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Lecture 1 A Review of Goal-Oriented Error Estimation and Adaptive Methods Serge Prudhomme D epartement de math ematiques et de g enie industriel Polytechnique Montr eal DCSE Fall School 2019 TU Delft, The Netherlands, November

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Lecture 1 A Review of Goal-Oriented Error Estimation and Adaptive Methods

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 Error Estimation November 4-8, 2019 1 / 38

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Plan of Lectures

Plan of Lectures

Lecture 1: A Review of Goal-Oriented Error Estimation and Adaptive Methods Lecture 2: Adaptive Methods for Problems with Uncertain Coefficients and Bayesian Inference Lecture 3: Goal-oriented formulation of boundary-value problems Lecture 4: Adaptive Construction of PGD reduced-order models with respect to QoI’s

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 2 / 38

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Outline

Outline

1) Introduction 2) Estimation and control of discretization errors in quantities of interest Linear problems (adjoint and error representation). Extension to nonlinear problems. Extension to time-dependent problems. Multiphysics coupled problems: Micro-fluidics application. 3) Conclusions

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 3 / 38

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Introduction

Introduction

Error estimation is useful for two purposes:

  • 1. To provide a measure of the accuracy in approximations.
  • 2. To control errors in those approximations (for example via mesh

adaptation in the case of finite element solutions). Error estimates have been constructed with respect to: a) Norms associated with the solution function spaces. b) Quantities of interest ⇒ goal-oriented error estimation.

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 4 / 38

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Introduction

Flow around obstacle (Stokes flow)

Top plots: Residual-based Error Estimation and adaptivity Bottom plots: Goal-Oriented Error Estimation and adaptivity QoI = averaged vorticity in lower left corner.

3 4 5 6 1 2 3 3 4 5 6 1 2 3 3 4 5 6 1 2 3 3 4 5 6 1 2 3

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 5 / 38

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Introduction

Catenary (linearized) model: −Tu′′ = f, in Ω = (0, 1) u = u0, at x = 0 u = u1, at x = 1 Exact solution: u(x) = f 2T x(1 − x) + u0(1 − x) + u1x Quantity of interest: Deflection at center point: Q(u) = u(0.5) = u0 + u1 2

  • + f

8T Average deflection: Q(u) = 1 u(x)dx = u0 + u1 2

  • +

f 12T Slope at origin: Q(u) = u′(0) =

  • u1 − u0
  • + f

2T

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 6 / 38

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Introduction

Green’s function

Suppose here that u0 = u1 = 0. Strong form of problem: −Tu′′ = f, in Ω = (0, 1) u = 0, at x = 0 u = 0, at x = 1 Weak formulation: Given f ∈ L2(Ω), Find u ∈ V = H1

0(Ω) s.t.

B(u, v) = F(v) ∀v ∈ V where          B(u, v) = 1 Tu′v′dx F(v) = 1 fv dx

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 7 / 38

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Introduction

Green’s function

Let Q(u) = u(x0). Note that: Q(u) = u(x0) = 1 u(x)δ(x − x0)dx The Green function is the function G0 = G0(x) ∈ V that yields: Q(u) = u(x0) = 1 fG0 dx = F(G0) that is: Q(u) = F(G0) = B(u, G0)

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 8 / 38

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Introduction

Green’s function

Q(u) = B(u, G0) = F(G0) Primal problem: −Tu′′ = f in (0, 1) u = 0 at x = 0, 1 Weak form: Given f ∈ L2(Ω), find u ∈ V s.t. B(u, v) = F(v) ∀v ∈ V Provide QoI: Q(u) = 1 u(x)δ(x − x0)dx Adjoint (dual) problem: Find G0 ∈ V s.t. B(v, G0) = Q(v) ∀v ∈ V Strong form: 1 Tv′G′

0dx =

1 vδdx so that: −TG′′

0 = δ

in (0, 1) G0 = 0 at x = 0, 1

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 9 / 38

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Introduction

Green’s function

Adjoint problem: −TG′′

0 = δ

in (0, 1) G0 = 0 at x = 0, 1 Analytical solution: G0(x) =      (1 − x0)x 2T , 0 ≤ x ≤ x0 x0(1 − x) 2T , x0 ≤ x ≤ 1

