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Towards automated model reduction:exact error bounds and simultaneous

finite-element reduced-basis adaptive refinement

Masayuki Yano

University of Toronto

Acknowledgment:Anthony Patera;

COST EU-MORNET, AFOSR, ONR

MoRePaS 2015Trieste, Italy

14 October 2015

Introduction

MotivationModel problem

Parametrized PDEs

µPDE: given µ ∈ D ⊂ RP , find u(µ) ∈ V s.t. ∞ ops

a(u(µ), v;µ) = `(v;µ) ∀v ∈ V

and evaluate the output

s(µ) ≡ `o(u(µ);µ).

FE: given µ ∈ D, find uN (µ) ∈ VN ⊂ V s.t. O(N ·) ops

a(uN (µ), v;µ) = `(v;µ) ∀v ∈ VN

and evaluate the output

sN (µ) ≡ `o(uN (µ);µ).

1

Certified reduced basis (CRB) method [review: Rozza et al 08]

CRB: introduce RB space N � N

VN ≡ span {uN (µn)}Nn=1︸︷︷︸FE snapshots

⊂ VN ;

given µ ∈ D, find uN(µ) ∈ VN s.t. O(N ·) ops

a(uN(µ), v;µ) = `(v;µ) ∀v ∈ VN ,

evaluate the outputsN(µ) ≡ `o(uN(µ);µ),

and bound the error wrt FE output

|sN (µ)− sN(µ)| ≤ ∆NN (µ).

CRB provides rapid and reliable solution of µPDEassuming the “truth” space VN is well chosen.

2

Issues

⇒ Objectives

“Truth” space VN may betoo coarse ⇒ unreliable solution wrt exact µPDE;too fine ⇒ unnecessarily expensive offline “truth” solves.

Goal: eliminate the issue of “truth” [cf. Ali, Steih, & Urban;Ohlberger & Schindler]

1. provide error bounds wrt exact µPDE,

|s(µ)− sN(µ)| ≤ ∆N(µ).

2. provide adaptivity in physical and parameter spaces

optimal VN and VN ⊂ VN .

3. maintain O(N · � N ·) online complexity.

3

Issues ⇒ Objectives

“Truth” space VN may betoo coarse ⇒ unreliable solution wrt exact µPDE;too fine ⇒ unnecessarily expensive offline “truth” solves.

Goal: eliminate the issue of “truth” [cf. Ali, Steih, & Urban;Ohlberger & Schindler]

1. provide error bounds wrt exact µPDE,

|s(µ)− sN(µ)| ≤ ∆N(µ).

2. provide adaptivity in physical and parameter spaces

optimal VN and VN ⊂ VN .

3. maintain O(N · � N ·) online complexity.

3

Introduction

MotivationModel problem

Model problem

Given µ ∈ D, find u(µ) ∈ V ≡ H10 (Ω) such that∫Ω

∇v · κ(µ)∇u(µ)dx︸︷︷︸a(u(µ), v;µ)

=

∫Ω

vf(µ)dx︸︷︷︸`(v;µ)

∀v ∈ V .

Assumptions:1. coercivity: κ(µ)(x) > 0, ∀x ∈ Ω;2. affine parameter decomposition

κ(µ) =

Qκ∑q=1

µ-dependent︷︸︸︷Θκq (µ) κq︸︷︷︸

µ-independent

, f(µ) =

Qf∑q=1

Θfq (µ)fq,

κ−1(µ) ≡ c(µ) =Qc∑q=1

Θcq(µ)cq (compliance tensor).

4

Output bound: ingredients

Norm: for δ > 0, ‖v‖2Vµ = (v, v)Vµ

(w, v)Vµ ≡∫

Ω

∇v · κ(µ)∇wdx+ δ∫

Ω

vwdx.

Residual: linear form

r(v; ũ;µ) ≡ `(v;µ)− a(ũ, v;µ)

and the associated dual norm

‖r(·; ũ;µ)‖V ′µ ≡ supv∈V

r(v; ũ;µ)

‖v‖Vµ(over V , not VN ).

