Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Accelerating PDE-Constrained Optimization using Adaptive Reduced-Order Models Matthew J. Zahr Institute for Computational and Mathematical Engineering Farhat Research Group Stanford University Sandia National Laboratories July 8, 2015 Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Outline Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Application I: Shape Optimization of Vehicle in Turbulent Flow Volkswagen Passat Shape optimization Minimum drag configuration Unsteady effects Simulation 4M vertices, 24M dof Compressible Navier-Stokes Spalart-Allmaras Single forward simulation ≈ 1 day on 2048 CPUs Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Application II: Optimal Control Flapping Wing Biologically-inspired flight CFD Micro Aerial Vehicles (MAVs) Compressible Navier-Stokes Discontinuous Galerkin Mesh Shape optimization, control 43,000 vertices 231,000 tetra ( p = 3) unsteady effects 2,310,000 DOF min energy, const thrust Figure: Flapping Wing ( ? ) Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Application III: Topology Optimization Design of new lacrosse head 1 Desired: topology optimization Finer mesh (10-100x) Mesh Realistic material model 96,247 vertices 475,666 tetra 276,159 DOF Single forward simulation ≈ 5 minutes on 1 core 1 Collaboration with K. Washabaugh Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Application III: Topology Optimization Design of new lacrosse head 1 Desired: topology optimization Finer mesh (10-100x) Mesh Realistic material model 96,247 vertices 475,666 tetra 276,159 DOF Single forward simulation ≈ 5 minutes on 1 core 1 Collaboration with K. Washabaugh Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Application III: Topology Optimization Design of new lacrosse head 1 Desired: topology optimization Finer mesh (10-100x) Mesh Realistic material model 96,247 vertices 475,666 tetra 276,159 DOF Single forward simulation ≈ 5 minutes on 1 core 1 Collaboration with K. Washabaugh Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Reduced-Order Models (ROMs) ROMs as Enabling Technology Optimization: design, control Single objective, single-point Multiobjective, multi-point Unsteady effects Uncertainty Quantification Figure: Flapping Wing Optimization under uncertainty ( ? ) Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions 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 Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Outline Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Reduced-Order Model Model Order Reduction (MOR) assumption: state vector lies in low-dimensional affine subspace ∂ w ∂ µ ≈ ∂ w r ∂ µ = Φ ∂ y w ≈ w r = ¯ w + Φy = ⇒ ∂ µ where y ∈ R n are the reduced coordinates of w r in the basis Φ ∈ R N × n , and n ≪ N Substitute assumption into High-Dimensional Model (HDM), R ( w , µ ) = 0 R ( ¯ w + Φy , µ ) ≈ 0 Require projection of residual in low-dimensional left subspace , with basis Ψ ∈ R N × n to be zero R r ( y , µ ) = Ψ T R ( ¯ w + Φy , µ ) = 0 Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Reduced Optimization Problem ROM-Constrained Optimization minimize f ( ¯ w + Φy ( µ ) , µ ) µ ∈ R p Ψ T R ( ¯ subject to w + Φy , µ ) = 0 Issues that must be considered Construction of bases Speedup potential Sensitivity analysis (adjoint method) Training Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Offline-Online Approach Optimizer HDM HDM Compress ROB Φ , Ψ HDM ROM HDM Offline Figure: Schematic of Algorithm Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Offline-Online Approach (a) Idealized Optimization Trajectory: Parameter Space Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Offline-Online (Database) Approach Offline-Online Approach to ROM-Constrained Optimization Identify samples in offline phase to be used for training Space-fill sampling (i.e. latin hypercube) Greedy sampling Collect snapshots from HDM Build ROB Φ Solve optimization problem minimize f ( ¯ w + Φy , µ ) y ∈ R n , µ ∈ R p Ψ T R ( ¯ subject to w + Φy , µ ) = 0 ( ? ), ( ? ), ( ? ), ( ? ) Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Adaptive Approach Optimizer HDM HDM ROB Φ , Ψ Compress HDM ROM Figure: Schematic of Algorithm Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Adaptive Approach (a) Idealized Optimization Trajectory: Parameter Space Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Adaptive Approach Adaptive 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 ( ¯ 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 Use solution of above problem to enrich training and repeat until convergence ( ? ), ( ? ), ( ? ), ( ? ), ( ? ), ( ? ), ( ? ) Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Difficulty of Breaking Offline-Online Barrier Offline-Online Approach ROM ROM ROM ROM ROM ROM ROM ROM ROM HDM HDM HDM HDM ROB Figure: Offline-Online Approach Offline/Online Barrier + Enables large online speedups - Difficult to construct accurate, robust ROM ROM Minimize ! Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Difficulty of Breaking Offline-Online Barrier Progressive Approach ROM ROM ROM ROM ROM ROM ROM ROM ROM ROM ROM ROM ROM ROM ROM ROM HDM ROB HDM ROB Figure: Progressive Approach ROM Requires minimizing HDM , ROB , and ! Cost and Quantity Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Numerical Experiments Extensions Conclusion References Progressive Approach Ingredients of Proposed Approach ( ? ) Minimum-residual ROM (LSPG) and minimum-residual sensitivities f r ( µ ) = f ( µ ) and d f r d µ ( µ ) = d f d µ ( µ ) for training parameters µ Reduced optimization (sub)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 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 Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Shape Optimization: Airfoil Design Numerical Experiments Minimum Compliance: 2D Cantilever Extensions Minimum Compliance: 3D Trestle Conclusion References Outline Zahr Adaptive ROM-Constrained Optimization
Motivation ROM-Constrained Optimization Shape Optimization: Airfoil Design Numerical Experiments Minimum Compliance: 2D Cantilever Extensions Minimum Compliance: 3D Trestle 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 Adaptive ROM-Constrained Optimization
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