

Large-scale SVD (N × nsnap) replaced by small SVD (independent of N)

Error incurred by using truncated basis ∝ σn+1

Usually small in MOR applications




Airfoil DesignRocket Nozzle Design

Compressible, Inviscid Airfoil Inverse Design

(a) NACA0012: Pressure field(M∞ = 0.5, α = 0.0◦)

(b) RAE2822: Pressure field (M∞ = 0.5,α = 0.0◦)

Pressure discrepancy minimization (Euler equations)Initial Configuration: NACA0012Target Configuration: RAE2822





Initial/Target Airfoils: Scaled





Shape Parametrization

(a) µ(1) = 0.1 (b) µ(2) = 0.1

(c) µ(3) = 0.1 (d) µ(4) = 0.1

Figure: Shape parametrization of a NACA0012 airfoil using a cubic design element





Shape Parametrization

(a) µ(5) = 0.1 (b) µ(6) = 0.1

(c) µ(7) = 0.1 (d) µ(8) = 0.1

Figure: Shape parametrization of a NACA0012 airfoil using a cubic design element





Optimization Results

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Distance along airfoil

-Cp

InitialTarget

HDM-based optimizationROM-based optimization

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Dis

tance

Tra

nsv

erse

toC

ente

rlin

e






0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Number of HDM queries

1 2||p

(w+

Φky

(µ))−

p(w

(µRAE2822))||2 2

1 2||p

(w+

Φky

(0))−

p(w

(µRAE2822))||2 2

HDM-based optimizationROM-based optimization






0 20 40 60 80 100 120 140 160

10−17

10−15

10−13

10−11

10−9

10−7

10−5

10−3

10−1

101

Reduced optimization iterations

1 2||p

(w+

Φky

(µ))−

p(w

(µRAE2822))||2 2

1 2||p

(w+

Φky

(0))−

p(w

(µRAE2822))||2 2

HDM sample0

10

20

30

40

50

60

70

RO

Msi

ze






0 20 40 60 80 100 120 140 16010−13

10−10

10−7

10−4

10−1

102

105

108

1011

Reduced optimization iterations

1 2||R

(w+

Φky

)||2 2

HDM sampleResidual norm

Residual norm bound (ε)






HDM-basedoptimization

ROM-basedoptimization

# of HDM Evaluations 29 7# of ROM Evaluations - 346||µ∗ − µRAE2822||||µRAE2822|| 2.28× 10−3% 4.17× 10−6%

Table: Performance of the HDM- and ROM-based optimization methods





Quasi-1D Euler Flow

Quasi-1D Euler equations:

∂U

∂t+

1

A

∂(AF)

∂x= Q

where

U =

ρρue

, F =

ρuρu2 + p(e+ p)u

, Q =

0pA∂A∂x0

Semi-discretization

Finite Volume Method: constant reconstruction, 500 cellsRoe flux and entropy correction

Full discretization

Backward EulerPseudo-transient integration to steady state





Nozzle Parametrization

Nozzle parametrized with cubic splines using 13 control points and constraintsrequiring

convexity A′′(x) ≥ 0

bounds on A(x) Al(x) ≤ A(x) ≤ Au(x)

bounds on A′(x) at inlet/outlet A′(xl) ≤ 0, A′(xr) ≥ 0

0 0.05 0.1 0.15 0.2 0.250

0.01

0.02

0.03

0.04

0.05

0.06

0.07

x

Nozzle

Heig

ht

Nozz l e Parametri zation

Al(x)

Au(x)

A(x)

Spl ine Points

Student Version of MATLAB





Parameter Estimation/Inverse Design

For this problem, the goal is to determine the parameter µ∗ such that the flowachieves some optimal or desired state w∗

minimizew∈RN , µ∈Rp

||w(µ)−w∗||

subject to R(w,µ) = 0

c(w,µ) ≤ 0

where c are the nozzle constraints.





Objective Function Convergence

(a) Convergence (# HDM Evals)

0 5 10 15 20 25 3010

0

101

102

103

104

105

106

# HDM Evaluation s

Objectiv

eFunction

HDM -based opt

HROM -based opt

(b) Convergence (CPU Time)

0 500 1000 1500 2000 2500 3000 350010

0

101

102

103

104

105

106

CPU T ime (se c )

Obje

ctiveFunction

HDM -B ased Opt

HROM -B ased Opt





Hyper-Reduced Optimization Progression

Figure: Parameter (µ) Progression

0 0.05 0.1 0.15 0.2 0.250

0.01

0.02

0.03

0.04

0.05

0.06

0.07

x

Nozzle

Heig

ht

Desi red Optimal

Ini tial Guess

HROM-Based Iterates

HROM-Based Optimal

Sample Mesh





Optimization Summary

HDM-Based Opt HROM-Based Opt

Rel. Error in µ∗ (%) 1.82 5.26

Rel. Error in w∗ (%) 0.11 0.12

# HDM Evals 27 8

# HROM Evals 0 161

CPU Time (s) 3361.51 2001.74




Summary

Summary

Introduced progressive, nonlinear trust region framework for reducedoptimization

Demonstrated approach on canonical problem from aerodynamic shapeoptimization

Factor of 4 fewer queries to HDM than standard PDE-constrainedoptimization approaches

Preliminary results on toy problem regarding extension of framework tohyperreduction




References I

Afanasiev, K. and Hinze, M. (2001).Adaptive control of a wake flow using proper orthogonal decomposition.Lecture Notes in Pure and Applied Mathematics, pages 317–332.

