A progressive RB/EIM for nonlinear parabolic problemsaminabenaceur.mit.edu/sites/default/files/documents/...Modelproblem Standardmethod PREIM Numericalresults References...

Model problemStandard method

PREIMNumerical results

A progressive RB/EIM for nonlinear parabolicproblems

Amina Benaceur1,2

V. Ehrlacher1, A. Ern1 and S. Meunier2

me1University Paris-Est, CERMICS (ENPC) and INRIA Paris, France

2EDF R&D, France

MORTech, November 8th, 2017

Amina Benaceur PREIM for nonlinear parabolic problems 1/18



Outline

1 Model problem

2 Standard method

3 PREIM

4 Numerical results




Industrial contextNonlinear heat transfer problem

Industrial context

• Motivation : Behavior of valve components.• Complex and costly simulations:

• Heat transfer (≈ 4h),• mechanics (≈ 12h).

• Non-affordable for parametrized studies.• Goal: Speed up simulations via reduced ordermodeling.





Nonlinear heat transfer problem

Ω : bounded domain in Rd.I : time interval.P: parameter set.Parametric thermal conductivity.Nonlinear reaction term.

For many values λ ∈ P: find uλ : I × Ω→ R such that∂uλ∂t−∇ ·

(κ(λ)∇uλ

)+ Γ(λ, uλ) = f, in I × Ω,

−κ(λ)∂uλ∂n

= φe, on I × ∂Ω,

uλ(t = 0, ·) = u0(·), in Ω,





Discrete High-Fidelity scheme

Discrete time nodes (tk)1≤k≤K , FE space X ⊂ H1(Ω).

Backward-Euler semi-implicit scheme: Given u0λ = u0, find

(ukλ)1≤k≤K such that ∀v ∈ X,

m(ukλ, v)+∆tkκ(λ)a0

(ukλ, v

)+∆tknΓ

(λ, uk−1

λ , v)

= m(uk−1λ , v)+∆tklk(v)

m(v, w) =

∫Ω

vw, a0(v, w) =

∫Ω

∇v · ∇w,

lk(v) =

∫Ω

fkv +

∫∂Ω

φkev,

nΓ(λ, v, w) =

∫Ω

Γ(λ, v)w,




Online stageEmpirical Interpolation Method

Standard method

• Steady nonlinear PDEs addressed using the EIMRB : Machiels, Maday, Patera, 2001.EIM : Barrault, Maday, Nguyen, and Patera, 2004.

• Extension to time-dependent parabolic PDEsGrepl, Maday, Nguyen, and Patera, 2007.Grepl, 2012.

Two-stage procedure:

• Online: multi-query or real-time.• Offline:

• Potentially costly.• Compute RB functions.• Approximate the nonlinearity using the EIM.





Online stage

• XN = spanθ1, · · · , θN ⊂ X (N dim(X)) is a RB space:

ukλ ≈ ukλ =

N∑n=1

ukλ,n θn

→ Component vector ukλ = (ukλ,n)1≤n≤N .→ Mass and stiffness matrices M and A0 of size N .

• Given u0λ = u0 ∈ RN , find (ukλ)1≤k≤K ∈ (RN )K such that

(M + ∆tkκ(λ)A0)ukλ = ∆tkfk + Muk−1λ −∆tkG(uk−1

λ ),

with

G(uk−1λ ) =

( N∑n=1

uk−1λ,n

∫Ω

Γ(λ, uk−1λ ) θp

)1≤p≤N

.

Problem: How to deal efficiently with Γ(λ, uk−1λ ) ?





Empirical Interpolation Method

For the HF solution ukλ(x)

γ(λ, k, x) := Γ(ukλ(x), λ),

the EIM allows us to separate (λ, k) from x

γ(λ, k, x) ≈ γM (λ, k, x) =

M∑j=1

ϕkλ,jqj(x).

