Motivation PDE-Constrained Optimization ROM-Constrained Optimization Numerical Experiments Conclusion References A Nonlinear Trust Region Framework for PDE-Constrained Optimization Using Progressively-Constructed Reduced-Order Models Matthew J. Zahr and Charbel Farhat Institute for Computational and Mathematical Engineering Farhat Research Group Stanford University SIAM Conference on Computational Science and Engineering MS4: Adaptive Model Order Reduction Salt Lake City, UT March 14, 2015 Zahr and Farhat Progressive ROM-Constrained Optimization
Motivation PDE-Constrained Optimization ROM-Constrained Optimization Numerical Experiments Conclusion References 1 Motivation 2 PDE-Constrained Optimization 3 ROM-Constrained Optimization 4 Numerical Experiments Airfoil Design Rocket Nozzle Design 5 Conclusion Zahr and Farhat Progressive ROM-Constrained Optimization
Motivation PDE-Constrained Optimization ROM-Constrained Optimization Numerical Experiments Conclusion References Reduced-Order Models (ROMs) ROMs as Enabling Technology Many-query analyses Optimization: design, control Single objective, single-point Multiobjective, multi-point Uncertainty Quantification Optimization under uncertainty Real-time analysis Figure: Flapping Wing (Persson et al., 2012) Model Predictive Control (MPC) Zahr and Farhat Progressive ROM-Constrained Optimization
Motivation PDE-Constrained Optimization ROM-Constrained Optimization Numerical Experiments Conclusion References Application I: Compressible, Turbulent Flow over Vehicle Benchmark in automotive industry Mesh 2,890,434 vertices 17,017,090 tetra 17,342,604 DOF (a) Ahmed Body: Geometry (Ahmed et al, 1984) CFD Compressible Navier-Stokes DES + Wall func Single forward simulation ≈ 0 . 5 day on 512 cores Desired: shape optimization unsteady effects minimize average drag (b) Ahmed Body: Mesh (Carlberg et al, 2011) Zahr and Farhat Progressive ROM-Constrained Optimization
Motivation PDE-Constrained Optimization ROM-Constrained Optimization Numerical Experiments Conclusion References Application II: Turbulent Flow over Flapping Wing CFD Biologically-inspired flight Compressible Navier-Stokes Micro aerial vehicles Discontinuous Galerkin Mesh Desired: shape optimization + 43,000 vertices control 231,000 tetra ( p = 3) unsteady effects 2,310,000 DOF maximize thrust Figure: Flapping Wing (Persson et al., 2012) Zahr and Farhat Progressive ROM-Constrained Optimization
Motivation PDE-Constrained Optimization ROM-Constrained Optimization Numerical Experiments Conclusion References Problem Formulation Goal: Rapidly solve PDE-constrained optimization problems of the form minimize f ( w , µ ) w ∈ R N , µ ∈ R p Discretize-then-optimize subject to R ( w , µ ) = 0 where R : R N × R p → R N is the discretized (steady, nonlinear) PDE, w is the PDE state vector, µ is the vector of parameters, and N is assumed to be very large . Zahr and Farhat Progressive ROM-Constrained Optimization
Motivation PDE-Constrained Optimization ROM-Constrained Optimization Numerical Experiments Conclusion References Definition of Φ : Proper Orthogonal Decomposition MOR assumption ∂ w ∂ µ ≈ Φ ∂ y w − ¯ w ≈ Φy = ⇒ ∂ µ State-Sensitivity 1 POD Collect state and sensitivity snapshots by sampling HDM � w ( µ 1 ) − ¯ w � X = w ( µ 2 ) − ¯ · · · w ( µ n ) − ¯ w w � � ∂ w ∂ w ∂ w Y = ∂ µ ( µ 1 ) ∂ µ ( µ 2 ) · · · ∂ µ ( µ n ) Use Proper Orthogonal Decomposition to generate reduced bases from each individually Φ X = POD( X ) Φ Y = POD( Y ) Concatenate to get ROB � Φ X � Φ = Φ Y 1 (Washabaugh and Farhat, 2013),(Zahr and Farhat, 2014) Zahr and Farhat Progressive ROM-Constrained Optimization
Motivation PDE-Constrained Optimization ROM-Constrained Optimization Numerical Experiments Conclusion References ROM-Constrained Optimization ROM-constrained optimization: minimize f ( ¯ w + Φy , µ ) y ∈ R n , µ ∈ R p Ψ T R ( ¯ subject to w + Φy , µ ) = 0 where R r ( y , µ ) = Ψ T R ( ¯ w + Φy , µ ) = 0 is the reduced-order model Zahr and Farhat Progressive ROM-Constrained Optimization
