a nonlinear trust region framework for pde-constrained...
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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
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
1 Motivation
2 PDE-Constrained Optimization
3 ROM-Constrained Optimization
4 Numerical ExperimentsAirfoil DesignRocket Nozzle Design
5 Conclusion
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
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!
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
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)
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
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)
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
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!
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
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)
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
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
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
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
minimizey∈Rn, µ∈Rp
f(w + Φy,µ)
subject to ΨTR(w + Φy,µ) = 0
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)
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
Progressive Approach
HDM
HDM
ROBΦ,ΨCompress
ROM
OptimizerHDM
Figure: Schematic of Algorithm
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
Progressive Approach
(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
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
Progressive Approach
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
minimizey∈Rn, µ∈Rp
f(w + Φy,µ)
subject to ΨTR(w + Φy,µ) = 0
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
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
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
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
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
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
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
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
Airfoil DesignRocket Nozzle Design
Initial/Target Airfoils: Scaled
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
Airfoil DesignRocket Nozzle Design
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
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
Airfoil DesignRocket Nozzle Design
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
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
Airfoil DesignRocket Nozzle Design
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
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
Airfoil DesignRocket Nozzle Design
Optimization Results
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
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
Airfoil DesignRocket Nozzle Design
Optimization Results
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
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
Airfoil DesignRocket Nozzle Design
Optimization Results
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)
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
Airfoil DesignRocket Nozzle Design
Optimization Results
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
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
Airfoil DesignRocket Nozzle Design
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
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
Airfoil DesignRocket Nozzle Design
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
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
Airfoil DesignRocket Nozzle Design
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.
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
Airfoil DesignRocket Nozzle Design
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
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
Airfoil DesignRocket Nozzle Design
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
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
Airfoil DesignRocket Nozzle Design
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
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
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
Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
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Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
References II
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MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
References III
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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.
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Zahr and Farhat Progressive ROM-Constrained Optimization
MotivationPDE-Constrained OptimizationROM-Constrained Optimization
Numerical ExperimentsConclusionReferences
References IV
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