Introduction Optimization via Adaptive Model Reduction Large-Scale, Constrained Optimization Conclusion Accelerating PDE-Constrained Optimization Problems using Adaptive Reduced-Order Models Matthew J. Zahr Advisor: Charbel Farhat Computational and Mathematical Engineering Stanford University Sandia National Laboratories, Albuquerque, NM January 11-12, 2016 Zahr PDE-Constrained Optimization with Adaptive ROMs
• ‒ ‒ ‒ • ‒ • Introduction Optimization via Adaptive Model Reduction Large-Scale, Constrained Optimization Conclusion Multiphysics Optimization Key Player in Next-Gen Problems Current interest in computational physics reaches far beyond analysis of a single configuration of a physical system into design (shape and topology 1 ), control , and uncertainty quantification Micro-Aerial Vehicle EM Launcher Engine System 1 Emergence of additive manufacturing technologies has made topology optimization increasingly relevant, particularly in DOE. Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Optimization via Adaptive Model Reduction Large-Scale, Constrained Optimization Conclusion Topology Optimization and Additive Manufacturing 2 Emergence of AM has made TO an increasingly relevant topic AM+TO lead to highly efficient designs that could not be realized previously Challenges: smooth topologies require very fine meshes and modeling of complex manufacturing process 2 MIT Technology Review , Top 10 Technological Breakthrough 2013 Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Model Order Reduction Optimization via Adaptive Model Reduction Non-Quadratic Trust-Region Solver Large-Scale, Constrained Optimization Shape Optimization: Airfoil Design Conclusion PDE-Constrained Optimization I Goal: Rapidly solve PDE-constrained optimization problem of the form minimize J ( u , µ ) u ∈ R n u , µ ∈ R n µ subject to r ( u , µ ) = 0 where r : R n u × R n µ → R n u is the discretized partial differential equation J : R n u × R n µ → R is the objective function u ∈ R n u is the PDE state vector µ ∈ R n µ is the vector of parameters red indicates a large-scale quantity, O ( mesh ) Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Model Order Reduction Optimization via Adaptive Model Reduction Non-Quadratic Trust-Region Solver Large-Scale, Constrained Optimization Shape Optimization: Airfoil Design Conclusion Nested Approach to PDE-Constrained Optimization Virtually all expense emanates from primal/dual PDE solvers Optimizer Primal PDE Dual PDE Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Model Order Reduction Optimization via Adaptive Model Reduction Non-Quadratic Trust-Region Solver Large-Scale, Constrained Optimization Shape Optimization: Airfoil Design Conclusion Nested Approach to PDE-Constrained Optimization Virtually all expense emanates from primal/dual PDE solvers Optimizer µ Primal PDE Dual PDE Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Model Order Reduction Optimization via Adaptive Model Reduction Non-Quadratic Trust-Region Solver Large-Scale, Constrained Optimization Shape Optimization: Airfoil Design Conclusion Nested Approach to PDE-Constrained Optimization Virtually all expense emanates from primal/dual PDE solvers Optimizer J ( u , µ ) Primal PDE Dual PDE Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Model Order Reduction Optimization via Adaptive Model Reduction Non-Quadratic Trust-Region Solver Large-Scale, Constrained Optimization Shape Optimization: Airfoil Design Conclusion Nested Approach to PDE-Constrained Optimization Virtually all expense emanates from primal/dual PDE solvers Optimizer µ J ( u , µ ) u Primal PDE Dual PDE Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Model Order Reduction Optimization via Adaptive Model Reduction Non-Quadratic Trust-Region Solver Large-Scale, Constrained Optimization Shape Optimization: Airfoil Design Conclusion Nested Approach to PDE-Constrained Optimization Virtually all expense emanates from primal/dual PDE solvers Optimizer d J d µ ( u , µ ) J ( u , µ ) Primal PDE Dual PDE Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Model Order Reduction Optimization via Adaptive Model Reduction Non-Quadratic Trust-Region Solver Large-Scale, Constrained Optimization Shape Optimization: Airfoil Design Conclusion Projection-Based Model Reduction to Reduce PDE Size Model Order