Stability constant:

α(µ) ≡ infv∈V

a(v, v;µ)

‖v‖2Vµ(over V , not VN ).

5

Output bound

Proposition. For an output estimate [Piece & Giles; . . . ]

sN(µ) ≡ `o(uN(µ))︸︷︷︸“raw” output

+ rpr(ψN(µ);uN(µ);µ)︸︷︷︸adjoint correction

,

the error is bounded by

|s(µ)− sN(µ)| ≤1

α(µ)︸︷︷︸stability

‖rpr(·;uN(µ);µ)‖V ′µ︸︷︷︸primal residual dual-norm

‖radj(·;ψN(µ);µ)‖V ′µ︸︷︷︸adjoint residual dual-norm

.

Agenda: provide

1. rapidly computable upper bound of ‖rpr/adj(·; ũ;µ)‖V ′µ ;2. rapidly computable lower bound of α(µ);

3. automatic spatio-parameter adaptivity.

6

Residual bound

Bound formFinite dimensional approximations

Residual bound

Introduce a “dual” (or “flux”) space

Q ≡ H(div; Ω) ≡ {q ∈ (L2(Ω))d | ∇ · v ∈ L2(Ω)}.

Define a bound form ‖ · ‖ ≡ ‖ · ‖L2(Ω)

F (ũ, p;µ) ≡ ‖κ1/2(µ)∇ũ− c1/2(µ)p‖2︸︷︷︸constitutive relation

+δ−1 ‖f(µ) +∇ · p‖2︸︷︷︸conservation law

.

Proposition. For any ũ ∈ V

‖r(·; ũ;µ)‖V ′µ ≤ (F (ũ, p;µ))1/2 ∀p ∈ Q.

Remark: bound holds for any approximations ũ and p

⇒ may be used with FE and RB approximations.7

Residual bound: proof

Proof. Note ∀p ∈ Q,

r(v; ũ;µ)

=

∫Ω

vfdx−∫

Ω

∇v · κ∇ũdx+

≡ 0 by Green’s theorem︷︸︸︷∫Ω

∇v · pdx+∫

Ω

v∇ · pdx

=

∫Ω

v(f +∇ · p)dx−∫

Ω

∇v · (κ∇ũ− p)dx

≤ (δ−1‖f +∇ · p‖2L2(Ω) + ‖κ1/2∇ũ− c1/2p‖2L2(Ω))1/2

(δ‖v‖2L2(Ω) + ‖κ1/2∇v‖2L2(Ω))1/2

= (F (ũ, p;µ))1/2‖v‖Vµ .

It follows

‖r(·; ũ;µ)‖V ′µ ≡ supv∈V

r(v; ũ;µ)

‖v‖Vµ≤ (F (ũ, p;µ))1/2.

8

Bound form decomposition

Bound form can be decomposed as

F (w, p;µ) = A((w, p), (w, p);µ)︸︷︷︸quadratic

+B((w, p);µ)︸︷︷︸linear

+ C(µ)︸︷︷︸constant

,

where (·, ·) ≡ (·, ·)L2(Ω)

A((w, p), (v, q);µ) = (κ1/2(µ)∇w, κ1/2(µ)∇v) + (c1/2(µ)p, c1/2(µ)q)

− (∇w, q)− (p,∇v) + δ−1(∇ · p,∇ · q)

B((w, p);µ) = 2δ−1(f(µ),∇ · p)

C(µ) = δ−1(f(µ), f(µ)).

Remark: forms A(·, ·;µ), B(·;µ), and C(µ) inheritaffine parameter decomposition.

9

Residual bound

Bound formFinite dimensional approximations

Minimum-residual mixed finite element method

IntroduceVN ≡ {v ∈ V | v|κ ∈ Pp, κ ∈ Th},QN ≡ {q ∈ Q | q|κ ∈ RTp−1, κ ∈ Th}.

Min-res: given µ ∈ D, find (uN (µ), pN (µ)) ∈ VN ×QN such that

(uN (µ), pN (µ)) = arg infw∈VNVq∈QNQ

F (w, q;µ).