Amsallem, D., Zahr, M. J., and Farhat, C. (2012).Nonlinear model order reduction based on local reduced-order bases.International Journal for Numerical Methods in Engineering.

Arian, E., Fahl, M., and Sachs, E. W. (2000).Trust-region proper orthogonal decomposition for flow control.Technical report, DTIC Document.

Brand, M. (2006).Fast low-rank modifications of the thin singular value decomposition.Linear algebra and its applications, 415(1):20–30.

Bui-Thanh, T., Willcox, K., and Ghattas, O. (2008).Model reduction for large-scale systems with high-dimensional parametric input space.SIAM Journal on Scientific Computing, 30(6):3270–3288.

Carlberg, K. (2014).Adaptive h-refinement for reduced-order models.arXiv preprint arXiv:1404.0442.

Carlberg, K., Bou-Mosleh, C., and Farhat, C. (2011).Efficient non-linear model reduction via a least-squares petrov–galerkin projection and compressive tensorapproximations.International Journal for Numerical Methods in Engineering, 86(2):155–181.

Carlberg, K. and Farhat, C. (2008).A compact proper orthogonal decomposition basis for optimization-oriented reduced-order models.AIAA Paper, 5964:10–12.




References II

Carlberg, K. and Farhat, C. (2011).A low-cost, goal-oriented compact proper orthogonal decompositionbasis for model reduction of static systems.International Journal for Numerical Methods in Engineering, 86(3):381–402.

Carlberg, K., Ray, J., and Waanders, B. v. B. (2012).Decreasing the temporal complexity for nonlinear, implicit reduced-order models by forecasting.arXiv preprint arXiv:1209.5455.

Dihlmann, M., Drohmann, M., and Haasdonk, B. (2011).Model reduction of parametrized evolution problems using the reduced basis method with adaptive timepartitioning.Proc. of ADMOS, 2011.

Drohmann, M. and Carlberg, K. (2014).The romes method for statistical modeling of reduced-order-model error.SIAM Journal on Uncertainty Quantification.

Fahl, M. (2001).Trust-region methods for flow control based on reduced order modelling.PhD thesis, Universitatsbibliothek.

Golub, G. H. and Van Loan, C. F. (2012).Matrix computations, volume 3.JHU Press.

Halko, N., Martinsson, P.-G., and Tropp, J. A. (2011).Finding structure with randomness: Probabilistic algorithms for constructing approximate matrixdecompositions.SIAM review, 53(2):217–288.

Hay, A., Borggaard, J. T., and Pelletier, D. (2009).Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition.Journal of Fluid Mechanics, 629:41–72.




References III

Hinze, M. and Matthes, U. (2013).Model order reduction for networks of ode and pde systems.In System Modeling and Optimization, pages 92–101. Springer.

Kunisch, K. and Volkwein, S. (2008).Proper orthogonal decomposition for optimality systems.ESAIM: Mathematical Modelling and Numerical Analysis, 42(1):1.

Lassila, T. and Rozza, G. (2010).Parametric free-form shape design with pde models and reduced basis method.Computer Methods in Applied Mechanics and Engineering, 199(23):1583–1592.

LeGresley, P. A. and Alonso, J. J. (2000).Airfoil design optimization using reduced order models based on proper orthogonal decomposition.In Fluids 2000 conference and exhibit, Denver, CO.

Manzoni, A. (2012).Reduced models for optimal control, shape optimization and inverse problems in haemodynamics.PhD thesis, EPFL.

Manzoni, A., Quarteroni, A., and Rozza, G. (2012).Shape optimization for viscous flows by reduced basis methods and free-form deformation.International Journal for Numerical Methods in Fluids, 70(5):646–670.

Persson, P.-O., Willis, D., and Peraire, J. (2012).Numerical simulation of flapping wings using a panel method and a high-order navier–stokes solver.International Journal for Numerical Methods in Engineering, 89(10):1296–1316.

Rozza, G. and Manzoni, A. (2010).Model order reduction by geometrical parametrization for shape optimization in computational fluid dynamics.In Proceedings of ECCOMAS CFD.




References IV

Sirovich, L. (1987).Turbulence and the dynamics of coherent structures. i-coherent structures. ii-symmetries and transformations.iii-dynamics and scaling.Quarterly of applied mathematics, 45:561–571.

Washabaugh, K. and Farhat, C. (2013).A family of approaches for the reduction of discrete steady nonlinear aerodynamic models.Technical report, Stanford University.

Yue, Y. and Meerbergen, K. (2013).Accelerating optimization of parametric linear systems by model order reduction.SIAM Journal on Optimization, 23(2):1344–1370.

Zahr, M. J. and Farhat, C. (2014).Progressive construction of a parametric reduced-order model for pde-constrained optimization.International Journal for Numerical Methods in Engineering, Special Issue on ModelReduction(http://arxiv.org/abs/1407.7618).

Zahr, M. J., Washabaugh, K., and Farhat, C. (2014).Basis updating in model reduction.International Journal for Numerical Methods in Engineering.


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