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 10 / 38

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Adjoints and QoIs

Generalized Green Function

Abstract linear BVP: Find u ∈ V, B(u, v) = F(v), ∀v ∈ V Quantity of interest: Q(u) =

  • Ω

u(x)k(x)dx Adjoint (dual) problem: Find p ∈ V, B(v, p) = Q(v), ∀v ∈ V FE approximation: Let V h ⊂ V Find uh ∈ V h, B(uh, vh) = F(vh), ∀vh ∈ V h Error equation: Find e ∈ V, B(e, vh) = R(uh; v) ≡ F(v) − B(uh, v), ∀v ∈ V

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 11 / 38

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Error Representation

Error Representation

Goal is to estimate E = Q(u) − Q(uh) E = B(u, p) − B(uh, p) (From adjoint problem) = F(p) − B(uh, p) (From primal problem) = R(uh; p) (From definition of residual) = R(uh; p − ph) (From orthogonality property) E = Q(u − uh) = Q(e) (From linearity of Q) = B(e, p) (From adjoint problem) = B(e, p − ph) (From orthogonality property) = R(uh; p − ph) (From error equation)

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 12 / 38

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Adaptive strategies

Adaptive strategies

Let ˜ p be a higher-order approximation of the adjoint solution on same mesh as uh, i.e. Find ˜ p ∈ V h,p+1, B(v, ˜ p) = Q(v), ∀v ∈ V h,p+1 Approach 1: Using only ˜ p and ˜ ph = Πh,p˜ p E ≈ η = R(uh; ˜ p − ˜ ph) =

  • K

RK(uh; ˜ p − ˜ ph) Approach 2: Using ˜ p, but also ˜ u ∈ V h,p+1, E ≈ η = B(˜ u − uh, ˜ p − ˜ ph) =

  • K

BK(˜ u − uh, ˜ p − ˜ ph)

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 13 / 38

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Adaptive strategies

Adaptive Strategies

Adaptive scheme (catenary exemple): η ≈ R(uh; ˜ p − ˜ ph) = F(˜ p − ˜ ph) − B(uh, ˜ p − ˜ ph) = 1 f (˜ p − ˜ ph) dx − 1 Tu′

h (˜

p − ˜ ph)′ dx =

Ne

  • K=1

K

f (˜ p − ˜ ph) + Tu′′

h (˜

p − ˜ ph) dx −

  • ∂K

Tu′

h(˜

p − ˜ ph) ds

  • =

Ne

  • K=1

K

(f + Tu′′

h )(˜

p − ˜ ph) dx −

  • i=1,2

1 2[ [Tu′

h]

](˜ p − ˜ ph)|xK

i

  • ηK

Refinement criterion: If |ηK| maxK |ηK| ≥ γtol, then refine element K

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 14 / 38

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Adaptive strategies

Example: Elliptic problem

0.00 0.25 0.50 0.75 1.00

X

0.00 0.25 0.50 0.75 1.00

Y

0.00 0.25 0.50 0.75 1.00

X

0.00 0.25 0.50 0.75 1.00

Y

Optimal?

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 15 / 38

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Nonlinear Problems

Nonlinear Problems

Abstract nonlinear problem: Find u ∈ V, B(u; v) = F(v), ∀ v ∈ V where: V = Banach space B(·; ·) = differentiable semilinear form. Q(u) = a possibly nonlinear differentiable functional on V.