E-L: find (uN (µ), pN (µ)) ∈ VN ×QN such that N ×N SPD

2A((uN (µ), pN (µ)), (v, q);µ) = B((v, q);µ)

∀(v, q) ∈ VNV ×QNQ .Built-in residual bound:

‖r(·;uN (µ);µ)‖2V ′µ ≤ F (uN (µ), pN (µ);µ).

10

Minimum-residual mixed reduced basis method

IntroduceVN ≡ span{uN (µn)}Nn=1QN ≡ span{qN (µn)}Nn=1.

Min-res: given µ ∈ D, find (uN(µ), pN(µ)) ∈ VN ×QN such that

(uN(µ), pN(µ)) = arg infw∈VNq∈QN

F (w, q;µ).

E-L: find (uN(µ), pN(µ)) ∈ VN ×QN such that 2N × 2N SPD

2A((uN(µ), pN(µ)), (v, q);µ) = B((v, q);µ)

∀(v, q) ∈ VN ×QN .Built-in residual bound:

‖r(·;uN(µ);µ)‖2V ′µ ≤ F (uN(µ), pN(µ);µ).11

Offline-online computational decomposition

Recall A(·, ·;µ) and B(·;µ) inherit affine parameter decomposition.

Offline:

Snapshots: VN = span{uN (µn)}Nn=1, QN = span{qN (µn)}Nn=1;Dataset: Aq(·, ·) and Bq(·) evaluated wrt VN ×QN .

Operation count: O(N ·, N ·, Q·)

Online:

µ ∈ D → uN(µ)︸︷︷︸primal

, pN(µ)︸︷︷︸dual

, F (uN(µ), pN(µ);µ)︸︷︷︸built-in residual bound

.

Operation count: O(N3) +O(N2Q2)12

Stability constant

Problem description and SCMLower bound of minimum eigenvalueFinite dimensional approximationsRelationship to complementary variational principle

Preliminary

Sketch: stability constant ⇒ least stable mode ⇒ eigenproblem.In our case ‖v‖2Vµ ≡ ‖κ

1/2(µ)∇v‖2 + δ‖v‖2

α(µ) ≡ infv∈V

a(v, v;µ)

‖v‖2Vµ≥(

1 +δ

τ(µ)

)−1where

τ(µ) ≡ infv∈V

‖κ1/2(µ)∇v‖2

‖v‖2H1(over V , not VN ),

which is the minimum (i.e. first) eigenvalue of∫Ω

∇v · κ(µ)∇zdx = λ1(µ)∫

Ω

∇v · ∇z + vzdx ∀v ∈ V .

Goal: online-efficient evaluation of a lower bound of τ(µ) ≡ λ1(µ).13

Approach: exact Successive Constraint Method (SCM)

Offline: ingredients [Huynh et al 2006; . . . ]1. Bounding boxes: ∀q = 1, . . . , Qκ,

γq ≡ supv∈V

|∫

Ω∇v · κq∇vdx|‖v‖2H1

(over V , not VN ).

⇒ evaluate ‖κq‖L∞(Ω) by inspection.

2. Stability constant at select points

τ(µ′) ≡ infv∈V

∫Ω∇v · κ(µ′)∇vdx‖v‖2H1

(over V , not VN ).

⇒ need a lower bound of the minimum eigenvalue.

Online: same as the standard SCM.

14

Stability constant


Eigenvalue bounds

Given an approximate eigenpair (z̃, λ̃) ∈ V × R with ‖z̃‖H1 = 1,introduce residual

r(v; z̃, λ̃;µ) ≡∫

Ω

∇v · κ∇z̃dx︸︷︷︸LHS of eigenproblem

− λ̃∫

Ω

∇v · ∇z̃ + vz̃dx︸︷︷︸RHS of eigenproblem

.

Proposition. Closest eigenvalue is bounded by [Weinstein]

minj|λj − λ̃| ≤ ‖r(·; z̃, λ̃;µ)‖(H1)′ ≡ sup

v∈V

r(v; z̃, λ̃;µ)

‖v‖H1.

