Lecture 4: Adaptive Construction of PGD reduced-order models with - PowerPoint PPT Presentation

Lecture 4: Adaptive Construction of PGD reduced-order models with respect to Quantities of Interest Serge Prudhomme D´ epartement de math´ ematiques et de g´ enie industriel Polytechnique Montr´ eal DCSE Fall School 2019 TU Delft, The Netherlands, November 4-8, 2019 S. Prudhomme (Polytechnique Montr´ eal) PGD reduced-order models November 4-8, 2019 1 / 16

Motivation Motivation: EIT problem for composite materials S. Prudhomme (Polytechnique Montr´ eal) PGD reduced-order models November 4-8, 2019 2 / 16

Outline Outline Introduction PGD Approximations Goal-oriented formulation for PGD Approximations Perspectives and Conclusions S. Prudhomme (Polytechnique Montr´ eal) PGD reduced-order models November 4-8, 2019 3 / 16

Model order reduction Reduced-order models/Surrogate models Model order reduction (MOR) is a technique for reducing the computational complexity of mathematical models in numerical simulations: ∞ N � � u ( t, x ) = ψ i ( t ) φ i ( x ) ≈ ψ i ( t ) φ i ( x ) = u N ( x, t ) i =1 i =1 Proper Orthogonal Decomposition methods (PCA, etc.) Reduced-basis methods: Peraire, Patera, Maday, Rozza, etc. Proper Generalized Decomposition methods (low-rank approx.): Ladev` eze, Nouy, Chinesta, Mattis, Le Maˆ ıtre, etc. F. Chinesta, R. Keunings, A. Leygue, The Proper Generalized Decomposition for Advanced Numerical Simulations, Springer International Publishing, 2014. S. Prudhomme (Polytechnique Montr´ eal) PGD reduced-order models November 4-8, 2019 4 / 16

PGD Approximations Proper Generalized Decomposition method (PGD) Model problem: Find u ∈ V such that B ( u, v ) = F ( v ) , ∀ v ∈ V with B ( u, v ) symmetric and positive-definite bilinear form. � 1 � u = argmin J ( v ) = argmin 2 B ( v, v ) − F ( v ) v ∈ V v ∈ V Separated representation: Find an approximation u m of u in the form m � u ( x, θ ) ≈ u m ( x, θ ) = ψ i ( x ) φ i ( θ ) , ∀ ( x, θ ) ∈ D × Ω i =1 S. Prudhomme (Polytechnique Montr´ eal) PGD reduced-order models November 4-8, 2019 5 / 16

Goal-oriented PGD formulation Proper Generalized Decomposition method (PGD) Progressive (iterative) approach: Given u m − 1 , find next mode ψφ such that u m is a better approximation to u : u m ( x, θ ) = u m − 1 ( x, θ ) + ψ ( x ) φ ( θ ) Optimization problem: u m = argmin J ( u m − 1 + ψφ ) ψφ J ( u m − 1 + ψφ + ǫδ ( ψφ )) − J ( u m − 1 + ψφ ) J ′ ( u m − 1 + ψφ ; δ ( ψφ )) = lim ǫ ǫ → 0 = B ( u m − 1 + ψφ, δ ( ψφ )) − F ( δ ( ψφ )) � � = B ( ψφ, δ ( ψφ )) − F ( δ ( ψφ )) − B ( u m − 1 , δ ( ψφ )) � �� � R ( δ ( ψφ )) S. Prudhomme (Polytechnique Montr´ eal) PGD reduced-order models November 4-8, 2019 6 / 16