In practice:• Compute and store (q1, · · · , qM ) ‘offline’ using a greedyalgorithm based on precomputed HF trajectories.

• Find (x1, · · · , xM ) using a greedy algorithm.• Store B =

(qj(xi)

)ij.





Reduced order problem

Find (ukλ)1≤k≤K ∈ (RN )K such that

(M + ∆tkκ(λ)A0)ukλ = Muk−1λ + ∆tk(fk −Dγk−1

λ ),

where

D = B−1C, C =

(∫Ωqmθp

)1≤p,m≤N,M

,

and

γk−1λ :=

(Γ(λ, uk−1

λ (xi)))

1≤i≤M.





Standard EIM

Algorithm 1 Standard EIM reduced modeling strategy

Offline stage :Input : Ptr ⊂ P, Ktr

= 0, · · · ,K and εPOD > 0

1: Compute S = (ukλ)λ∈Ptr,k∈Ktr

2: Compute (θ1, · · · , θN ) = POD(S, εPOD)3: Compute u0, (f1, · · · , fk), M, and A0

4: Compute D

Online stage :5: Choose λ ∈ P; set k = 1 and u0

λ = u0

6: while k ∈ Ktr do7: Compute γk−1

λ from uk−1λ

8: Solve the reduced system to obtain ukλ9: Set k = k + 1

10: end whileOutput : (u1

λ, · · · , uKλ )




Offline efficiencyGreedy selection

Offline efficiency

Standard EIM:• Precise EIM : Card(Ptr)M HF solutions.• RB selection : N HF solutions.• EIM functions based on HF solutions .

SER (Daversin, Prud’homme - 2015 ) for steady nonlinear PDEs:

PREIM (Progressive RB/EIM):• Simultaneous construction of EIM and RB functions.• Greedy selection based on both RB and HF solutions.• EIM functions based on HF solutions .• Total cost of at most M HF solutions.[DP15, GMNP07, Gre12, QMN16, HRS16, BMNP04]





Greedy selection (1)

At iteration m ≥ 2:• We introduce

um,kλ :=

ukλ if λ ∈ PHF

m ,

um,kλ otherwise,

and the nonlinear function

γm(λ, k, x) := Γ(λ, um,kλ (x)).

• Goals:• Produce (q1, · · · , qM ) to approximate γm by

γmm(λ, k, x) :=

m∑j=1

(ϕm)kλ,j qj(x).

• Produce RB functions (θm1 , · · · , θmNm).We have PHF

m−1 and γm−1m−1 from the previous stage.Amina Benaceur PREIM for nonlinear parabolic problems 12/18




Greedy selection (2)

• For any m ≥ 2, select a new pair

(λm, km) ∈ argmax(λ,k)∈Ptr×Ktr

‖Γ(λ, um−1,k

λ (·))−γm−1

m−1(λ, k, ·)‖`∞(Ωtr),

If λm /∈ PHFm−1

PHFm := PHF

m−1 ∪ λm, Sm = (ukλm)k∈Ktr .

• Update (λm, km) using the HF-based re-selection criterion

(λm, km) ∈ argmax(λ,k)∈PHF

m ×Ktr

‖Γ(λ, ukλ(·)

)− γm−1

m−1(λ, k, ·)‖`∞(Ωtr).

Compute the EIM functions (q1, · · · , qM ).• Update the RB functions

(θm1 , · · · , θmNm) = POD((S1, · · · ,Sm), εPOD

)Amina Benaceur PREIM for nonlinear parabolic problems 13/18



Study case

Computational domain and mesh

• 2-D perforated plate, ≈ 1500 mesh nodes.• u0 = 293K(20oC).• HF solutions using the software code_aster.




Test case: Nonlinear reaction term

∂uλ∂t−∇ ·

(κ(λ)∇uλ

)+ Γ(λ, uλ) = 0,

−κ(λ)∂uλ∂n

= φe.

