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The Standard POD Method Streamline Diffusion POD Models Optimization with TRPOD Conclusion Proper Orthogonal Decomposition in Optimization Bret Kragel and Ekkehard W. Sachs Surrogate Modelling and Space Mapping for Engineering Optimization

  1. The Standard POD Method The Standard POD Method Streamline Diffusion POD Models An Example Problem Optimization with TRPOD Failure of the Standard POD-Based Model Conclusion Galerkin POD-Based Model Assume boundary velocity separable: g ( t , x ) = γ ( t ) h ( x ) Expand velocity in terms of POD basis: Note separation of variables t and x m � ˜ u ( t , x ) = u n ( x ) + γ ( t ) u c ( x ) + y i ( t )Ψ i ( x ) . i = 1 Project momentum equation onto basis for POD modes: y j ( t ) = − ν ( ∇ ˜ ˙ γ ( t ) ( u c , Ψ j ) − (˜ u · ∇ ˜ u , ∇ Ψ j ) − ˙ u , Ψ j ) , j = 1 , . . . , m . Expansion of bilinear terms leads to ODE system for modes: y ( t ) = M 0 + γ ( t ) M 1 + γ 2 ( t ) M 2 + ˙ ˙ γ ( t ) M c + M 3 y + γ ( t ) M 4 y + M 5 ( y , y ) with M 0 , M 1 , M 2 , M c ∈ R m , M 3 , M 4 ∈ R m , m , M 5 ( y , y ) : R m , m → R m Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  2. The Standard POD Method The Standard POD Method Streamline Diffusion POD Models An Example Problem Optimization with TRPOD Failure of the Standard POD-Based Model Conclusion Galerkin POD-Based Model Assume boundary velocity separable: g ( t , x ) = γ ( t ) h ( x ) Expand velocity in terms of POD basis: Note separation of variables t and x m � ˜ u ( t , x ) = u n ( x ) + γ ( t ) u c ( x ) + y i ( t )Ψ i ( x ) . i = 1 Project momentum equation onto basis for POD modes: y j ( t ) = − ν ( ∇ ˜ ˙ γ ( t ) ( u c , Ψ j ) − (˜ u · ∇ ˜ u , ∇ Ψ j ) − ˙ u , Ψ j ) , j = 1 , . . . , m . Expansion of bilinear terms leads to ODE system for modes: y ( t ) = M 0 + γ ( t ) M 1 + γ 2 ( t ) M 2 + ˙ ˙ γ ( t ) M c + M 3 y + γ ( t ) M 4 y + M 5 ( y , y ) with M 0 , M 1 , M 2 , M c ∈ R m , M 3 , M 4 ∈ R m , m , M 5 ( y , y ) : R m , m → R m Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  3. The Standard POD Method The Standard POD Method Streamline Diffusion POD Models An Example Problem Optimization with TRPOD Failure of the Standard POD-Based Model Conclusion Driven Cavity Velocity Profile γ ( t ) = 1 2 + 3 π [ sin ( π t / 20 ) + 1 3 sin ( 3 π t / 10 ) + 1 5 sin ( 5 π t / 10 ) + 1 7 sin ( 7 π t / 10 ) + 1 9 sin ( 9 π t / 10 )] Reynolds number: Re = ν − 1 = 400 Simulation time: T = 20 sec. 100 snapshots Mesh: 49 × 49 Truncation: 99.9% of system energy Direct projection of snapshots onto POD basis: ˆ y i ( t ) = ( u − u n ( x ) − γ ( t ) u c ( x ) , Ψ i ) Projected velocity model: m � u ( t , x ) = u n ( x ) + γ ( t ) u c ( x ) + ˆ ˆ y i ( t )Ψ i ( x ) . i = 1 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  4. The Standard POD Method The Standard POD Method Streamline Diffusion POD Models An Example Problem Optimization with TRPOD Failure of the Standard POD-Based Model Conclusion Projected and Predicted Modes at Re = 400 −4 Absolute Simulation Error (n=100 snapshots; m=9 basis functions) x 10 1.4 Prediction Projection 1.2 9 basis functions for 99.9% of energy (9 modes) 1 Compare 0.8 u � 2 and projection error � u − ˆ Error u � 2 prediction error � u − ˜ 0.6 Predicted error slightly greater as 0.4 expected, but both ≤ 10 − 4 Excellent approximation at Re=400 0.2 0 0 2 4 6 8 10 12 14 16 18 20 Time Limits of POD in optimization POD basis derived from numerical data. Model fidelity dependent on problem data (boundary conditions, Reynolds number, etc.) Model must be reset repeatedly during optimization process. Computationally expensive! Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  5. The Standard POD Method The Standard POD Method Streamline Diffusion POD Models An Example Problem Optimization with TRPOD Failure of the Standard POD-Based Model Conclusion Projected and Predicted Modes at Re = 400 −4 Absolute Simulation Error (n=100 snapshots; m=9 basis functions) x 10 1.4 Prediction Projection 1.2 9 basis functions for 99.9% of energy (9 modes) 1 Compare 0.8 u � 2 and projection error � u − ˆ Error u � 2 prediction error � u − ˜ 0.6 Predicted error slightly greater as 0.4 expected, but both ≤ 10 − 4 Excellent approximation at Re=400 0.2 0 0 2 4 6 8 10 12 14 16 18 20 Time Limits of POD in optimization POD basis derived from numerical data. Model fidelity dependent on problem data (boundary conditions, Reynolds number, etc.) Model must be reset repeatedly during optimization process. Computationally expensive! Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  6. The Standard POD Method The Standard POD Method Streamline Diffusion POD Models An Example Problem Optimization with TRPOD Failure of the Standard POD-Based Model Conclusion Projected and Predicted Modes at Re = 400 −4 Absolute Simulation Error (n=100 snapshots; m=9 basis functions) x 10 1.4 Prediction Projection 1.2 9 basis functions for 99.9% of energy (9 modes) 1 Compare 0.8 u � 2 and projection error � u − ˆ Error u � 2 prediction error � u − ˜ 0.6 Predicted error slightly greater as 0.4 expected, but both ≤ 10 − 4 Excellent approximation at Re=400 0.2 0 0 2 4 6 8 10 12 14 16 18 20 Time Limits of POD in optimization POD basis derived from numerical data. Model fidelity dependent on problem data (boundary conditions, Reynolds number, etc.) Model must be reset repeatedly during optimization process. Computationally expensive! Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  7. The Standard POD Method The Standard POD Method Streamline Diffusion POD Models An Example Problem Optimization with TRPOD Failure of the Standard POD-Based Model Conclusion Projected and Predicted Modes at Re = 400 −4 Absolute Simulation Error (n=100 snapshots; m=9 basis functions) x 10 1.4 Prediction Projection 1.2 9 basis functions for 99.9% of energy (9 modes) 1 Compare 0.8 u � 2 and projection error � u − ˆ Error u � 2 prediction error � u − ˜ 0.6 Predicted error slightly greater as 0.4 expected, but both ≤ 10 − 4 Excellent approximation at Re=400 0.2 0 0 2 4 6 8 10 12 14 16 18 20 Time Limits of POD in optimization POD basis derived from numerical data. Model fidelity dependent on problem data (boundary conditions, Reynolds number, etc.) Model must be reset repeatedly during optimization process. Computationally expensive! Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  8. The Standard POD Method The Standard POD Method Streamline Diffusion POD Models An Example Problem Optimization with TRPOD Failure of the Standard POD-Based Model Conclusion Projected and Predicted Modes at Re = 400 −4 Absolute Simulation Error (n=100 snapshots; m=9 basis functions) x 10 1.4 Prediction Projection 1.2 9 basis functions for 99.9% of energy (9 modes) 1 Compare 0.8 u � 2 and projection error � u − ˆ Error u � 2 prediction error � u − ˜ 0.6 Predicted error slightly greater as 0.4 expected, but both ≤ 10 − 4 Excellent approximation at Re=400 0.2 0 0 2 4 6 8 10 12 14 16 18 20 Time Limits of POD in optimization POD basis derived from numerical data. Model fidelity dependent on problem data (boundary conditions, Reynolds number, etc.) Model must be reset repeatedly during optimization process. Computationally expensive! Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  9. The Standard POD Method The Standard POD Method Streamline Diffusion POD Models An Example Problem Optimization with TRPOD Failure of the Standard POD-Based Model Conclusion Standard POD on Rough Mesh at Re = 10 , 000 Suggestion: Combine POD with coarser grids Questions: Coarser grids and higher Reynolds numbers? Try Re = 10 , 000 on 13 × 13 mesh with γ ( t ) ≡ 1 . 0 ODE solver fails to converge Standard POD method useless Question: What to do? Distinguish between POD for physical model and POD for numerical model. POD should conform to numerical model. Stability problems well-known from standard numerical approximation methods for Navier-Stokes equations Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  10. The Standard POD Method The Standard POD Method Streamline Diffusion POD Models An Example Problem Optimization with TRPOD Failure of the Standard POD-Based Model Conclusion Standard POD on Rough Mesh at Re = 10 , 000 Suggestion: Combine POD with coarser grids Questions: Coarser grids and higher Reynolds numbers? Try Re = 10 , 000 on 13 × 13 mesh with γ ( t ) ≡ 1 . 0 ODE solver fails to converge Mode Amplitudes (n=100 snapshots; m=13 basis functions) 0.3 Direct Projection POD Modes Standard POD method useless 0.25 Question: What to do? 0.2 Distinguish between POD for physical 0.15 Mode amplitude model and POD for numerical model. 0.1 POD should conform to numerical 0.05 model. 0 Stability problems well-known from standard numerical approximation −0.05 methods for Navier-Stokes equations −0.1 0 2 4 6 8 10 12 14 16 18 20 Time (seconds) Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  11. The Standard POD Method The Standard POD Method Streamline Diffusion POD Models An Example Problem Optimization with TRPOD Failure of the Standard POD-Based Model Conclusion Standard POD on Rough Mesh at Re = 10 , 000 Suggestion: Combine POD with coarser grids Questions: Coarser grids and higher Reynolds numbers? Try Re = 10 , 000 on 13 × 13 mesh with γ ( t ) ≡ 1 . 0 ODE solver fails to converge Mode Amplitudes (n=100 snapshots; m=13 basis functions) 0.3 Direct Projection POD Modes Standard POD method useless 0.25 Question: What to do? 0.2 Distinguish between POD for physical 0.15 Mode amplitude model and POD for numerical model. 0.1 POD should conform to numerical 0.05 model. 