Two-Level Approximation Error-Aware Trust-Region Model Management Numerical Experiments Adaptive Stochastic Collocation for PDE-Constrained Optimization under Uncertainty using Sparse Grids and Model Reduction Matthew J. Zahr Advisor: Charbel Farhat Computational and Mathematical Engineering Stanford University Joint work with: Kevin Carlberg (Sandia CA), Drew Kouri (Sandia NM) SIAM Conference on Uncertainty Quantification MS104: Reduced-Order Modeling in Uncertainty Quantification Lausanne, Switzerland April 7, 2016 Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level Approximation Error-Aware Trust-Region Model Management Numerical Experiments PDE-Constrained Optimization under Uncertainty Goal: Efficiently solve stochastic PDE-constrained optimization problems minimize E [ J ( u ( µ , · ) , µ , · )] µ ∈ R n µ subject to r ( u ( µ , ξ ) , µ , ξ ) = 0 ∀ ξ ∈ Ξ r : R n u × R n µ × R n ξ → R n u is the discretized stochastic PDE J : R n u × R n µ × R n ξ → R is a quantity of interest u ∈ R n u is the PDE state vector µ ∈ R n µ is the vector of (deterministic) optimization parameters ξ ∈ R n ξ is the vector of stochastic parameters Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level Approximation Error-Aware Trust-Region Model Management Numerical Experiments Literature Review: Stochastic Optimal Control Stochastic collocation Dimension-adaptive sparse grids – globalized with trust-region method [Kouri et al., 2013, Kouri et al., 2014] Generalized polynomial chaos – sequential quadratic programming [Tiesler et al., 2012] + Orders of magnitude improvement over isotropic sparse grids - Still requires many PDE solves for even moderate dimensional problems Model order reduction Goal-oriented, dimension-adaptive, weighted greedy algorithm for training stochastic reduced-order model [Chen and Quarteroni, 2015] Extension to optimal control [Chen and Quarteroni, 2014] + Reduction in number of PDE solves at cost of large number of ROM solves - Restriction to offline-online framework may lead to unnecessay PDE solves and large reduced bases Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level Approximation Error-Aware Trust-Region Model Management Numerical Experiments Proposed Approach Introduce two levels of inexactness to obtain an inexpensive, approximate version of the stochastic optimization problem; manage inexactness with trust-region-like method Anisotropic sparse grids used for inexact integration of risk measures Reduced-order models used for inexact evaluations at collocation nodes Error indicators introduced to account for both sources of inexactness Refinement of integral approximation and reduced-order model via dimension-adaptive sparse grids and a greedy method over collocation nodes Embedded in globally convergent trust-region-like algorithm with a strong connection to error indicators and refinement mechanism Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level Approximation Sparse Grids Error-Aware Trust-Region Model Management Model Reduction Numerical Experiments Anisotropic Sparse Grids [Gerstner and Griebel, 2003] 1D Quadrature Rules : Define the difference operator k − E j − 1 ∆ j k ≡ E j k k ≡ 0 and E j where E 0 k as the level- j 1d quadrature rule for dimension k Anisotropic Sparse Grid : i n ξ � ∆ i 1 E I ≡ 1 ⊗ · · · ⊗ ∆ n ξ i ∈I Forward Neighbors : N ( I ) = { k + e j | k ∈ I} \ I Truncation Error : [Gerstner and Griebel, 2003, Kouri et al., 2013] i n ξ i n ξ � ∆ i 1 � ∆ i 1 E − E I = 1 ⊗ · · · ⊗ ∆ n ξ ≈ 1 ⊗ · · · ⊗ ∆ n ξ = E N ( I ) ∈I i ∈N ( I ) i / Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level Approximation Sparse Grids Error-Aware Trust-Region Model Management Model Reduction Numerical Experiments Stochastic Collocation via Anisotropic Sparse Grids Stochastic collocation using anisotropic sparse grid nodes used to approximate integral with summation E [ J ( u , µ , · )] minimize u ∈ R n u , µ ∈ R n µ ∀ ξ ∈ Ξ subject to r ( u , µ , ξ ) = 0 ⇓ minimize E I [ J ( u , µ , · )] u ∈ R n u , µ ∈ R n µ subject to r ( u , µ , ξ ) = 0 ∀ ξ ∈ Ξ I Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level Approximation Sparse Grids Error-Aware Trust-Region Model Management Model Reduction Numerical Experiments Projection-Based Model