Goal-oriented PGD formulation Proper Generalized Decomposition method (PGD) Note that the variation δ ( ψφ ) is given by δ ( ψφ ) = ψ ( δφ ) + ( δψ ) φ := ψφ ∗ + ψ ∗ φ Nonlinear weak form for PGD: B ( ψφ, ψφ ∗ + ψ ∗ φ ) = R ( ψφ ∗ + ψ ∗ φ ) , ∀ ψ ∗ , ∀ φ ∗ or ∀ ψ ∗ = ψ ∗ ( x ) B ( ψφ, ψ ∗ φ ) = R ( ψ ∗ φ ) , ∀ φ ∗ = φ ∗ ( θ ) B ( ψφ, ψφ ∗ ) = R ( ψφ ∗ ) , Use Alternated Directions scheme: φ (0) → ψ (1) → φ (1) → ψ (2) → φ (2) . . . S. Prudhomme (Polytechnique Montr´ eal) PGD reduced-order models November 4-8, 2019 7 / 16

Goal-oriented PGD formulation PGD method with constraint Nonlinear mixed-weak PGD formulation: B ( ψφ, ψ ∗ φ ) + λ · Q ( ψ ∗ φ ) = R ( ψ ∗ φ ) , ∀ ψ ∗ B ( ψφ, ψφ ∗ ) + λ · Q ( ψφ ∗ ) = R ( ψφ ∗ ) , ∀ φ ∗ ∀ µ ∈ R k µ · Q ( ψφ ) = µ · ( Q ( u ) − Q ( u m − 1 )) , Issue: Separation of variables decouples the dimensions of the problem while constraints should be applied globally. Iterative approach ( Uzawa or Augmented Lagrangian ): Given λ i − 1 , solve for ψ i and φ i using Alternated directions. Update the Lagrange multiplier λ i . Repeat until convergence. Kergrene, Prudhomme, Chamoin, and Laforest, “Approximation of constrained problems using the PGD method with application to pure Neumann problems”, CMAME, 2017. S. Prudhomme (Polytechnique Montr´ eal) PGD reduced-order models November 4-8, 2019 8 / 16

Goal-oriented PGD formulation Example: Bar model � E 1 , x ∈ (0 , L/ 2) , N 1 ( x ) N 2 ( x ) E ( x ) = E 2 , x ∈ ( L/ 2 , L ) , � � 1 E 1 E 2 F = 1 Q 1 ( u ) = u ( L/ 2) , | D | D 1 D 2 x = 0 x = L/ 2 x = L � � 1 Q 2 ( u ) = u ( L ) , | D | D 1 D 2 L L u ( x, E 1 , E 2 ) = ( N 1 ( x ) + N 2 ( x )) + N 2 ( x ) , 2 E 1 2 E 2 m � u m ( x, E 1 , E 2 ) = ϕ i ( x ) φ 1 i ( E 1 ) φ 2 i ( E 2 ) i =1 S. Prudhomme (Polytechnique Montr´ eal) PGD reduced-order models November 4-8, 2019 9 / 16

Goal-oriented PGD formulation Example: Bar model 3 0.65 0.6 Classical 0.6 Penalization β = 10 2 0.55 2.5 0.55 Penalization β = 10 5 0.5 0.5 Uzawa 2 0.45 0.45 φ 1 φ 2 ϕ 1.5 0.4 0.4 0.35 0.35 1 0.3 0.3 0.25 0.5 0.25 0.2 0 0.15 0.2 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 E 1 E 2 x 0.6 0.6 1.4 0.4 1.2 0.4 0.2 1 0.2 0 0.8 φ 1 -0.2 φ 2 ϕ 0 0.6 -0.4 0.4 -0.2 -0.6 0.2 -0.8 -0.4 0 -1 -0.6 -1.2 -0.2 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 E 1 E 2 x 0.7 1.5 0.6 0.4 0.6 1 0.2 0.5 0.5 0 0.4 -0.2 0 0.3 φ 1 φ 2 ϕ -0.4 0.2 -0.5 -0.6 0.1 -0.8 -1 0 -1 -1.5 -0.1 -1.2 -0.2 -2 -1.4 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x E 1 E 2 First three PGD: ϕ ( x ) (left), φ 1 ( E 1 ) , φ 2 ( E 2 ) . S. Prudhomme (Polytechnique Montr´ eal) PGD reduced-order models November 4-8, 2019 10 / 16