Observation: if |λ1 − λ̃| ≤ |λ2 − λ̃|, then

λ̃− ‖r(·; z̃, λ̃;µ)‖(H1)′ ≤ λ1.

15

Eigenvalue residual bound

Introduce a “dual” (of “flux”) space Q ≡ H(div; Ω).

Define an eigenvalue bound form

G(z̃, λ̃, p;µ) ≡ ‖κ1/2(µ)∇z̃ − λ̃∇z̃ − p‖+ ‖λ̃z̃ +∇ · p‖.

Proposition. For any (z̃, λ̃) ∈ V × R,

‖r(·; z̃, λ̃;µ)‖(H1)′ ≤ (G(z̃, λ̃, p;µ))1/2 ∀p ∈ Q.

Remark: bound holds for any approximations (z̃, λ̃) and p

⇒ may be used with FE approximations.

16

Eigenvalue residual bound: proof

Proof. Note ∀p ∈ Q,

r(v; z̃, λ̃;µ)

=

∫Ω

∇v · κ(µ)∇z̃dx− λ̃∫

Ω

∇v · ∇z̃ − λ̃∫

Ω

vz̃dx

−∫

Ω

∇v · p−∫

Ω

v∇ · pdx}≡ 0 by Green’s theorem

=

∫Ω

v(−λ̃z̃ −∇ · p)dx+∫

Ω

∇v · (κ∇ũ− λ̃∇z̃ − p)dx

≤ (‖λ̃z̃ +∇ · p‖2L2(Ω) + ‖κ1/2∇ũ− λ̃∇z̃ − p‖2L2(Ω))1/2

(‖v‖2L2(Ω) + ‖∇v‖2L2(Ω))1/2

= (G(z̃, λ̃, p;µ))1/2‖v‖Vµ .It follows

‖r(·; z̃, λ̃;µ)‖(H1)′ ≡ supv∈V

r(v; ũ, λ̃;µ)

‖v‖H1≤ (G(z̃, λ̃, p;µ))1/2.

17

Stability constant


Finite element approximation

Introduce FE spaces VN ⊂ V and QN ⊂ Q.

Galerkin: find (zN , λN1 ) ∈ VN × R such that∫Ω

∇v · κ∇zNdx = λN1∫

Ω

∇v · ∇zN + vzNdx ∀v ∈ VN .

Min-res: find pN ∈ QN such that

pN = arg infq∈QN

G(zN , λN , q;µ).

Bounds: assuming |λN1 − λ1| < |λN1 − λ2|,

λN1 − (G(zN , λN , pN ;µ))1/2︸︷︷︸lower bound: min-res

≤ λ1 ≤ λN1︸︷︷︸upper bound: Galerkin

.

18

Exact SCM: offline-online computational decomposition

Offline:

1. bounding boxesγq ≡ ‖κq‖L∞(Ω)

2. stability constants: for µ′ ∈ ΞSCMtrain ,

τLB(µ′) = λN1 (µ′)− (G(zN , λN , pN ;µ′))1/2.

⇒ Online SCM dataset.

Online:

µ ∈ D → τLB(µ) → αLB(µ) ≡(

1 +δ

τLB(µ)

)−1.

19

Stability constant


Role of δ in ‖v‖2Vµ ≡ ‖κ1/2(µ)∇v‖2 + δ‖v‖2

For δ = 0, “classical” complementary variational principle [Ladevèze 83]

(u, p) = arg infw∈Vq∈Q̂µ

‖κ1/2∇w − c1/2p‖2︸︷︷︸constitutive relation

where Q̂µ ≡ {q ∈ H(div; Ω) | f(µ) +∇ · q = 0︸︷︷︸exact conservation

}.

⇒ α(µ) ≡ 1, but requires µ-dependent space Q̂µ (non-trivial RB).

In practiceδ =

1

10minµ∈Ξ⊂D

τLB(µ) > 0,

to “relax” the dual-feasibility condition but to ensure αLB(µ) > 0.9.

⇒ pessimistic SCM bound of τ(µ) does not affect effectivity.

