Lecture 3: Goal-oriented Formulation of Boundary-value Problems - PowerPoint PPT Presentation

Lecture 3: Goal-oriented Formulation of Boundary-value Problems 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) Goal-Oriented Formulation November 4-8, 2019 1 / 21

Outline Outline Introduction Goal-Oriented Formulation for FE approximations Error Estimation and Adaptive Scheme Numerical Example Conclusions S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 2 / 21

Introduction Main Idea Classical goal-oriented error control: Primal Problem Estimate Errors and Adapt Mesh Dual Problem One then hopes that discrete space is optimized with respect to QoI. Suggested approach: Consider a perturbed primal problem tailored for QoI optimization Estimate Errors Constrained Dual Problem Primal Problem and Adapt Mesh Kergrene, Prudhomme, Chamoin, and Laforest, “A New Goal-Oriented Formulation of the Finite Element Method”, CMAME , 2017. S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 3 / 21

Introduction Notation and model problem Consider the model problem: Find u ∈ V such that B ( u, v ) = F ( v ) , ∀ v ∈ V Finite Element approximation: Let V h ⊂ V be a FE subspace of V , then Find u h ∈ V h such that B ( u h , v h ) = F ( v h ) , ∀ v h ∈ V h If B = symmetric and coercive bilinear form, problem can be recast as a minimization problem: � 1 � u h = argmin 2 B ( v h , v h ) − F ( v h ) v h ∈ V h S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 4 / 21

Introduction Review of literature Chaudhry, Cyr, Liu, Manteuffel, Olson, and Tang, “Enhancing Least-Squares FEM through a QoI”, SINUM , 2014. Penalization approach: � 2 � 1 � + β 2 � u h = argmin 2 B ( v h , v h ) − F ( v h ) Q ( v h ) − Q ( u ) 2 v h ∈ V h leads to the problem: Find u h ∈ V h s.t. B ( u h , v h ) + β 2 Q ( u h ) Q ( v h ) = F ( v h ) + β 2 Q ( u ) Q ( v h ) , ∀ v h ∈ V h S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 5 / 21

Introduction Review of literature Ben Dhia, Chamoin, Oden, Prudhomme, “A new adaptive modeling strategy based on optimal control for atomic-to-continuum coupling simulations”, CMAME , 2011. Optimal control problem: Find the set of “discretization” parameters m ∗ that minimizes the error in QoI Q � 2 With constraint on ¯ u ( m ) : 1 � m ∗ = argmin Q ( u ) − Q (¯ u ( m )) 2 B m (¯ u ; v ) − F ( v ) = 0 , ∀ v ∈ V m f F P 1 P 2 S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 6 / 21

New Goal-Oriented Formulation Constrained Problem Objective: Minimize energy functional such that relative error on QoI is below a given tolerance ǫ , � 1 � w h = argmin 2 B ( v h , v h ) − F ( v h ) v h ∈ V h | Q ( v h ) − Q ( u ) | ≤ ǫ | Q ( u ) | Inequality constraint ⇒ KKT (Karush-Kuhn-Tucker conditions) Instead, we consider: � 1 � w h = argmin 2 B ( v h , v h ) − F ( v h ) v h ∈ V h Q ( v h ) − Q ( u ) = 0 S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 7 / 21

New Goal-Oriented Formulation Illustrative example 1D Problem: − u ′′ + αu = f, in (0 , 1) u (0) = 0 u (1) = 0 Quantity of interest: Q ( u ) = u ( x 0 ) = u (5 / 8) Manufactured solution: u ( x ) = x (1 − x ) f ( x ) = 2 + αx (1 − x ) Parameters: α = 0 . 1 , ǫ = 0 . 05 and 0 . 01 S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 8 / 21

New Goal-Oriented Formulation Goal-oriented formulation: Constrained minimization Objective: Given α ≈ Q ( u ) , minimize energy functional subject to constraint on QoI, � 1 � w h = argmin 2 B ( v h , v h ) − F ( v h ) v h ∈ V h Q ( v h ) = α Straightforward extension to multiple QoI’s. Lagrange formulation: Let v h ∈ V h and µ ∈ R k , L ( v h , µ ) = 1 2 B ( v h , v h ) − F ( v h ) + µ · ( Q ( v h ) − α ) Vector α is user-specified and should reflect the QoI’s . . . S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 9 / 21

New Goal-Oriented Formulation Goal-oriented formulation: Constrained minimization Adjoint problems: p i ∈ ˜ V ⊃ V h For each Q i , compute higher-order adjoint solutions ˜ v ∈ ˜ B (˜ v, ˜ p i ) = Q i (˜ v ) , ∀ ˜ V Compute enhanced quantities of interest: α i = F (˜ p i ) (= Q i (˜ u )) Constrained FE formulation: Find ( w h , λ ) ∈ V h × R k such that � B ( w h , v h ) + λ i Q i ( v h ) = F ( v h ) , ∀ v h ∈ V h i � � ∀ µ ∈ R k µ i Q i ( w h ) = µ i α i , i i Theorem: The mixed problem is well-posed. S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 10 / 21