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 16 / 38

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Nonlinear Problems

Nonlinear Problems

Linearization (Taylor with exact remainder): B(u + w; v) = B(u; v) + B′(u; w, v) + ∆B(u, w, v) Q(u + w) = Q(u) + Q′(u; w) + ∆Q(u, w) where B′(u; v, z) = lim

θ→0

1 θ[B(u + θv; z) − B(u; z)] Q′(u; v) = lim

θ→0

1 θ[Q(u + θv) − Q(u)] ∆B(u, w, v) = 1 B′′(u + sw; w, w, v)(1 − s)ds ∆Q(q, w) = 1 Q′′(u + sw; w, w)(1 − s)ds

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 17 / 38

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Nonlinear Problems

Secant Form (exact) Representation

B(u; v) = B(uh; v) + 1 B′(su + (1 − s)uh; e, v)ds

  • ≡Bs(u,uh;e,v)=secant form of B

, ∀v ∈ V Then Q(u) − Q(uh) = Q(e) = Bs(u, uh; e, p) = B(u; p) − B(uh; p) = F(p) − B(uh; p) = R(uh; p) where p is the solution of the dual problem: Find p ∈ V such that Bs(u, uh; v, p) = Q(v), ∀v ∈ V

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 18 / 38

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Nonlinear Problems

Linearization approach: Error Equation, Adjoint problem

Error Equation: Let e = u − uh B(u; v) = B(uh + e; v) = B(uh; v) + B′(uh; e, v) + ∆B(uh, e, v) = F(v) So Find e ∈ V, B′(uh; e, v) + ∆B(uh, e, v) = R(uh; v), ∀v ∈ V Dropping higher-order terms, error can be approximated by: Find ˆ e ∈ V, B′(uh; ˆ e, v) = R(uh; v), ∀v ∈ V Adjoint problem: Find p ∈ V such that B′(uh; v, p) = Q′(uh; v), ∀v ∈ V

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 19 / 38

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Nonlinear Problems

Linearization approach

Representation of the error: E = Q(u) − Q(uh) = Q′(uh; e) + ∆Q(uh, e) = B′(uh; e, p) + ∆Q(uh, e) = R(uh; p) + ∆Q(uh, e) − ∆B(uh, e, p) = R(uh; p − ph)

  • Discretization error

+ ∆Q(uh, e) − ∆B(uh, e, p)

  • Linearization error

with ∆B(uh, e, p) = 1 B′′(uh + se; e, e, p)(1 − s)ds ∆Q(uh, e) = 1 Q′′(uh + se; e, e)(1 − s)ds

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 20 / 38

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Nonlinear Problems

Tumor Growth Model (Cahn-Hilliard)

Mixture theory: diffuse-interface phase-field model The tumor cell concentration u satisfies ut = ∆µ(u) + g(σ, u) in Ω µ(u) = f′(u) − ǫ2∆u in Ω ∂nu = ∂nµ = 0

  • n ∂Ω

u(0) = u0 in Ω Nonlinear free energy density f(u) drives phase separation f(u) = 1

4(u2 − 1)2

−1 1 1

Lowengrub et al., NL (2010), Cristini et al., JMB (2009), Oden et al, M3AS (2010)

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 21 / 38

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Nonlinear Problems

Model is sensitive!

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 22 / 38

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Nonlinear Problems

Model is sensitive!

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 23 / 38

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Nonlinear Problems

Theorem (Extension of C´ ea’s Lemma to Nonlinear Problems)

Assume: B is differentiable. u is a nonsingular solution. Residual is Lipschitz continuous with constant cL Linearized problem is well posed with discrete inf-sup constant γB′

h(u)

u − uhV ≤

  • 1 + cB′ (u)

γB′

h(u)

  • inf

φ∈V h u − φV +

cL 2γB′

h(u)u − uh2

V

where cB′ (u) is the continuity constant for B′(u; ·, ·) Corollary: if u − uhV ≤ γB′

h(u)/cL

u − uhV ≤ 2

  • 1 + cB′ (u)

γB′

h(u)

  • inf

φ∈V h u − φV

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 24 / 38

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Nonlinear Problems

Burgers’ problem Let Ω = (−1, 1). Function u(x) = − tanh(x/2µ) is exact solution of: µuxx + uu′ = 0, x ∈ Ω u = u− at x = −1 u = u+ at x = +1 Find u ∈ V such that B(u; v) = F(v), ∀v ∈ V Find z ∈ V such that B′(u; v, z) = Q′(u; v), ∀v ∈ V where B(u; v) =