20

Role of δ in ‖v‖2Vµ ≡ ‖κ1/2(µ)∇v‖2 + δ‖v‖2

For δ = 0, “classical” complementary variational principle [Ladevèze 83]

(u, p) = arg infw∈Vq∈Q̂µ

‖κ1/2∇w − c1/2p‖2︸︷︷︸constitutive relation

where Q̂µ ≡ {q ∈ H(div; Ω) | f(µ) +∇ · q = 0︸︷︷︸exact conservation

}.

⇒ α(µ) ≡ 1, but requires µ-dependent space Q̂µ (non-trivial RB).

In practiceδ =

1

10minµ∈Ξ⊂D

τLB(µ) > 0,

to “relax” the dual-feasibility condition but to ensure αLB(µ) > 0.9.

⇒ pessimistic SCM bound of τ(µ) does not affect effectivity.20

Spatio-parameter adaptivity

AlgorithmRemarks

Recap

Given µ ∈ D, we compute

uN ∈ VprN , pN ∈ QprN , (and ψN ∈ V

adjN , qN ∈ Q

adjN )

and evaluate the output and an error bound

|s(µ)− sN(µ)| ≤ ∆N(µ)

≡ 1αLB(µ)︸︷︷︸

stability bound

F pr(uN , pN ;µ)1/2︸︷︷︸

primal residual bound

F adj(ψN , qN ;µ)1/2︸︷︷︸

adjoint residual bound

.

Goal: choose RB spaces VprN , QprN , (and V

adjN , Q

adjN ) that achieve

rapid convergence.

21

Spatio-parameter Greedy

Input: initial FE space VNN=0error tolerance �toltraining set Ξtrain ⊂ D.

Output: online dataset.

For N = 1, . . . , Nmax1. Find least-well approximated parameter

µN+1 = arg supµ∈Ξtrain⊂D

∆N(µ)

2. Compute using adaptive FE

uN (µN+1) ∈ VNN+1

such that ∆N (µN+1) ≤ �tol.3. Augment the reduced basis space

VN+1 = span{VN , uN (µN+1)}22

Step 2: finite element mesh adaptation

Employ

Solve→ Estimate→ Mark→ Refine.

Solve: minimum residual mixed FEM.

Estimate: built-in local error indicator

ηκ(µ) ≡ ‖κ(µ)∇uN (µ)− pN (µ)‖2L2(κ) + δ−1‖f(µ) +∇ · pN (µ)‖2L2(κ) ;

note F (uN (µ), pN (µ);µ) =∑

κ∈Th ηκ(µ).

Mark: top 5% of elements with largest ηκ(µ).

Refine: quad-tree based adaptation.

23

Step 3: quad-tree solution representation

Two spaces: NN ≤ Nmaster,N1. Working mesh (NN): approximate current u(µN).2. Master mesh (Nmaster,N): represent all reduced-basis functions.

Remark: rapid update of master mesh by a quad-tree structure.

+ →

old master mesh working mesh new master mesh

24

Spatio-parameter adaptivity

AlgorithmRemarks

Remark

ECRB provides in Online stage

i. ∀µ ∈ D, exact error bound wrt u ∈ V (not uN ∈ VN );ii. ∀µ ∈ Ξtrain, exact error bound ∆N(µ) ≤ �tol wrt u (not uN ).

ECRB is different from a two-step approach:

1. perform the standard RB Greedy with error bounds wrt “truth”to identify the worst parameter;

2. compute each snapshot using adaptive FEM.

In Online stage, this approach only provide

i. ∀µ ∈ D, bound wrt FE “truth”;ii. bound wrt exact PDE for snapshot parameters.

25

Remark: ECRB 6= (RB with adaptive FE snapshots)

ECRB provides in Online stage

i. ∀µ ∈ D, exact error bound wrt u ∈ V (not uN ∈ VN );ii. ∀µ ∈ Ξtrain, exact error bound ∆N(µ) ≤ �tol wrt u (not uN ).

ECRB is different from a two-step approach:

1. perform the standard RB Greedy with error bounds wrt “truth”to identify the worst parameter;

2. compute each snapshot using adaptive FEM.