0 Stability problems well-known from standard numerical approximation −0.05 methods for Navier-Stokes equations −0.1 0 2 4 6 8 10 12 14 16 18 20 Time (seconds) Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  12. The Standard POD Method The Standard POD Method Streamline Diffusion POD Models An Example Problem Optimization with TRPOD Failure of the Standard POD-Based Model Conclusion Use of POD Concept Problematic Approach Generate snap shots using a numerical scheme with stabilization Generate POD from Galerkin approximation based on physical model Suggested Approach Generate snap shots using a numerical scheme with stabilization Generate POD from Galerkin approximation based on numerical model Upwinding not suitable in this context. Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  13. The Standard POD Method Streamline Diffusion POD Models Sources POD Failure Optimization with TRPOD The SDPOD Model Conclusion Sources of Instabilities in Navier-Stokes Solvers 1 Dominant convective terms Exact solution h=1/10 h= ν Example: − ν u ′′ + u ′ = 0 in Ω = ( 0 , 1 ) 0.5 u ( 0 ) = 0, u ( 1 ) = 1, ν = 1 / 100 FE discretization leads to central difference scheme for convective term u ′ 0 Common solutions: Upwinding −0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Streamline diffusion Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  14. The Standard POD Method Streamline Diffusion POD Models Sources POD Failure Optimization with TRPOD The SDPOD Model Conclusion Streamline diffusion Find u h ∈ V h and p h ∈ Q h , such that ( u h , v h ) + ν a ( u h , v h ) + ˜ n ( u h , u h , v h ) + b ( v h , p h ) = 0 ∀ v h ∈ V h b ( u h , q h ) = 0 ∀ q h ∈ Q h , Modified convective term � ˜ n ( u h , v h , w h ) = n ( u h , v h , w h ) + δ T ( u h · ∇ v h , u h · ∇ w h ) | T T ∈T h Local damping parameter: h T 2 Re T δ T = δ · � u � ∞ · , Re T = � u � T · h T /ν 1 + Re T Only δ T = δ T ( u , h ) is nonlinear in u Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  15. The Standard POD Method Streamline Diffusion POD Models Sources POD Failure Optimization with TRPOD The SDPOD Model Conclusion The Streamline Diffusion POD Method Harmonize POD-based model and POD basis functions by adding streamline diffusion regularization to the Galerkin POD projection: ˙ y j ( t ) = − ν ( ∇ u , ∇ Ψ j ) − ˙ γ ( t ) ( u c , Ψ j ) − ( u · ∇ u , Ψ j ) − δ T ( u · ∇ u , u · ∇ Ψ j ) For large Reynolds numbers 2 h T δ T ≈ δ m T = δ · � u � ∞ SDPOD ODE System Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  16. The Standard POD Method Streamline Diffusion POD Models Sources POD Failure Optimization with TRPOD The SDPOD Model Conclusion The Streamline Diffusion POD Method Harmonize POD-based model and POD basis functions by adding streamline diffusion regularization to the Galerkin POD projection: ˙ y j ( t ) = − ν ( ∇ u , ∇ Ψ j ) − ˙ γ ( t ) ( u c , Ψ j ) − ( u · ∇ u , Ψ j ) − δ T ( u · ∇ u , u · ∇ Ψ j ) For large Reynolds numbers 2 h T δ T ≈ δ m T = δ · � u � ∞ SDPOD ODE System Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  17. The Standard POD Method Streamline Diffusion POD Models Sources POD Failure Optimization with TRPOD The SDPOD Model Conclusion The Streamline Diffusion POD Method Harmonize POD-based model and POD basis functions by adding streamline diffusion regularization to the Galerkin POD projection: ˙ y j ( t ) = − ν ( ∇ u , ∇ Ψ j ) − ˙ γ ( t ) ( u c , Ψ j ) − ( u · ∇ u , Ψ j ) − δ T ( u · ∇ u , u · ∇ Ψ j ) For large Reynolds numbers 2 h T δ T ≈ δ m T = δ · � u � ∞ SDPOD ODE System y ( t ) = M 0 + γ ( t ) M 1 + γ 2 ( t ) M 2 + ˙ ˙ γ ( t ) M c + M 3 y + γ ( t ) M 4 y + M 5 ( y , y ) Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  18. The Standard POD Method Streamline Diffusion POD Models Sources POD Failure Optimization with TRPOD The SDPOD Model Conclusion The Streamline Diffusion POD Method Harmonize POD-based model and POD basis functions by adding streamline diffusion regularization to the Galerkin POD projection: y j ( t ) = − ν ( ∇ u , ∇ Ψ j ) − ˙ ˙ γ ( t ) ( u c , Ψ j ) − ( u · ∇ u , Ψ j ) − δ T ( u · ∇ u , u · ∇ Ψ j ) For large Reynolds numbers 2 h T δ T ≈ δ m T = δ · � u � ∞ SDPOD ODE System y ( t ) = M 0 + γ ( t ) M 1 + γ 2 ( t ) M 2 + ˙ γ ( t ) M c + M 3 y + γ ( t ) M 4 y + M 5 ( y , y ) ˙ + δ m M δ 0 + γ ( t ) M δ 1 + γ 2 ( t ) M δ 2 + M δ 3 y + γ ( t ) M δ � 4 y T + M δ 5 ( y , y ) + γ 3 ( t ) M δ 6 + γ 2 M δ 7 y + γ ( t ) M δ 8 ( y , y ) + M δ � 9 ( y , y , y ) Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  19. The Standard POD Method Streamline Diffusion POD Models Sources POD Failure Optimization with TRPOD The SDPOD Model Conclusion POD vs. SDPOD at Re = 10 , 000 ( δ = 1 . 