Reduction (MOR) assumption: state vector lies in low-dimensional subspace ∂ u ∂ u r u ≈ Φ u u r ∂ µ ≈ Φ u ∂ µ where ∈ R n u × k u is the reduced basis � φ 1 φ k u � Φ u = · · · u u u r ∈ R k u are the reduced coordinates of u n u ≫ k u Substitute assumption into High-Dimensional Model (HDM), r ( u , µ ) = 0, and project onto test subspace Ψ u ∈ R n u × k u T r ( Φ u u r , µ ) = 0 Ψ u Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Model Order Reduction Optimization via Adaptive Model Reduction Non-Quadratic Trust-Region Solver Large-Scale, Constrained Optimization Shape Optimization: Airfoil Design Conclusion Connection to Finite Element Method: Hierarchical Subspaces S S - infinite-dimensional trial space Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Model Order Reduction Optimization via Adaptive Model Reduction Non-Quadratic Trust-Region Solver Large-Scale, Constrained Optimization Shape Optimization: Airfoil Design Conclusion Connection to Finite Element Method: Hierarchical Subspaces S h S S - infinite-dimensional trial space S h - (large) finite-dimensional trial space Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Model Order Reduction Optimization via Adaptive Model Reduction Non-Quadratic Trust-Region Solver Large-Scale, Constrained Optimization Shape Optimization: Airfoil Design Conclusion Connection to Finite Element Method: Hierarchical Subspaces S k h S h S S - infinite-dimensional trial space S h - (large) finite-dimensional trial space S k h - (small) finite-dimensional trial space S k h ⊂ S h ⊂ S Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Model Order Reduction Optimization via Adaptive Model Reduction Non-Quadratic Trust-Region Solver Large-Scale, Constrained Optimization Shape Optimization: Airfoil Design Conclusion Few Global, Data-Driven Basis Functions v. Many Local Ones Instead of using traditional local shape functions (e.g., FEM), use global shape functions Instead of a-priori, analytical shape functions, leverage data-rich computing environment by using data-driven modes Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Model Order Reduction Optimization via Adaptive Model Reduction Non-Quadratic Trust-Region Solver Large-Scale, Constrained Optimization Shape Optimization: Airfoil Design Conclusion Definition of Φ u : Data-Driven Reduction State-Sensitivity Proper Orthogonal Decomposition (POD) Collect state and sensitivity snapshots by sampling HDM � � u ( µ 1 ) u ( µ 2 ) · · · u ( µ n ) X = � � ∂ u ∂ u ∂ u ∂ µ ( µ 1 ) ∂ µ ( µ 2 ) · · · ∂ µ ( µ n ) Y = Use Proper Orthogonal Decomposition to generate reduced basis for each individually Φ X = POD( X ) Φ Y = POD( Y ) Concatenate to get reduced-order basis � Φ X � Φ u = Φ Y Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Model Order Reduction Optimization via Adaptive Model Reduction Non-Quadratic Trust-Region Solver Large-Scale, Constrained Optimization Shape Optimization: Airfoil Design Conclusion Definition of Ψ u : Minimum-Residual ROM Least-Squares Petrov-Galerkin (LSPG) 3 projection Ψ u = ∂ r ∂ u Φ u Minimum-Residual Property A ROM possesses the minimum-residual property if Ψ u r ( Φ u u r , µ ) = 0 is equivalent to the optimality condition of (Θ ≻ 0) minimize || r ( Φ u u r , µ ) || Θ u r ∈ R k u Implications Recover exact solution when basis not truncated (consistent 3 ) Monotonic improvement of solution as basis size increases Ensures sensitivity information in Φ cannot degrade state approximation 4 LSPG possesses minimum-residual property 3 [Bui-Thanh et al., 2008] 4 [Fahl, 2001] Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Model Order Reduction Optimization via Adaptive Model Reduction Non-Quadratic Trust-Region Solver Large-Scale, Constrained Optimization Shape Optimization: Airfoil Design Conclusion Definition of ∂ u r ∂ µ : Minimum-Residual Reduced Sensitivities Traditional sensitivity analysis − 1 � ∂ r � T ∂ r N � ∂ u r T ∂ r j ∂ µ = − ∂ u ∂ u Φ u + r j Φ u ∂ u Φ u ∂ u Φ u j =1 � ∂ r � T ∂ r � N T ∂ 2 r j r j Φ u ∂ u ∂ µ + ∂ u Φ u ∂ µ j =1 + Guaranteed to give rise to exact derivatives of ROM quantities of interest - Requires 2nd derivatives of r - Φ u ∂ u r ∂ µ not guaranteed to be good approximate to full sensitivity ∂ u ∂ µ Zahr PDE-Constrained Optimization with Adaptive ROMs
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