Reduction to Reduce PDE Size Model Order Reduction (MOR) assumption: state vector lies in low-dimensional subspace ∂ u ∂ µ ≈ Φ ∂ y u ≈ Φ y ∂ µ where ∈ R n u × k u is the reduced basis φ k u � φ 1 � Φ = · · · u u y ∈ R k u are the reduced coordinates of u n u ≫ k u Substitute assumption into High-Dimensional Model (HDM), r ( u , µ , ξ ) = 0, and use a Galerkin projection to obtain the square system Φ T r ( Φ y , µ , ξ ) = 0 Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level Approximation Sparse Grids Error-Aware Trust-Region Model Management Model Reduction Numerical Experiments Definition of Φ : Data-Driven Reduction State-Sensitivity Proper Orthogonal Decomposition (SSPOD) Collect state and sensitivity snapshots by sampling HDM � � X = u ( µ 1 , ξ 1 ) u ( µ 2 , ξ 2 ) · · · u ( µ n , ξ n ) � � ∂ u ∂ u ∂ u Y = ∂ µ ( µ 1 , ξ 1 ) ∂ µ ( µ 2 , ξ 2 ) · · · ∂ µ ( µ n , ξ n ) Use Proper Orthogonal Decomposition to generate reduced basis for each individually Φ X = POD( X ) Φ Y = POD( Y ) Concatenate and orthogonalize to get reduced-order basis �� Φ X �� Φ = QR Φ Y Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level Approximation Sparse Grids Error-Aware Trust-Region Model Management Model Reduction Numerical Experiments Reduced-Order Stochastic Collocation via Anisotropic Sparse Grids Stochastic collocation of the reduced-order model over anisotropic sparse grid nodes used to approximate integral with cheap summation E [ J ( u , µ , · )] minimize u ∈ R n u , µ ∈ R n µ subject to r ( u , µ , ξ ) = 0 ∀ ξ ∈ Ξ ⇓ minimize E I [ J ( u , µ , · )] u ∈ R n u , µ ∈ R n µ ∀ ξ ∈ Ξ I subject to r ( u , µ , ξ ) = 0 ⇓ minimize E I [ J ( Φ y , µ , · )] y ∈ R k u , µ ∈ R n µ Φ T r ( Φ y , µ , ξ ) = 0 subject to ∀ ξ ∈ Ξ I Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level Approximation Primal and Sensitivity Error Indicators Error-Aware Trust-Region Model Management Trust-Region Algorithm Numerical Experiments Two-Level Model Refinement ROM-Based Trust-Region Framework for Optimization Schematic µ -space Breakdown of Computational Effort Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level Approximation Primal and Sensitivity Error Indicators Error-Aware Trust-Region Model Management Trust-Region Algorithm Numerical Experiments Two-Level Model Refinement ROM-Based Trust-Region Framework for Optimization HDM ROB Φ Compress HDM Schematic µ -space HDM ROB Breakdown of Computational Effort Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level Approximation Primal and Sensitivity Error Indicators Error-Aware Trust-Region Model Management Trust-Region Algorithm Numerical Experiments Two-Level Model Refinement ROM-Based Trust-Region Framework for Optimization Optimizer HDM ROB Φ Compress HDM ROM Schematic µ -space ROM ROM ROM ROM ROM · · · HDM ROB Breakdown of Computational Effort Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level Approximation Primal and Sensitivity Error Indicators Error-Aware Trust-Region Model Management Trust-Region Algorithm Numerical Experiments Two-Level Model Refinement ROM-Based Trust-Region Framework for Optimization Optimizer HDM HDM ROB Φ Compress HDM ROM Schematic µ -space ROM ROM ROM ROM ROM · · · HDM ROB HDM ROB Breakdown of Computational Effort Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level Approximation Primal and Sensitivity Error Indicators Error-Aware Trust-Region Model Management Trust-Region Algorithm Numerical Experiments Two-Level Model Refinement ROM-Based Trust-Region Framework for Optimization Optimizer HDM HDM ROB Φ Compress HDM ROM Schematic µ -space ROM ROM ROM ROM ROM ROM ROM ROM ROM ROM · · · · · · HDM ROB HDM ROB Breakdown of Computational Effort Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
Two-Level Approximation Primal and Sensitivity Error Indicators Error-Aware Trust-Region Model Management Trust-Region Algorithm Numerical Experiments Two-Level Model Refinement ROM-Based Trust-Region Framework for Optimization Optimizer HDM HDM ROB Φ Compress HDM ROM Schematic µ -space ROM ROM ROM ROM ROM ROM ROM ROM ROM ROM · · · · · · · · · HDM ROB HDM ROB Breakdown of Computational Effort Zahr, Carlberg, Kouri Stochastic PDE-Optimization with Adaptive SG/ROMs
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