• Nonlinear heat conductivity with• κ0 = 0.01m2.K−2.s−1.• Γ(λ, z) := exp

(− λ z

u0

)z.

• Neumann boundary conditionswith φe = 3K.m.s−1.

• I = [0, 1], ∆tk = 0.2s,P = [0, 20] and Ptr = [[0, 20]] .

HF solutions for λ = 0.5 (top) and λ = 9.5(bottom) at t = 0.2s (left) and t = 1s (right).




Offline stage

Standard EIM:• Offline HF computations for all parameters in Ptr.• Card Ptr = 21 , K = 5 =⇒ 105 space functions.• POD with εPOD = 10−1 extracts N = 15 RB functions.• For εEIM = 10−2, M = 12 EIM functions.

PREIM:

m 1 2 3 4 5 6 7 8 9 10 11 12

PREIM λ 0 0 1 2 4 1 2 0 2 7 1 0k 4 1 4 0 4 4 4 2 3 4 1 3

Test case (a): Selected parameter and time node values in PREIM.The red cells correspond to a new parameterselection and, therefore, to a new HF computation.




Approximation error

EIM approximation error ‖rm‖`∞(Ωtr×Itr×Ptr). RB approximation error ‖uλ − uλ‖`1(Itr;`2(Ωtr))




Conclusion

• PREIM:• Significant computational savings.• Good accuracy.

• Preprint available on arXiv:• A progressive reduced basis/empirical interpolation method for

nonlinear parabolic problems.

Thank you for your attention




References

M. Barrault, Y. Maday, N. C. Nguyen, and A. T. Patera.An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partialdifferential equations.2004.

C. Daversin and C. Prud’homme.Simultaneous empirical interpolation and reduced basis method for non-linear problems.2015.

M. A. Grepl, Y. Maday, N. C. Nguyen, and A. T. Patera.Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations.2007.

M. A. Grepl.Certified reduced basis methods for nonaffine linear time-varying and nonlinear parabolic partialdifferential equations.2012.

J. S. Hesthaven, G. Rozza, and B. Stamm.Certified reduced basis methods for parametrized partial differential equations.2016.

A. Quarteroni, A. Manzoni, and F. Negri.Reduced basis methods for partial differential equations.2016.


A variant of PREIM : PREIM-NRAn Unsteady SER : U-SER

A variant of PREIM : PREIM-NR

For any m ≥ 2, select a new pair


‖Γ(λ, um−1,k

λ (·))− γm−1

m−1(λ, k, ·)‖`∞(Ωtr).

• Greedy selection based on both HF and RB solutions.• No re-selection step.• EIM functions derived from HF solutions .



An Unsteady SER : U-SER

For any m ≥ 2, select a new pair


‖Γ(λ, um−1,k

λ (·))− γm−1

m−1(λ, k, ·)‖`∞(Ωtr).

• Greedy selection based on RB solutions only .• No re-selection step.• EIM functions derived from RB solutions .



PREIM: Initialization

We recall that γ(λ, k, x) := Γ(λ, ukλ(x)).For a random λ1, compute S1 = (ukλ1)

k∈Ktr .Compute the EIM quantities

k1 = argmaxk∈Ktr

‖Γ(λ1, u

kλ1(·)

)‖`∞(Ωtr), r1(·) = Γ

(λ1, u

k1λ1

(·)),

x1 = argmaxx∈Ωtr

|r1(x)|, and q1 =r1

r1(x1).

Initialize the EIM :

γ11(λ, k, x) = γ1(λ, k, x1)q1(x).

Build the RB space

XN1 = Spanθ11, · · · , θ1

N1 = SpanPOD(S1, εPOD).

for later computations of u1,kλ and set PHF

1 = λ1.Amina Benaceur PREIM for nonlinear parabolic problems 22/18

A progressive RB/EIM for nonlinear parabolic problemsaminabenaceur.mit.edu/sites/default/files/documents/...Modelproblem Standardmethod PREIM Numericalresults References...

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