New Goal-Oriented Formulation Goal-oriented formulation: Constrained minimization Theorem: (Relation between u h and w h ) k � λ i p i,h ∈ V h u h − w h = i =1 Proof: � B ( w h , v h ) + λ i Q i ( v h ) = F ( v h ) , ∀ v h ∈ V h i � B ( w h , v h ) + λ i B ( v h , ˜ p i ) = B ( u h , v h ) , ∀ v h ∈ V h i � B ( w h , v h ) − B ( u h , v h ) + λ i B ( p i,h , v h ) = 0 , ∀ v h ∈ V h i � � � B u h − w h − λ i p i,h , v h = 0 , ∀ v h ∈ V h i S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 11 / 21

New Goal-Oriented Formulation Goal-oriented formulation: Constrained minimization Remark: Lagrange multipliers λ i indicate how much w h moves away from global minimizer u h , i.e. what the “sacrifice” on energy is to satisfy constraints. Theorem: (Near-optimality in energy norm) � u − w h � E ≤ C � u − u h � E S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 12 / 21

Error Estimation and Adaptation Error estimation for constrained approach Error in energy norm: Discretization error Error due to constraint � �� � � �� � ⇒ Additional E 2 h = � u − w h � 2 � u − u h � 2 � u h − w h � 2 E = E + E error term k scales with � λ � 2 � λ i p i,h � 2 = R h ( u h ; u − u h ) + � E i =1 Error in QoI: Discretization error Error due to constraint � � � �� � � �� � ⇒ Additional � � E i = Q i ( u − u h ) + Q i ( u h − w h ) � � error term k scales with � λ � � � � � R h ( u h ; p i − p i,h ) + � � = λ j B ( p j,h , p i,h ) � j =1 S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 13 / 21

Error Estimation and Adaptation Algorithm p i ∈ ˜ 1. For each QoI, solve higher-order adjoint problem for ˜ V 2. For each QoI, compute enhanced value α i = F (˜ p i ) 3. Solve constrained primal problem using Q i ( w h ) = α i as constraints 4. Estimate errors 5. Adapt mesh 6. Iterate S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 14 / 21

Numerical Examples Numerical example: Poisson problem � 1 Q 1 ( u ) = udx | ω 1 | in Ω = (0 , 1) 2 ω 1 − ∆ u = 1 , � 1 u = 0 , on ∂ Ω Q 2 ( u ) = u x dx | ω 2 | ω 2 Computational domain Ω Adjoint solution for Q 2 S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 15 / 21

Numerical Examples Numerical example: Poisson problem Sequence of uniform refinements: Exact errors (left) and effectivity indices (right) as functions of h − 1 . S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 16 / 21

Numerical Examples Numerical example: Poisson problem with singularities −∇ · ( a ∇ u ) = 1 , in Ω u = 0 , on ∂ Ω where a = piecewise constant (“double L-shaped problem”). a = 100 0.02 0.015 u a = 1 0.01 ω 1 0.005 ω 2 0 a = 100 1 1 0.5 0.5 y 0 0 x Computational domain Ω Primal solution S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 17 / 21

Numerical Examples Adjoint solutions 0.2 1 0.15 0.5 p 1 p 2 0.1 0 0.05 -0.5 0 -1 1 1 1 1 0.5 0.5 0.5 0.5 y 0 0 y 0 0 x x � � 1 1 Q 1 ( u ) = u d Ω Q 2 ( u ) = u x d Ω | ω 1 | | ω 2 | ω 1 ω 2 S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 18 / 21

Numerical Examples Meshes after 20 adaptive steps u h w h Adapt w.r.t. energy norm Adapt w.r.t. two QoIs S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 19 / 21

Numerical Examples Convergence analysis 10 0 10 0 Adapt u h in norm Adapt u h in norm Adapt u h in QoI Adapt u h in QoI Adapt w h in norm Adapt w h in norm Adapt w h in QoI Adapt w h in QoI 10 -2 − 1 E h E 1 − 1 / 3 10 -1 10 -4 − 2 10 -6 − 1 / 2 10 -2 10 -8 10 1 10 2 10 3 10 4 10 5 10 1 10 2 10 3 10 4 10 5 # dofs # dofs Error in energy norm Error in QoI #1 S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 20 / 21

Conclusions Concluding Remarks Reformulation of the problem to directly take into account QoI’s. Extension to multiple Quantities of Interest (Multi-objective optimization). Extension to PGD-type model reduction methods to optimize modes with respect to QoI’s. S. Prudhomme (Polytechnique Montr´ eal) Goal-Oriented Formulation November 4-8, 2019 21 / 21