In Online stage, this approach only provide

i. ∀µ ∈ D, bound wrt FE “truth”;ii. bound wrt exact PDE for snapshot parameters.

25

Remark: generalizations

1. More sophisticated RB selection

Examples: hp (in parameter), two-stage, online adaptive, . . .

2. More sophisticated FE mesh adaptivity

Examples: hp (in space), anisotropic, . . .

...

26

Numerical results: linear elasticity

Problem descriptionUniform refinementAdaptive refinement

Linear elasticity problem with cracks

Governing equation: linear elasticity.

Parameter domain: D ≡ [0.25, 0.4]× [0.3, 0.7].Output: compliance

∫Γt · uds

7271

1

1

t = 1

27

Linear elasticity problem with cracks

Governing equation: linear elasticity.

Parameter domain: D ≡ [0.25, 0.4]× [0.3, 0.7].Output: compliance

∫Γt · uds

µ = (0.25, 0.3) µ = (0.4, 0.7)

27

Setup

Spatial mesh: P3-RT2 uniform meshes

⇒ ⇒

Snapshot parameters: uniform over D

0.25 0.3 0.35 0.471

0.3

0.4

0.5

0.6

0.7

72

1 2

3 4

⇒

0.25 0.3 0.35 0.471

0.3

0.4

0.5

0.6

0.7

72

1 2 3

4 5 6

7 8 9

⇒

0.25 0.3 0.35 0.471

0.3

0.4

0.5

0.6

0.7

72

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

Stability constant: αmin = 0.9 for δ = 10−4. (Deduced from SCM.)28

Uniform refinement

Parameter refinement:

exponential convergence with N (for N very large);spatial error limits convergence (N−1/2 ∼ h1).

0 5 10 15 20 25 30N

10-2

10-1

100

101

max

72%"

rel

N(7

)

N = 1008N = 3744N = 14400N = 56448N = 223488

29



SetupSCM constructionRB constructionOnline

Setup

µ-training set: 1271 uniform points over D ≡ [0.25, 0.4]× [0.3, 0.7].

Initial mesh: 16-elements P3-RT2 tensor mesh

.

Relative error tolerance: �reltol = 0.01.

(Relative error bound: ∆relN (µ) ≡∆N(µ)

JN −∆N(µ)).

Remark: we need not worry about the fidelity of the “truth.”

30

Offline: N = 1

Parameter: µ1 = (0.32, 0.50)Working mesh: N1 = 4606Master mesh: Nmaster,1 = 4606Resolution: hmin = 5× 10−4

0.25 0.3 0.35 0.471

0.3

0.4

0.5

0.6

0.7

72 1

102 103 104

dof

-0.01

0

0.01

0.02

=(7

=(0

.33,

0.50

))upper boundlower bound

31

Offline: N = 2

Parameter: µ2 = (0.39, 0.57)Working mesh: N2 = 5592Master mesh: Nmaster,2 = 5592

0.25 0.3 0.35 0.471

0.3

0.4

0.5

0.6

0.7

72 1

2

102 103 104

dof

-0.01

0

0.01

0.02

=(7

=(0

.39,

0.57


32

Offline: N = 3

Parameter: µ3 = (0.40, 0.70)Working mesh: N3 = 7776Master mesh: Nmaster,3 = 7776Resolution: hmin = 1.2× 10−4

0.25 0.3 0.35 0.471

0.3

0.4

0.5

0.6

0.7

72 1

2

3

102 103 104

dof

-0.01

0

0.01

0.02

=(7

=(0

.40,

0.70


33

Offline: N = 11


0.25 0.3 0.35 0.471

0.3

0.4

0.5

0.6

0.7

72 1

2

3

4

5

6 7

8

9

10

11

102 103 104

dof

-0.01

0

0.01

0.02

=(7

=(0

.25,

0.30


34

Offline: N = Nmax = 40

Final master mesh: Nmaster,40 = 8502Minimum τ : min

µ∈Ξ⊂DτLB(µ) = 0.00188

⇒ choose δ = 10−4 to obtain αLBmin ≥ 0.95.