Motivation PDE-Constrained Optimization ROM-Constrained Optimization Numerical Experiments Conclusion References 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 minimize f ( ¯ w + Φy , µ ) y ∈ R n , µ ∈ R p Ψ T R ( ¯ subject to w + Φy , µ ) = 0 1 w + Φy , µ ) || 2 2 || R ( ¯ 2 ≤ ǫ Use solution of above problem to enrich training and repeat until convergence (Arian et al., 2000), (Fahl, 2001), (Afanasiev and Hinze, 2001), (Kunisch and Volkwein, 2008), (Hinze and Matthes, 2013), (Yue and Meerbergen, 2013), (Zahr and Farhat, 2014) Zahr and Farhat Progressive ROM-Constrained Optimization
Motivation PDE-Constrained Optimization ROM-Constrained Optimization Numerical Experiments Conclusion References Progressive Approach Optimizer HDM HDM ROB Φ , Ψ Compress HDM ROM Figure: Schematic of Algorithm Zahr and Farhat Progressive ROM-Constrained Optimization
Motivation PDE-Constrained Optimization ROM-Constrained Optimization Numerical Experiments Conclusion References Progressive Approach (a) Idealized Optimization Trajectory: Parameter Space ROM ROM ROM ROM ROM ROM ROM ROM ROM ROM ROM ROM HDM HDM HDM (b) Breakdown of Computational Effort Zahr and Farhat Progressive ROM-Constrained Optimization
Motivation PDE-Constrained Optimization ROM-Constrained Optimization Numerical Experiments Conclusion References Progressive Approach Ingredients of Proposed Approach (Zahr and Farhat, 2014) Minimum-residual ROM (LSPG) and minimum-error sensitivities f r ( µ ) = f ( µ ), d f r d µ ( µ ) = d f d µ ( µ ) for training parameters µ Reduced optimization (sub)problem f ( ¯ minimize w + Φy , µ ) y ∈ R n , µ ∈ R p Ψ T R ( ¯ subject to w + Φy , µ ) = 0 1 w + Φy , µ ) || 2 2 || R ( ¯ 2 ≤ ǫ Efficiently update ROB with additional snapshots or new translation vector Without re-computing SVD of entire snapshot matrix Adaptive selection of ǫ → trust-region approach Zahr and Farhat Progressive ROM-Constrained Optimization
Motivation PDE-Constrained Optimization ROM-Constrained Optimization Numerical Experiments Conclusion References Adaptive Selection of Trust-Region Radius Let − 1 = µ (0) µ ∗ = initial condition for PDE-constrained optimization 0 µ ∗ j = solution of j th reduced optimization problem Define f ( w ( µ ∗ j ) , µ ∗ j ) − f ( w ( µ ∗ j − 1 ) , µ ∗ j − 1 ) ρ j = f ( w r ( µ ∗ j ) , µ ∗ j ) − f ( w r ( µ ∗ j − 1 ) , µ ∗ j − 1 ) Trust-Region Radius 1 τ ǫ ρ k ∈ [0 . 5 , 2] ǫ ′ = ρ k ∈ [0 . 25 , 0 . 5) ∪ (2 , 4] ǫ τǫ otherwise Zahr and Farhat Progressive ROM-Constrained Optimization
Motivation PDE-Constrained Optimization ROM-Constrained Optimization Numerical Experiments Conclusion References 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 − ˜ w1 T ) w1 T ) given Φ = POD( X − ¯ Compute new basis using singular factors of existing basis complete without complete recomputation Fast, Low-Rank Updates to ROB Compute (Brand, 2006) Φ ′ = POD( X + AB T ) given Φ = POD( X ) Large-scale SVD ( N × n snap ) replaced by small SVD (independent of N ) Error incurred by using truncated basis ∝ σ n +1 Usually small in MOR applications Zahr and Farhat Progressive ROM-Constrained Optimization
Motivation PDE-Constrained Optimization ROM-Constrained Optimization Airfoil Design Numerical Experiments Rocket Nozzle Design Conclusion References Compressible, Inviscid Airfoil Inverse Design (a) NACA0012: Pressure field (b) RAE2822: Pressure field ( M ∞ = 0 . 5, ( M ∞ = 0 . 5, α = 0 . 0 ◦ ) α = 0 . 0 ◦ ) Pressure discrepancy minimization (Euler equations) Initial Configuration: NACA0012 Target Configuration: RAE2822 Zahr and Farhat Progressive ROM-Constrained Optimization
Motivation PDE-Constrained Optimization ROM-Constrained Optimization Airfoil Design Numerical Experiments Rocket Nozzle Design Conclusion References Initial/Target Airfoils: Scaled Zahr and Farhat Progressive ROM-Constrained Optimization
Motivation PDE-Constrained Optimization ROM-Constrained Optimization Airfoil Design Numerical Experiments Rocket Nozzle Design Conclusion References 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
Motivation PDE-Constrained Optimization ROM-Constrained Optimization Airfoil Design Numerical Experiments Rocket Nozzle Design Conclusion References 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
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