Goal-oriented PGD formulation EIT Example Quantities of interest: � 1 1 1 � � � � Q 1 ( u ) = u − u | D | | Γ 2 | | Γ 3 | D Γ 2 Γ 3 � 1 Γ 1 Γ 2 Γ 3 Γ 4 1 1 � � � � Q 2 ( u ) = u − u σ = 1 | D | | Γ 2 | | Γ 5 | D Γ 2 Γ 5 � 1 1 1 � � � � Q 3 ( u ) = u − u σ = σ b | D | | Γ 2 | | Γ 6 | D Γ 2 Γ 6 σ = 1 where D = D a × D b with: σ = σ a σ a ∈ D a = [1 , 10] σ b ∈ D b = [0 . 1 , 1] σ = 1 Γ 5 Γ 6 Separation of variables: m � u m ( x, y, σ a , σ b ) = f i ( x, y ) g i ( σ a ) h i ( σ b ) i =1 S. Prudhomme (Polytechnique Montr´ eal) PGD reduced-order models November 4-8, 2019 11 / 16

Goal-oriented PGD formulation EIT Example 10 -1 0.22 Classical PGD Classical PGD Goal-Oriented PGD Goal-Oriented PGD 0.21 10 -2 0.2 0.19 ǫ 1 ǫ 10 -3 0.18 0.17 10 -4 0.16 0.15 10 -5 0.14 0 10 20 30 40 50 60 0 10 20 30 40 50 60 m m (a) Error in the energy norm, (b) Error in Q 1 , with respect to # of modes S. Prudhomme (Polytechnique Montr´ eal) PGD reduced-order models November 4-8, 2019 12 / 16

Goal-oriented PGD formulation EIT Example 10 -1 10 -1 Classical PGD Classical PGD Goal-Oriented PGD Goal-Oriented PGD 10 -2 10 -2 ǫ 3 ǫ 2 10 -3 10 -3 10 -4 10 -4 10 -5 10 -5 0 10 20 30 40 50 60 0 10 20 30 40 50 60 m m (c) Error in Q 2 , (d) Error in Q 3 , with respect to # of modes S. Prudhomme (Polytechnique Montr´ eal) PGD reduced-order models November 4-8, 2019 13 / 16

Goal-oriented PGD formulation EIT Example Model Problem: Γ 1 Γ 2 Γ 3 Γ 4 σ = 1 −∇ · ( σ ∇ u ) = 0 , in Ω , σ = σ b n · σ ∇ u = g, on ∂ Ω . σ = 1 This is a 5D problem: σ = σ a 2 space variables ( x, y ) σ = 1 Diffusivities σ a and σ b Γ 5 Γ 6 Position x 1 of electrode Γ 1 Input/Output: Load g corresponds to the difference of potential between Γ 1 and Γ 4 . QoI’s are 3 differences of potential between other pairs of electrodes. S. Prudhomme (Polytechnique Montr´ eal) PGD reduced-order models November 4-8, 2019 14 / 16

Goal-oriented PGD formulation Adapted meshes and error convergence From top to bottom: 2d mesh in x and y , 1d meshes in σ a and σ b , 1d mesh in x 1 S. Prudhomme (Polytechnique Montr´ eal) PGD reduced-order models November 4-8, 2019 15 / 16

Conclusions Concluding Remarks Reformulation of the problem to directly take into account QoI’s. Extension to multiple QoI’s (Multi-objective optimization). Extension to other ROM methods to optimize modes wrt QoI’s. Adaptivity both in number of modes m and in mesh size h . Further research work: Extension to non-linear problems and quantities of interest. Development of robust error estimators S. Prudhomme (Polytechnique Montr´ eal) PGD reduced-order models November 4-8, 2019 16 / 16