  • Ω

µu′v′ − 1 2u2v′ dx Q(u) = 1 2

  • Ω

u2dx B′(u; w, v) = −

  • Ω

uwv′ dx Q′(u; w) =

  • Ω

uwdx

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 25 / 38

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Nonlinear Problems

Burgers’ problem – Convergence estimates

If z′ is continuous, |∆B(uh, e, z)| =

  • 1
  • Ω

e2z′dx

  • (1 − s)ds
  • ≤ 1

2

  • Ω
  • e2

z′ dx ≤ e2

L2(Ω)z′L∞

∼ O(h4) |∆Q(uh, e)| =

  • 1
  • 2
  • Ω

e2dx

  • (1 − s)ds
  • ≤
  • Ω

|e|2 dx ≤ e2

L2(Ω)

∼ O(h4)

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 26 / 38

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Nonlinear Problems

Burgers’ problem – Convergence of error terms

10−1 100 101 h 10−10 10−8 10−6 10−4 10−2 100

h2 h4

|Q(u) − Q(uh)| R(uh; z+) ∆B(uh, e, z) ∆Q(uh, e)

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 27 / 38

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Nonlinear Problems

Nonlinear diffusion problem

Let Ω = (0, 1) −

  • A(u′)u′′ = f

u(0) = u(1) = 0 A(u′) = 1 + 1 10u′2

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 u(x)

10−3 10−2 10−1 100 h 10−12 10−10 10−8 10−6 10−4 10−2 100

h2 h4

|Q(u) − Q(uh)| R(uh; z+) ∆B(uh, e, z) ∆Q(uh, e)

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 28 / 38

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Nonlinear Problems

Nonlinear diffusion - Convergence estimates

Observe that for small h, (uh + se)′L2 ∼ (uh)′L2, |∆B(uh, e, z)| =

  • 3

5 1

  • Ω

(uh + se)′ e′2 z′dx

  • (1 − s)ds
  • ≤ 3

5 1 (uh + se)′L2e′2

L2z′L∞(1 − s)ds

∼ O(h2) |∆Q(uh, e)| =

  • 1
  • 2
  • Ω

e2dx

  • (1 − s)ds
  • ≤
  • Ω

|e|2 dx ≤ e2

L2

∼ O(h4)

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 29 / 38

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Extension to Time-dependent Problems

Extension to Time-dependent Problems: Cahn-Hilliard

ut = ∆µ µ = f′(u) − ǫ2∆u

  • in Ω + BCs

Weak formulation: Find (u, µ) ∈ W0 × V s.t. B

  • (u, µ); (v, η)
  • = 0

∀(v, η) ∈ V × V

B

  • U, V
  • =

T

  • ut, v + (∇µ, ∇v)
  • +
  • (µ, η) − (f ′(u), η) − ǫ2(∇u, ∇η)
  • dt

W0 =

  • u ∈ V ; ut ∈ V ∗, u(0) = u0
  • V = L2(0, T; H1(Ω))
  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 30 / 38

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Extension to Time-dependent Problems

Adjoint of the Cahn-Hilliard problem

Goal: Q(u, µ) =

  • kT , u(T)
  • +

T

  • (ku, u) + (kµ, µ)
  • dt

Adjoint: (Backward-in-time linearized-adjoint problem) −∂tpu + ǫ2∆pµ − f′′(uh) pµ = ku in Ω pµ − ∆pu = kµ in Ω pu(T) = kT (Final condition) + [Natural BCs]

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 31 / 38

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Extension to Time-dependent Problems

Example: merging bubbles

van der Zee, Oden, Prudhomme, Hawkins, NMPDE 2011

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 32 / 38

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Extension to Coupled Problems

Multiphysics Coupled Problems: Micro-fluidics

Structure of the Electric Double Layer (EDL) between the fluid and the wall. Under the effect of an electric field tangent to the wall, charged particles are subjected to an electric force and thus move in the direction of the electric field. Garg, Prudhomme, van der Zee, Carey, 2014.

