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

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eal) PGD reduced-order models November 4-8, 2019 1 / 16

Motivation

Motivation: EIT problem for composite materials

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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

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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: u(t, x) =

∞

• i=1

ψi(t)φi(x) ≈

N

• i=1

ψi(t)φi(x) = uN(x, t) 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.

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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. u = argmin

v∈V

J(v) = argmin

v∈V

1 2B(v, v) − F(v)

• Separated representation:

Find an approximation um of u in the form u(x, θ) ≈ um(x, θ) =

m

• i=1

ψi(x)φi(θ), ∀(x, θ) ∈ D × Ω

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Goal-oriented PGD formulation

Proper Generalized Decomposition method (PGD)

Progressive (iterative) approach: Given um−1, find next mode ψφ such that um is a better approximation to u: um(x, θ) = um−1(x, θ) + ψ(x)φ(θ) Optimization problem: um = argmin

ψφ

J(um−1 + ψφ) J′(um−1 + ψφ; δ(ψφ)) = lim

ǫ→0

J(um−1 + ψφ + ǫδ(ψφ)) − J(um−1 + ψφ) ǫ = B(um−1 + ψφ, δ(ψφ)) − F(δ(ψφ)) = B(ψφ, δ(ψφ)) −

• F(δ(ψφ)) − B(um−1, δ(ψφ))
• R(δ(ψφ))
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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∗

• r

B(ψφ, ψ∗φ) = R(ψ∗φ), ∀ψ∗ = ψ∗(x) B(ψφ, ψφ∗) = R(ψφ∗), ∀φ∗ = φ∗(θ) Use Alternated Directions scheme: φ(0) → ψ(1) → φ(1) → ψ(2) → φ(2) . . .

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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 · Q(ψφ) = µ · (Q(u) − Q(um−1)), ∀µ ∈ Rk Issue: Separation of variables decouples the dimensions of the problem while constraints should be applied globally. Iterative approach (Uzawa or Augmented Lagrangian): Given λi−1, solve for ψi and φi using Alternated directions. Update the Lagrange multiplier λi. Repeat until convergence.

Kergrene, Prudhomme, Chamoin, and Laforest, “Approximation of constrained problems using the PGD method with application to pure Neumann problems”, CMAME, 2017.

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Goal-oriented PGD formulation

Example: Bar model

N1(x) E1 E2 N2(x) x = 0 x = L/2 x = L F = 1 E(x) =

• E1,

x ∈ (0, L/2), E2, x ∈ (L/2, L), Q1(u) = 1 |D|

• D1
• D2

u(L/2), Q2(u) = 1 |D|

• D1
• D2

u(L),

u(x, E1, E2) = L 2E1 (N1(x) + N2(x)) + L 2E2 N2(x), um(x, E1, E2) =

m

• i=1

ϕi(x)φ1i(E1)φ2i(E2)

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Goal-oriented PGD formulation

Example: Bar model

0.5 1

x

0.5 1 1.5 2 2.5 3

ϕ Classical Penalization β = 102 Penalization β = 105 Uzawa

0.2 0.4 0.6 0.8 1

E1

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

φ1

0.2 0.4 0.6 0.8 1

E2

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

φ2

0.5 1

x

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6

ϕ

0.2 0.4 0.6 0.8 1

E1

• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6

φ1

0.2 0.4 0.6 0.8 1

E2

• 0.2

0.2 0.4 0.6 0.8 1 1.2 1.4

φ2

0.5 1

x

• 0.2
• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7

ϕ

0.2 0.4 0.6 0.8 1

E1

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5

φ1

0.2 0.4 0.6 0.8 1

E2

• 1.4
• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6

φ2

First three PGD: ϕ(x) (left), φ1(E1), φ2(E2).

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Goal-oriented PGD formulation

EIT Example

σ = 1 σ = σa σ = 1 σ = σb σ = 1 Γ1 Γ2 Γ3 Γ4 Γ5 Γ6

Quantities of interest:

Q1(u) = 1 |D|

• D

1 |Γ2|

• Γ2

u − 1 |Γ3|

• Γ3

u

• Q2(u) =

1 |D|

• D

1 |Γ2|

• Γ2

u − 1 |Γ5|

• Γ5

u

• Q3(u) =

1 |D|

• D

1 |Γ2|

• Γ2

u − 1 |Γ6|

• Γ6

u

• where D = Da × Db with:

σa ∈ Da = [1, 10] σb ∈ Db = [0.1, 1] Separation of variables: um(x, y, σa, σb) =

m

• i=1

fi(x, y)gi(σa)hi(σb)

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Goal-oriented PGD formulation

EIT Example

10 20 30 40 50 60

m

0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22

ǫ

Classical PGD Goal-Oriented PGD

10 20 30 40 50 60

m

10 -5 10 -4 10 -3 10 -2 10 -1

ǫ1

Classical PGD Goal-Oriented PGD

(a) Error in the energy norm, (b) Error in Q1, with respect to # of modes

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Goal-oriented PGD formulation

EIT Example

10 20 30 40 50 60

m

10 -5 10 -4 10 -3 10 -2 10 -1

ǫ2

Classical PGD Goal-Oriented PGD

10 20 30 40 50 60

m

10 -5 10 -4 10 -3 10 -2 10 -1

ǫ3

Classical PGD Goal-Oriented PGD

(c) Error in Q2, (d) Error in Q3, with respect to # of modes

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Goal-oriented PGD formulation

EIT Example

Model Problem: −∇ · (σ∇u) = 0, in Ω, n · σ∇u = g,

• n ∂Ω.

This is a 5D problem: 2 space variables (x, y) Diffusivities σa and σb Position x1 of electrode Γ1

σ = 1 σ = σa σ = 1 σ = σb σ = 1 Γ1 Γ2 Γ3 Γ4 Γ5 Γ6

Input/Output: Load g corresponds to the difference of potential between Γ1 and Γ4. QoI’s are 3 differences of potential between other pairs of electrodes.

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Goal-oriented PGD formulation

Adapted meshes and error convergence

From top to bottom: 2d mesh in x and y, 1d meshes in σa and σb, 1d mesh in x1

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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

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eal) PGD reduced-order models November 4-8, 2019 16 / 16