Introduction Optimization via Adaptive Model Reduction Large-Scale, Constrained Optimization Conclusion A Nonlinear Trust-Region Framework for PDE-Constrained Optimization Using Adaptive Model Reduction Matthew J. Zahr Institute for Computational and Mathematical Engineering Farhat Research Group Stanford University West Coast ROM Workshop Sandia National Laboratories, Livermore, CA November 19, 2015 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. Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Optimization via Adaptive Model Reduction Model Order Reduction Large-Scale, Constrained Optimization Nonlinear Trust-Region Solver 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 blue indicates a small quantity Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Optimization via Adaptive Model Reduction Model Order Reduction Large-Scale, Constrained Optimization Nonlinear Trust-Region Solver Conclusion Projection-Based Model Reduction Model Order Reduction (MOR) assumption: state vector lies in low-dimensional subspace u ≈ Φ u u r 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 apply Galerkin (or Petrov-Galerkin) projection T r ( Φ u u r , µ ) = 0 Φ u Method of Snapshots and Proper Orthogonal Decomposition used to construct reduced-order basis, Φ u Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Optimization via Adaptive Model Reduction Model Order Reduction Large-Scale, Constrained Optimization Nonlinear Trust-Region Solver Conclusion Nonlinear Trust-Region Framework with Adaptive ROMs Optimizer HDM HDM ROB Φ u Compress HDM ROM [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 PDE-Constrained Optimization with Adaptive ROMs
Introduction Optimization via Adaptive Model Reduction Model Order Reduction Large-Scale, Constrained Optimization Nonlinear Trust-Region Solver Conclusion Nonlinear Trust-Region Framework with Adaptive ROMs Nonlinear Trust-Region Framework with Adaptive Model Reduction Collect snapshots from HDM at sparse sampling of the parameter space Build ROB Φ u from sparse training Solve optimization problem minimize J ( Φ u u r , µ ) u r ∈ R k u , µ ∈ R n µ Φ T subject to u r ( Φ u u r , µ ) = 0 || r ( Φ u u r , µ ) || ≤ ∆ Use solution of above problem to enrich training, adapt ∆ using standard trust-region methods, and repeat until convergence Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Reduction of High-Dimensional Parameter Space Optimization via Adaptive Model Reduction Elastic Nonlinear Constraints Large-Scale, Constrained Optimization Topology Optimization: 2D Cantilever Conclusion PDE-Constrained Optimization II 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 c ( 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 c : R n u × R n µ → R n c are the side constraints u ∈ R n u is the PDE state vector µ ∈ R n µ is the vector of parameters Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Reduction of High-Dimensional Parameter Space Optimization via Adaptive Model Reduction Elastic Nonlinear Constraints Large-Scale, Constrained Optimization Topology Optimization: 2D Cantilever Conclusion Problem Setup 16000 8-node brick elements, 77760 dofs Total Lagrangian form, finite strain, StVK 2 St. Venant-Kirchhoff material 25 Sparse Cholesky linear solver (CHOLMOD 3 ) Newton-Raphson nonlinear solver Minimum compliance optimization problem 40 T u minimize f ext u ∈ R n u , µ ∈ R n µ V ( µ ) ≤ 1 subject to 2 V 0 r ( u , µ ) = 0 Gradient computations: Adjoint method Optimizer: SNOPT [Gill et al., 2002] 2 [Bonet and Wood, 1997, Belytschko et al., 2000] 3 [Chen et al., 2008] Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Reduction of High-Dimensional Parameter Space Optimization via Adaptive Model Reduction Elastic Nonlinear Constraints Large-Scale, Constrained Optimization Topology Optimization: 2D Cantilever Conclusion Restrict Parameter Space to Low-Dimensional Subspace Restrict parameter to a low-dimensional subspace µ ≈ Φ µ µ r � � ∈ R n µ × k µ is the reduced basis k µ φ 1 Φ µ = · · · φ µ µ µ r ∈ R k µ are the reduced coordinates of µ n µ ≫ k µ Substitute restriction into reduced-order model to obtain T r ( Φ u u r , Φ µ µ r ) = 0 Φ u Related work: [Maute and Ramm, 1995, Lieberman et al., 2010, Constantine et al., 2014] Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Reduction of High-Dimensional Parameter Space Optimization via Adaptive Model Reduction Elastic Nonlinear Constraints Large-Scale, Constrained Optimization Topology Optimization: 2D Cantilever Conclusion Restrict Parameter Space to Low-Dimensional Subspace Parameter space Background mesh Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Reduction of High-Dimensional Parameter Space Optimization via Adaptive Model Reduction Elastic Nonlinear Constraints Large-Scale, Constrained Optimization Topology Optimization: 2D Cantilever Conclusion Restrict Parameter Space to Low-Dimensional Subspace Parameter space Macroelements Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Reduction of High-Dimensional Parameter Space Optimization via Adaptive Model Reduction Elastic Nonlinear Constraints Large-Scale, Constrained Optimization Topology Optimization: 2D Cantilever Conclusion Optimality Conditions to Adapt Reduced-Order Basis, Φ µ Selection of Φ µ amounts to a restriction of the parameter space Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Reduction of High-Dimensional Parameter Space Optimization via Adaptive Model Reduction Elastic Nonlinear Constraints Large-Scale, Constrained Optimization Topology Optimization: 2D Cantilever Conclusion Optimality Conditions to Adapt Reduced-Order Basis, Φ µ Selection of Φ µ amounts to a restriction of the parameter space Adaptation of Φ µ should attempt to include the optimal solution in the restricted parameter space, i.e. µ ∗ ∈ col( Φ µ ) Adaptation based on first-order optimality conditions of HDM optimization problem Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Reduction of High-Dimensional Parameter Space Optimization via Adaptive Model Reduction Elastic Nonlinear Constraints Large-Scale, Constrained Optimization Topology Optimization: 2D Cantilever Conclusion Optimality Conditions to Adapt Reduced-Order Basis, Φ µ Lagrangian L ( µ , λ ) = J ( u ( µ ) , µ ) − λ T c ( u ( µ ) , µ ) Karush-Kuhn Tucker (KKT) Conditions 4 ∇ µ L ( µ , λ ) = 0 λ ≥ 0 λ i c i ( u ( µ ) , µ ) = 0 c ( u ( µ ) , µ ) ≥ 0 4 [Nocedal and Wright, 2006] Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Reduction of High-Dimensional Parameter Space Optimization via Adaptive Model Reduction Elastic Nonlinear Constraints Large-Scale, Constrained Optimization Topology Optimization: 2D Cantilever Conclusion Lagrangian Gradient Refinement Indicator From Lagrange multiplier estimates, only KKT condition not satisfied automatically: ∇ µ L ( µ , λ ) = 0 Use |∇ µ L ( µ , λ ) | as indicator for refinement of discretization of µ -space |∇ µ L ( µ , λ ) | µ Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction Reduction of High-Dimensional Parameter Space Optimization via Adaptive Model Reduction Elastic Nonlinear Constraints Large-Scale, Constrained Optimization Topology Optimization: 2D Cantilever Conclusion Constraints may lead to infeasible sub-problems Nonlinear Trust-Region MOR [Zahr and Farhat, 2014] minimize J ( Φ u u r , Φ µ µ r ) u r ∈ R k u , µ r ∈ R k µ subject to c ( Φ u u r , Φ µ µ r ) ≥ 0 r ( Φ u u r , Φ µ µ r ) = 0 || r ( Φ u u r , Φ µ µ r ) || ≤ ∆ Zahr PDE-Constrained Optimization with Adaptive ROMs
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