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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.

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Lecture 3: Goal-oriented Formulation

  • f 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

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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

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Introduction

Main Idea

Classical goal-oriented error control:

Primal Problem Dual Problem Estimate Errors and Adapt Mesh

One then hopes that discrete space is optimized with respect to QoI. Suggested approach: Consider a perturbed primal problem tailored for QoI optimization

Dual Problem Constrained Primal Problem Estimate Errors 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

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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 uh ∈ V h such that B(uh, vh) = F(vh), ∀vh ∈ V h If B = symmetric and coercive bilinear form, problem can be recast as a minimization problem: uh = argmin

vh ∈ V h

1 2B(vh, vh) − F(vh)

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-Oriented Formulation November 4-8, 2019 4 / 21

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Introduction

Review of literature

Chaudhry, Cyr, Liu, Manteuffel, Olson, and Tang, “Enhancing Least-Squares FEM through a QoI”, SINUM, 2014. Penalization approach: uh = argmin

vh ∈ V h

1 2B(vh, vh) − F(vh)

  • + β2

2

  • Q(vh) − Q(u)

2 leads to the problem: Find uh ∈ V h s.t. B(uh, vh) + β2Q(uh)Q(vh) = F(vh) + β2Q(u)Q(vh), ∀vh ∈ V h

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-Oriented Formulation November 4-8, 2019 5 / 21

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Introduction

Review of literature

Ben Dhia, Chamoin, Oden, Prudhomme, “A new adaptive modeling strategy based on

  • ptimal 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 m∗ = argmin

m

1 2

  • Q(u) − Q(¯

u(m)) 2 With constraint on ¯ u(m): Bm(¯ u; v) − F(v) = 0, ∀v ∈ V

f

P P

2 1

F

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-Oriented Formulation November 4-8, 2019 6 / 21

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New Goal-Oriented Formulation

Constrained Problem

Objective: Minimize energy functional such that relative error on QoI is below a given tolerance ǫ, wh = argmin

vh ∈ V h |Q(vh) − Q(u)| ≤ ǫ|Q(u)|

1 2B(vh, vh) − F(vh)

  • Inequality constraint ⇒ KKT (Karush-Kuhn-Tucker conditions)

Instead, we consider: wh = argmin

vh ∈ V h Q(vh) − Q(u) = 0

1 2B(vh, vh) − F(vh)

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-Oriented Formulation November 4-8, 2019 7 / 21

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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(x0) = 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

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New Goal-Oriented Formulation

Goal-oriented formulation: Constrained minimization

Objective: Given α ≈ Q(u), minimize energy functional subject to constraint on QoI, wh = argmin

vh ∈ V h Q(vh) = α

1 2B(vh, vh) − F(vh)

  • Straightforward extension to multiple QoI’s.

Lagrange formulation: Let vh ∈ V h and µ ∈ Rk, L(vh, µ) = 1 2B(vh, vh) − F(vh) + µ · (Q(vh) − 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

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New Goal-Oriented Formulation

Goal-oriented formulation: Constrained minimization

Adjoint problems: For each Qi, compute higher-order adjoint solutions ˜ pi ∈ ˜ V ⊃ V h B(˜ v, ˜ pi) = Qi(˜ v), ∀˜ v ∈ ˜ V Compute enhanced quantities of interest: αi = F(˜ pi) (= Qi(˜ u)) Constrained FE formulation: Find (wh, λ) ∈ Vh × Rk such that B(wh, vh) +

  • i

λiQi(vh) = F(vh), ∀vh ∈ Vh

  • i

µiQi(wh) =

  • i

µiαi, ∀µ ∈ Rk Theorem: The mixed problem is well-posed.