0) Discr. m %Energy E PROJ E SDPOD E POD 4 × 4 5 99.9 8.4037 e-6 9.7677 e-6 1.8121 e-1 7 × 7 10 99.9 1.1408 e-5 1.3565 e-5 9.1854 e-2 13 × 13 15 99.9 6.6566 e-6 9.1258 e-6 9.8077 e-2 25 × 25 20 99.9 4.5393 e-6 7.4863 e-6 1.1682 e-2 49 × 49 24 99.9 3.6355 e-6 5.0878 e-6 7.4593 e-3 97 × 97 30 99.9 6.4540 e-6 9.2616 e-6 1.0731 e-2 Observations SDPOD and projection errors order of 10 − 5 SDPOD error slightly larger, but excellent agreement POD error comparatively large - even on finer grids Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  20. The Standard POD Method Streamline Diffusion POD Models Sources POD Failure Optimization with TRPOD The SDPOD Model Conclusion Graphical Comparison: POD vs. SDPOD at Re = 10 , 000 ( δ = 1 . 0) Absolute Error (n=100 snapshots) −5 Absolute Error. (n=100 snapshots) x 10 0.5 3 4 × 4 Mesh 4 × 4 Mesh 7 × 7 Mesh 7 × 7 Mesh 0.45 13 × 13 Mesh 13 × 13 Mesh 25 × 25 Mesh 2.5 25 × 25 Mesh 0.4 49 × 49 Mesh 49 × 49 Mesh 97 × 97 Mesh 97 × 97 Mesh 0.35 2 0.3 0.25 1.5 0.2 1 0.15 0.1 0.5 0.05 0 0 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 POD Method SDPOD Method Similar phenomenon when underlying model data changes well-known from POD literature. Changing boundary condition leads to model deterioration ⇒ POD basis update Question: When to update? ⇒ Need framework for managing model updates. ( SMF) Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  21. The Standard POD Method Streamline Diffusion POD Models Sources POD Failure Optimization with TRPOD The SDPOD Model Conclusion POD Basis Functions for u 0 and u 5 (red) 30 30 30 30 25 25 25 25 20 20 20 20 15 15 15 15 10 10 10 10 5 5 5 5 5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30 30 30 30 30 25 25 25 25 20 20 20 20 15 15 15 15 10 10 10 10 5 5 5 5 5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  22. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Model Functions in Optimization R n , how to find a ’better’ point ? Given g ∈ I ’Better’: J ( g + ) < J ( g ) . Difficult problem, since J nonlinear. Approximate J ( · ) by model function: linear or quadratic or less complex than J . (POD models, multifidelity models, neural networks,...) Surrogate Optimization. J ( g + s ) ≈ m ( g + s ) = J ( g ) + ∇J ( g ) T s J ( g + s ) ≈ m ( g + s ) = J ( g ) + ∇J ( g ) T s + 1 2 s T ∇ 2 J ( g ) s Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  23. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Model Minimization Given a model function m ( g + s ) for J ( g + s ) . Replace minimize J ( u + s ) by minimize m ( u + s ) . Validity of model m ( g + s ) ? Restrict s to a region where we trust model. ’Democratic’ choice: s : � s � ≤ ∆ . (P) min m ( g + s ) � s � ≤ ∆ Trust Region Problem Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  24. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Model Adaption Choice of ∆ ? When to update trust region ? Let s (∆) denote the solution of trust region problem. If m ( g + s ) is a good model for φ , choose large ∆ , otherwise small ∆ . If predicted reduction m ( g + s (∆)) − m ( g ) is close to actual reduction J ( g + s (∆)) − J ( g ) , then accept s (∆) and keep or increase trust region radius ∆ , otherwise take smaller ∆ and recompute s (∆) . Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  25. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Inexact Gradient Information Carter 1991 Suppose that no exact gradient information is available for model function m ( g + s ) = J ( g ) + n T s + 1 2 s T Hs Exact information for value J ( g ) is not necessary. Gradient information has to improve as iteration progresses. Theorem Let J be twice cont. differentiable and bounded below on level set. Let sufficient decrease condition hold, and H k be bounded. If n k → 0 and �∇J ( g k ) − n k � ≤ ζ < 1 � n k � then lim inf k →∞ �∇J ( g k ) � = 0 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  26. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Nonlinear Model Function Toint 1988, Carter 1991, Fahl 2000 Nonlinear model functions m ( g + s ) . Sufficient decrease condition m ( g + s ) − m ( g ) ≤ c ( m ( g + s CP ) − m ( g )) where s CP is a descent step in direction of negative gradient. Need a replacement for second order information: measure of curvature 2 � s � 2 ( m ( g + s ) − m ( g ) − ∇ m ( g ) T s ) ω ( m , g , s ) = Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  27. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Nonlinear Model Function, continued Set n k = ∇ m k ( g k ) . Then previous theorems can be extended to. Theorem Let f be twice cont. differentiable and bounded below on level set. Let sufficient decrease condition hold, and ω ( m k , g k , s k ) be bounded. If n k → 0 and �∇J ( g k ) − n k � ≤ ζ < 1 � n k � then lim inf k →∞ �∇J ( g k ) � = 0 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  28. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Adaption of POD Model Arian, Fahl, Sachs Afanasiev, Hinze Kunisch, Volkwein Gunzburger Willcox Model Management: Alexandrov, Lewis Dennis Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  29. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Replace objective J ( g ) = J ( u ( g ) , g ) with cheaper objective m ( g ) := J (˜ u ( g ) , g ) during optimization – ˜ u (SD)POD-based model Constrain model m k ( g k + s k ) to trust-region � s k � ≤ δ k Compute snapshot set U + using g k + s k and accept or reject new iterate based on J ( g k ) − J ( g k + s k ) ρ k = m k ( g k ) − m k ( g k + s k ) Fahl [2000]: Convergence of Trust-Region POD (TRPOD) method under appropriate conditions. Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  30. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Replace objective J ( g ) = J ( u ( g ) , g ) with cheaper objective m ( g ) := J (˜ u ( g ) , g ) during optimization – ˜ u (SD)POD-based model Constrain model m k ( g k + s k ) to trust-region � s k � ≤ δ k Compute snapshot set U + using g k + s k and accept or reject new iterate based on J ( g k ) − J ( g k + s k ) ρ k = m k ( g k ) − m k ( g k + s k ) Fahl [2000]: Convergence of Trust-Region POD (TRPOD) method under appropriate conditions. Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  31. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Replace objective J ( g ) = J ( u ( g ) , g ) with cheaper objective m ( g ) := J (˜ u ( g ) , g ) during optimization – ˜ u (SD)POD-based model Constrain model m k ( g k + s k ) to trust-region � s k � ≤ δ k Compute snapshot set U + using g k + s k and accept or reject new iterate based on J ( g k ) − J ( g k + s k ) ρ k = m k ( g k ) − m k ( g k + s k ) Fahl [2000]: Convergence of Trust-Region POD (TRPOD) method under appropriate conditions. Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  32. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Replace objective J ( g ) = J ( u ( g ) , g ) with cheaper objective m ( g ) := J (˜ u ( g ) , g ) during optimization – ˜ u (SD)POD-based model Constrain model m k ( g k + s k ) to trust-region � s k � ≤ δ k Compute snapshot set U + using g k + s k and accept or reject new iterate based on J ( g k ) − J ( g k + s k ) ρ k = m k ( g k ) − m k ( g k + s k ) Fahl [2000]: Convergence of Trust-Region POD (TRPOD) method under appropriate conditions. Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  33. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Replace objective J ( g ) = J ( u ( g ) , g ) with cheaper objective m ( g ) := J (˜ u ( g ) , g ) during optimization – ˜ u (SD)POD-based model Constrain model m k ( g k + s k ) to trust-region � s k � ≤ δ k Compute snapshot set U + using g k + s k and accept or reject new iterate based on J ( g k ) − J ( g k + s k ) ρ k = m k ( g k ) − m k ( g k + s k ) Fahl [2000]: Convergence of Trust-Region POD (TRPOD) method under appropriate conditions. Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  34. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region POD (TRPOD) Goal: Replace objective J ( g ) = J ( u ( g ) , g ) with cheaper objective m ( g ) := J (˜ u ( g ) , g ) during optimization – ˜ u SDPOD-based model Initialization: Choose 0 < η 1 < η 2 < 1, 0 < γ 1 < γ 2 < 1 < γ 3 , an initial trust-region radius δ 0 > 0 and an initial iterate g 0 . Compute snapshot set U 0 and J ( g 0 ) . Set k = 0 Step 1: Compute POD basis and POD-based model m k ( g k ) Step 2: Compute approximately: min m k ( g k + s k ) s.t. � s k � ≤ δ 0 Step 3: Compute snapshot set U + using g k + s k and set J ( g k ) − J ( g k + s k ) ρ k = m k ( g k ) − m k ( g k + s k ) Step 4: Update Trust-Region Radius ρ k ≥ η 2 : U k + 1 = U + , g k + 1 = g k + s k , J ( g k + 1 ) = J ( g k + s k ) , δ k + 1 ∈ [ δ k , γ 3 δ k ) , k = k + 1, → Step 1 η 1 ≤ ρ k < η 2 : U k + 1 = U + , g k + 1 = g k + s k , J ( g k + 1 ) = J ( g k + s k ) , δ k + 1 ∈ [ γ 2 δ k , δ k ) , k = k + 1, → Step 1 ρ k < η 1 : U k + 1 = U k , δ k + 1 ∈ [ γ 1 δ k , γ 2 δ k ) , k = k + 1, → Step 2 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  35. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region POD (TRPOD) Goal: Replace objective J ( g ) = J ( u ( g ) , g ) with cheaper objective m ( g ) := J (˜ u ( g ) , g ) during optimization – ˜ u SDPOD-based model Initialization: Choose 0 < η 1 < η 2 < 1, 0 < γ 1 < γ 2 < 1 < γ 3 , an initial trust-region radius δ 0 > 0 and an initial iterate g 0 . Compute snapshot set U 0 and J ( g 0 ) . Set k = 0 Step 1: Compute POD basis and POD-based model m k ( g k ) Step 2: Compute approximately: min m k ( g k + s k ) s.t. � s k � ≤ δ 0 Step 3: Compute snapshot set U + using g k + s k and set J ( g k ) − J ( g k + s k ) ρ k = m k ( g k ) − m k ( g k + s k ) Step 4: Update Trust-Region Radius ρ k ≥ η 2 : U k + 1 = U + , g k + 1 = g k + s k , J ( g k + 1 ) = J ( g k + s k ) , δ k + 1 ∈ [ δ k , γ 3 δ k ) , k = k + 1, → Step 1 η 1 ≤ ρ k < η 2 : U k + 1 = U + , g k + 1 = g k + s k , J ( g k + 1 ) = J ( g k + s k ) , δ k + 1 ∈ [ γ 2 δ k , δ k ) , k = k + 1, → Step 1 ρ k < η 1 : U k + 1 = U k , δ k + 1 ∈ [ γ 1 δ k , γ 2 δ k ) , k = k + 1, → Step 2 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  36. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region POD (TRPOD) Goal: Replace objective J ( g ) = J ( u ( g ) , g ) with cheaper objective m ( g ) := J (˜ u ( g ) , g ) during optimization – ˜ u SDPOD-based model Initialization: Choose 0 < η 1 < η 2 < 1, 0 < γ 1 < γ 2 < 1 < γ 3 , an initial trust-region radius δ 0 > 0 and an initial iterate g 0 . Compute snapshot set U 0 and J ( g 0 ) . Set k = 0 Step 1: Compute POD basis and POD-based model m k ( g k ) Step 2: Compute approximately: min m k ( g k + s k ) s.t. � s k � ≤ δ 0 Step 3: Compute snapshot set U + using g k + s k and set J ( g k ) − J ( g k + s k ) ρ k = m k ( g k ) − m k ( g k + s k ) Step 4: Update Trust-Region Radius ρ k ≥ η 2 : U k + 1 = U + , g k + 1 = g k + s k , J ( g k + 1 ) = J ( g k + s k ) , δ k + 1 ∈ [ δ k , γ 3 δ k ) , k = k + 1, → Step 1 η 1 ≤ ρ k < η 2 : U k + 1 = U + , g k + 1 = g k + s k , J ( g k + 1 ) = J ( g k + s k ) , δ k + 1 ∈ [ γ 2 δ k , δ k ) , k = k + 1, → Step 1 ρ k < η 1 : U k + 1 = U k , δ k + 1 ∈ [ γ 1 δ k , γ 2 δ k ) , k = k + 1, → Step 2 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  37. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region POD (TRPOD) Goal: Replace objective J ( g ) = J ( u ( g ) , g ) with cheaper objective m ( g ) := J (˜ u ( g ) , g ) during optimization – ˜ u SDPOD-based model Initialization: Choose 0 < η 1 < η 2 < 1, 0 < γ 1 < γ 2 < 1 < γ 3 , an initial trust-region radius δ 0 > 0 and an initial iterate g 0 . Compute snapshot set U 0 and J ( g 0 ) . Set k = 0 Step 1: Compute POD basis and POD-based model m k ( g k ) Step 2: Compute approximately: min m k ( g k + s k ) s.t. � s k � ≤ δ 0 Step 3: Compute snapshot set U + using g k + s k and set J ( g k ) − J ( g k + s k ) ρ k = m k ( g k ) − m k ( g k + s k ) Step 4: Update Trust-Region Radius ρ k ≥ η 2 : U k + 1 = U + , g k + 1 = g k + s k , J ( g k + 1 ) = J ( g k + s k ) , δ k + 1 ∈ [ δ k , γ 3 δ k ) , k = k + 1, → Step 1 η 1 ≤ ρ k < η 2 : U k + 1 = U + , g k + 1 = g k + s k , J ( g k + 1 ) = J ( g k + s k ) , δ k + 1 ∈ [ γ 2 δ k , δ k ) , k = k + 1, → Step 1 ρ k < η 1 : U k + 1 = U k , δ k + 1 ∈ [ γ 1 δ k , γ 2 δ k ) , k = k + 1, → Step 2 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  38. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region POD (TRPOD) Goal: Replace objective J ( g ) = J ( u ( g ) , g ) with cheaper objective m ( g ) := J (˜ u ( g ) , g ) during optimization – ˜ u SDPOD-based model Initialization: Choose 0 < η 1 < η 2 < 1, 0 < γ 1 < γ 2 < 1 < γ 3 , an initial trust-region radius δ 0 > 0 and an initial iterate g 0 . Compute snapshot set U 0 and J ( g 0 ) . Set k = 0 Step 1: Compute POD basis and POD-based model m k ( g k ) Step 2: Compute approximately: min m k ( g k + s k ) s.t. � s k � ≤ δ 0 Step 3: Compute snapshot set U + using g k + s k and set J ( g k ) − J ( g k + s k ) ρ k = m k ( g k ) − m k ( g k + s k ) Step 4: Update Trust-Region Radius ρ k ≥ η 2 : U k + 1 = U + , g k + 1 = g k + s k , J ( g k + 1 ) = J ( g k + s k ) , δ k + 1 ∈ [ δ k , γ 3 δ k ) , k = k + 1, → Step 1 η 1 ≤ ρ k < η 2 : U k + 1 = U + , g k + 1 = g k + s k , J ( g k + 1 ) = J ( g k + s k ) , δ k + 1 ∈ [ γ 2 δ k , δ k ) , k = k + 1, → Step 1 ρ k < η 1 : U k + 1 = U k , δ k + 1 ∈ [ γ 1 δ k , γ 2 δ k ) , k = k + 1, → Step 2 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  39. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region POD (TRPOD) Goal: Replace objective J ( g ) = J ( u ( g ) , g ) with cheaper objective m ( g ) := J (˜ u ( g ) , g ) during optimization – ˜ u SDPOD-based model Initialization: Choose 0 < η 1 < η 2 < 1, 0 < γ 1 < γ 2 < 1 < γ 3 , an initial trust-region radius δ 0 > 0 and an initial iterate g 0 . Compute snapshot set U 0 and J ( g 0 ) . Set k = 0 Step 1: Compute POD basis and POD-based model m k ( g k ) Step 2: Compute approximately: min m k ( g k + s k ) s.t. � s k � ≤ δ 0 Step 3: Compute snapshot set U + using g k + s k and set J ( g k ) − J ( g k + s k ) ρ k = m k ( g k ) − m k ( g k + s k ) Step 4: Update Trust-Region Radius ρ k ≥ η 2 : U k + 1 = U + , g k + 1 = g k + s k , J ( g k + 1 ) = J ( g k + s k ) , δ k + 1 ∈ [ δ k , γ 3 δ k ) , k = k + 1, → Step 1 η 1 ≤ ρ k < η 2 : U k + 1 = U + , g k + 1 = g k + s k , J ( g k + 1 ) = J ( g k + s k ) , δ k + 1 ∈ [ γ 2 δ k , δ k ) , k = k + 1, → Step 1 ρ k < η 1 : U k + 1 = U k , δ k + 1 ∈ [ γ 1 δ k , γ 2 δ k ) , k = k + 1, → Step 2 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  40. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 1 x 0 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  41. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 1 Radius: δ 0 x 0 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  42. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 1 Radius: δ 0 ρ ≥ η 2 : Accept x ∗ x 0 x ∗ Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  43. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 1 Radius: δ 0 ρ ≥ η 2 : Accept x ∗ Choose δ 1 ∈ ( δ 0 , γ 3 δ 0 ) x 0 x 1 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  44. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 2 Radius: δ 1 x 1 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  45. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 2 Radius: δ 1 η 1 ≤ ρ ≤ η 2 : Accept x ∗ x 1 x ∗ Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  46. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 2 Radius: δ 1 η 1 ≤ ρ ≤ η 2 : Accept x ∗ Choose δ 2 ∈ ( γ 2 δ 1 , δ 1 ] x 1 x 2 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  47. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 3 Radius: δ 2 x 2 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  48. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 3 Radius: δ 2 η 1 ≤ ρ ≤ η 2 : Accept x ∗ x 2 x ∗ Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  49. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 3 Radius: δ 2 η 1 ≤ ρ ≤ η 2 : Accept x ∗ Choose δ 3 ∈ ( γ 2 δ 2 , δ 2 ] x 2 x 3 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  50. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 4 Radius: δ 3 x 3 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  51. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 4 Radius: δ 3 ρ ≤ η 1 : Reject x ∗ x 3 x ∗ Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  52. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 4 Radius: δ 3 ρ ≤ η 1 : Reject x ∗ Choose δ 3 ∈ ( γ 1 δ 3 , γ 2 δ 3 ] x 3 x ∗ Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  53. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 4 Radius: δ 3 ρ ≤ η 1 : Reject x ∗ Choose δ 3 ∈ ( γ 1 δ 3 , γ 2 δ 3 ] x 3 x ∗ Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  54. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 4 Radius: δ 3 x 3 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  55. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 4 Radius: δ 3 η 1 ≤ ρ ≤ η 2 : Accept x ∗ x 3 x ∗ Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  56. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 4 Radius: δ 3 η 1 ≤ ρ ≤ η 2 : Accept x ∗ Choose δ 4 ∈ ( γ 2 δ 3 , δ 3 ] x 3 x 4 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  57. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 5 Radius: δ 4 x 4 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  58. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 5 Radius: δ 4 η 1 ≤ ρ ≤ η 2 : Accept x ∗ x 4 x ∗ Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  59. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 5 Radius: δ 4 η 1 ≤ ρ ≤ η 2 : Accept x ∗ Choose δ 5 ∈ ( γ 2 δ 4 , δ 4 ] x 4 x 5 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  60. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 6 Radius: δ 5 x 5 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  61. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 17 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  62. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region POD Algorithm Theorem (Fahl, 2000) Objective J continuously differentiable, bounded below and ∇J Lipschitz continuous on U ⊂ R . Model differentiable on open convex set containing trust-region, and n k = ∇ m k ( g k ) approximates ∇J ( g k ) , such that �∇J k ( g k ) − n k � ≤ ζ ∈ ( 0 , 1 ) . � n k � Then k →∞ � n k � = 0 . lim Gradient consistency condition: � n k � → 0 ⇒ �∇J ( g k ) � → 0 Consistency condition difficult for multiple mesh levels Reduction in model on one grid does not guarantee reduction of model on another grid Need extension to multilevel case (Gratton etal.) Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  63. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Recursive Multilevel Trust-Region Algorithm (RMTR) Goal: Replace objective J ( g ) = J ( u ( g ) , g ) with cheaper objective m i ( g ) := J ( ˜ u i ( g ) , g ) during optimization – ˜ u i POD-based model Initialization: Choose 0 < η 1 < η 2 < 1, 0 < γ 1 < γ 2 < 1 < γ 3 , an initial trust-region radius δ 0 > 0 and an initial iterate g i , 0 . Compute snapshot set U i , 0 and J ( g i , 0 ) . Set k = 0 Step 1: Compute POD basis and POD-based model m i , k ( g i , k ) Step 2: Compute approximately: min m i , k ( g i , k + s i , k ) s.t. � s i , k � ≤ δ 0 Step 3: Compute snapshot set U + using g i , k + s i , k and set J ( g i , k ) − J ( g i , k + s i , k ) ρ i , k = m i , k ( g i , k ) − m i , k ( g i , k + s i , k ) Step 4: If ρ i , k ≥ η 1 , then g i , k + 1 = g i , k + s i , k , otherwise keep g i , k Step 5: Trust region update as usual, new snapshot set on finer grid for g i , k + 1 . Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  64. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Convergence of Multilevel Trust-Region Procedure for Quadratic Case Theorem (Gratton, etal. 2005) If the algorithm RMTR is called at the highest level r with the gradient tolerance ǫ g r = 0 , then global first-order convergence can be shown, that is, k →∞ � n r , k � = 0 lim provided we have quadratic model functions. Convergence results for quadratic model functions Extension to non-quadratic models requires additional assumptions Rigorous adaption of RMTR to POD-based model functions in work Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  65. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 1 x 0 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  66. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 1 Radius: δ 0 x 0 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  67. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 1 Radius: δ 0 ρ ≥ η 2 : Accept x ∗ x 0 x ∗ Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  68. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 1 Radius: δ 0 ρ ≥ η 2 : Accept x ∗ Choose δ 1 ∈ ( δ 0 , γ 3 δ 0 ) x 0 x 1 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  69. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 2 Radius: δ 1 x 1 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  70. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 2 Radius: δ 1 η 1 ≤ ρ ≤ η 2 : Accept x ∗ x 1 x ∗ Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  71. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 2 Radius: δ 1 η 1 ≤ ρ ≤ η 2 : Accept x ∗ Choose δ 2 ∈ ( γ 2 δ 1 , δ 1 ] x 1 x 2 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  72. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 3 Radius: δ 2 x 2 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  73. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 3 Radius: δ 2 η 1 ≤ ρ ≤ η 2 : Accept x ∗ x 2 x ∗ Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  74. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 3 Radius: δ 2 η 1 ≤ ρ ≤ η 2 : Accept x ∗ Choose δ 3 ∈ ( γ 2 δ 2 , δ 2 ] x 2 x 3 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  75. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 4 Radius: δ 3 x 3 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  76. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 4 Radius: δ 3 ρ ≤ η 1 : Reject x ∗ x 3 x ∗ Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  77. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 4 Radius: δ 3 ρ ≤ η 1 : Reject x ∗ Choose δ 3 ∈ ( γ 1 δ 3 , γ 2 δ 3 ] x 3 x ∗ Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  78. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 4 Radius: δ 3 ρ ≤ η 1 : Reject x ∗ Choose δ 3 ∈ ( γ 1 δ 3 , γ 2 δ 3 ] x 3 x ∗ Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  79. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 4 Radius: δ 3 x 3 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  80. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 4 Radius: δ 3 η 1 ≤ ρ ≤ η 2 : Accept x ∗ x 3 x ∗ Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  81. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 4 Radius: δ 3 η 1 ≤ ρ ≤ η 2 : Accept x ∗ Choose δ 4 ∈ ( γ 2 δ 3 , δ 3 ] x 3 x 4 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  82. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 5 Radius: δ 4 x 4 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  83. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 5 Radius: δ 4 η 1 ≤ ρ ≤ η 2 : Accept x ∗ x 4 x ∗ Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  84. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 5 Radius: δ 4 η 1 ≤ ρ ≤ η 2 : Accept x ∗ Choose δ 5 ∈ ( γ 2 δ 4 , δ 4 ] x 4 x 5 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  85. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 6 Radius: δ 5 x 5 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  86. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region Procedure [0 < η 1 ≤ η 2 < 1 , 0 < γ 1 ≤ γ 2 < 1 ≤ γ 3 ] Iteration: 17 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  87. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region: Computational Complexity Iteration: 1 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  88. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region: Computational Complexity Iteration: 2 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

  89. The Standard POD Method Trust-Region Methods Streamline Diffusion POD Models Recursive Multilevel Trust-Region Methods Optimization with TRPOD Multilevel Optimization with TRPOD Conclusion Trust-Region Optimization Standard Trust-Region: Computational Complexity Iteration: 3 Bret Kragel and Ekkehard W. Sachs Proper Orthogonal Decomposition in Optimization

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