Coupled model (simplified): −∇ · (σc∇φ) = 0 in Ω µ∆u + ∇p = 0 in Ω ∇ · u = 0 in Ω Boundary conditions: n · (σc∇φ) = 0

  • n Γw

φ = g

  • n Γio

u + λ∇φ = 0

  • n Γw

u · t = 0

  • n Γio

n · (σ · n) = 0

  • n Γio
  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 33 / 38

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Extension to Coupled Problems

Micro-Fluidic Flows

x y 1 2 3 4 5 0.2 0.4 0.6 0.8 1 1.2

phi:

  • 0.25
  • 0.2
  • 0.15
  • 0.1
  • 0.05

0.05 0.1 0.15 0.2 0.25 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −0.05 0.05 0.1 0.15 0.2 0.25 0.3 x distance along bottom boundary Computed Potential

Computed Adjoint Potential x distance along bottom boundary

u + λ∇φ = 0 ⇒ u · t + λ∂tφ = 0 u · n + λ∂nφ = 0 ⇒ u · t + λ∂tφ = 0 u · n = 0

x y 1 2 3 4 5 0.2 0.4 0.6 0.8 1 1.2

phi:

  • 0.1
  • 0.08
  • 0.06
  • 0.04
  • 0.02

0.02 0.04 0.06 0.08 0.1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −0.02 0.02 0.04 0.06 0.08 0.1 0.12

x distance along bottom boundary Computed Adjoint Potential

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 34 / 38

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Extension to Coupled Problems

Micro-Fluidic Flows

Strong form of adjoint problem: −∇ · (σc∇ϕ∗) = 0 in Ω −∆w∗ + ∇p∗ = kα in Ω −∇ · w∗ = 0 in Ω with three boundary conditions on Γio: ϕ∗ = 0

  • n Γio

w∗ · t = 0

  • n Γio

n · (σ∗ · n) = k

  • n Γio

and three BCs on Γw: n · (σc∇ϕ∗) + ∇Γw ·

  • λt · (σ∗ · n)
  • t
  • = 0
  • n Γw

w∗ = 0

  • n Γw
  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 35 / 38

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SLIDE 36

Extension to Coupled Problems

Micro-Fluidic Flows

x y

  • 1.5
  • 1
  • 0.5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5

v: -0.014 -0.012

  • 0.01
  • 0.008 -0.006 -0.004 -0.002

x y

  • 1.5
  • 1
  • 0.5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5

phi: -0.08 -0.06 -0.04 -0.02 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Adjoint velocity u∗ (in y direction). It is mainly different from zero inside the vertical channel indicating that the primal solution needs to be accurate in that region. Adjoint potential φ∗. Note that φ∗ almost vanishes everywhere except at the corners and along the middle section of the top wall.

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 36 / 38

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SLIDE 37

Extension to Coupled Problems

Micro-Fluidic Flows

x y

  • 1.5
  • 1
  • 0.5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 3 3.5 4 4.5 5 5.5 6 −3.2 −3 −2.8 −2.6 −2.4 −2.2 −2 −1.8

Log10(Dofs), Degrees of Freedom Log10(Relative Error) Convergence of the QoI error

Uniform Adjoint Residual

Adaptive mesh obtained using adjoint-based error estimates. Note that the elements get refined almost exclusively near the corners due to the singularities in the primal velocity and adjoint potential. Convergence plots for the relative error in QoI using uniform and adjoint based refinements.

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 37 / 38

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Summary

Conclusions

Goal-oriented error estimation is based on the notion of the adjoint/dual problem. The adjoint problem is “straightforwardly” derived from the weak formulation of the primal problem. The error in the quantity of interest is represented as the product of the residual by the adjoint solution. Error estimates and refinement indicators are obtained by solving for approximations to the adjoint problem. Approach is generic! Goal-oriented error estimates can be applied to other discretization methods (FD, FV, DGM, spectral methods, meshless methods, etc.).

  • S. Prudhomme (Polytechnique Montr´

eal) Goal-oriented Error Estimation November 4-8, 2019 38 / 38