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-Oriented Formulation November 4-8, 2019 10 / 21

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New Goal-Oriented Formulation

Goal-oriented formulation: Constrained minimization

Theorem: (Relation between uh and wh) uh − wh =

k

  • i=1

λi pi,h ∈ V h Proof: B(wh, vh) +

  • i

λiQi(vh) = F(vh), ∀vh ∈ Vh B(wh, vh) +

  • i

λiB(vh, ˜ pi) = B(uh, vh), ∀vh ∈ Vh B(wh, vh) − B(uh, vh) +

  • i

λiB(pi,h, vh) = 0, ∀vh ∈ Vh B

  • uh − wh −
  • i

λi pi,h, vh

  • = 0,

∀vh ∈ Vh

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-Oriented Formulation November 4-8, 2019 11 / 21

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New Goal-Oriented Formulation

Goal-oriented formulation: Constrained minimization

Remark: Lagrange multipliers λi indicate how much wh moves away from global minimizer uh, i.e. what the “sacrifice” on energy is to satisfy constraints. Theorem: (Near-optimality in energy norm) u − whE ≤ Cu − uhE

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-Oriented Formulation November 4-8, 2019 12 / 21

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Error Estimation and Adaptation

Error estimation for constrained approach

Error in energy norm: E2

h = u − wh2 E =

Discretization error

  • u − uh2

E +

Error due to constraint

  • uh − wh2

E

= Rh(uh; u − uh) +

k

  • i=1

λipi,h2

E

⇒ Additional error term scales with λ2 Error in QoI: Ei =

  • Discretization error
  • Qi(u − uh) +

Error due to constraint

  • Qi(uh − wh)
  • =
  • Rh(uh; pi − pi,h) +

k

  • j=1

λjB(pj,h, pi,h)

  • ⇒ Additional

error term scales with λ

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-Oriented Formulation November 4-8, 2019 13 / 21

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Error Estimation and Adaptation

Algorithm

  • 1. For each QoI, solve higher-order adjoint problem for ˜

pi ∈ ˜ V

  • 2. For each QoI, compute enhanced value αi = F(˜

pi)

  • 3. Solve constrained primal problem using Qi(wh) = αi as

constraints

  • 4. Estimate errors
  • 5. Adapt mesh
  • 6. Iterate
  • S. Prudhomme (Polytechnique Montr´

eal) Goal-Oriented Formulation November 4-8, 2019 14 / 21

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Numerical Examples

Numerical example: Poisson problem

−∆u = 1, in Ω = (0, 1)2 u = 0,

  • n ∂Ω

Q1(u) = 1 |ω1|

  • ω1

udx Q2(u) = 1 |ω2|

  • ω2

uxdx

Computational domain Ω Adjoint solution for Q2

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-Oriented Formulation November 4-8, 2019 15 / 21

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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

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Numerical Examples

Numerical example: Poisson problem with singularities

−∇ · (a∇u) = 1, in Ω u = 0,

  • n ∂Ω

where a = piecewise constant (“double L-shaped problem”).

ω1 ω2 a = 100 a = 100 a = 1

Computational domain Ω

1 0.005 1 0.01

u

0.015

y

0.5

x

0.5 0.02

Primal solution

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-Oriented Formulation November 4-8, 2019 17 / 21

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Numerical Examples

Adjoint solutions

1 0.05 1 0.1

p1

0.15

y

0.5

x

0.5 0.2

Q1(u) = 1 |ω1|

  • ω1

u dΩ

  • 1

1

  • 0.5

1

p2

0.5

y

0.5

x

0.5 1

Q2(u) = 1 |ω2|

  • ω2

ux dΩ

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-Oriented Formulation November 4-8, 2019 18 / 21

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Numerical Examples

Meshes after 20 adaptive steps

uh wh 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

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Numerical Examples

Convergence analysis

10 1 10 2 10 3 10 4 10 5

# dofs

10 -2 10 -1 10 0

Eh −1/2 −1/3

Adapt uh in norm Adapt uh in QoI Adapt wh in norm Adapt wh in QoI

Error in energy norm

10 1 10 2 10 3 10 4 10 5

# dofs

10 -8 10 -6 10 -4 10 -2 10 0

E1 −2 −1

Adapt uh in norm Adapt uh in QoI Adapt wh in norm Adapt wh in QoI

Error in QoI #1

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-Oriented Formulation November 4-8, 2019 20 / 21

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Conclusions

Concluding Remarks

Reformulation of the problem to directly take into account QoI’s. Extension to multiple Quantities of Interest (Multi-objective

  • ptimization).

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