lower bound (SCM) upper bound (Galerkin)

35

Offline: N = 1

Parameter: µ1 = (0.40, 0.30)Working mesh: N1 = 12852Master mesh: Nmaster,1 = 12852Resolution: hmin = 2× 10−3

Convergence: N−3 ∼ h6 0.25 0.3 0.35 0.471

0.3

0.4

0.5

0.6

0.7

72

1

103 104

dof

10-2

10-1

100

101

erro

r(7

=(0

:40;

0:3

0))

1%

"NjJref ! JN j

36

Offline: N = 2


0.25 0.3 0.35 0.471

0.3

0.4

0.5

0.6

0.7

72

1

2

103 104

dof

10-2

10-1

100

101

erro

r(7

=(0

:25;

0:7

0))

1%

"NjJref ! JN j

37

Offline: N = 3


0.25 0.3 0.35 0.471

0.3

0.4

0.5

0.6

0.7

72

1

2 3

103 104

dof

10-2

10-1

100

101

erro

r(7

=(0

:40;

0:7

0))

1%

"NjJref ! JN j

38

Offline: N = 4


0.25 0.3 0.35 0.471

0.3

0.4

0.5

0.6

0.7

72

1

2 3

4

103 104

dof

10-2

10-1

100

101

erro

r(7

=(0

:25;

0:3

0))

1%

"NjJref ! JN j

39

Offline: N = Nmax = 33

Final master mesh: Nmaster,33 = 17058Max relative error: max

µ∈Ξ⊂D∆relN (µ) = 0.0096 < 0.01

0 10 20 30N

10-2

10-1

100

max72%"

rel

N(7

)

convergence final error distribution

40

Online: rapid and reliable solution

Online evaluation and certification wrt exact PDE (N = 33)

µ = (0.35, 0.6) → JN(µ) = 6.7273∆relN (µ) = 0.0068 ≤ 0.01 ≡ �tol.

Online time: 0.007 seconds.

Field visualization: 0.2 seconds∗.

41

Summary

Exact-certificate RB: summary

Offline: automatic spatio-parameter adaptivity⇒ efficient approximations [Binev et al; Ainsworth & Oden; . . . ]⇒ reduced man-hour for ROM construction.

Online: error bounds wrt to u(µ) ∈ V (and not uN (µ) ∈ VN ).

Key ingredient: minimum-residual mixed formulation with built-inresidual bounds for PDEs and eigenproblems.

Limitation: error bounds only available for linear coercive equations(error estimate survives for more general PDEs).

0.25 0.3 0.35 0.471

0.3

0.4

0.5

0.6

0.7

72

1

2 3

4

5

6

78

9

10 11

12

13

1415

16

17

18

19

20 21

2223

24

25

26

27

2829

30

31

32

33

42

Backup

Exact-certificate RB: outlook

Automatic spatio-parameter adaptivity become increasingly importantfor complex systems with non-intuitive behaviors.

Large-scale systems [Maday & Rønquist 02; Huynh et al 13; . . . ]reduced-basis element (RBE) framework

Weakly stable and nonlinear dynamicserror bound requires coercivity,

but error estimate survives for more general PDEs

Akselos

43

Diffusion equation → linear elasticity

Norm:

‖v‖2Vµ ≡ ‖K1/2(µ)E∇v‖2 + δ(‖v‖2 + ‖v‖2Γ + ‖K

1/2ref ∇v‖

2).

Bound form:

F (ũ, p;µ) ≡

constitutive relation︷︸︸︷‖K1/2(µ)E∇ũ− C1/2(µ)p‖2

+ δ−1(‖f(µ) +∇ · p‖2︸︷︷︸body force

+ ‖g(µ)− n · p‖2Γ︸︷︷︸boundary force

+ ‖C1/2ref (I − E)p‖2︸︷︷︸

stress symmetrization

).

Stability constant:

α(µ) ≡ infv∈V

‖K1/2(µ)E∇v‖‖v‖2Vµ

=

(1 +

δ

τ(µ)

)−1,

where (c.f. Korn’s inequality)

τ(µ) ≡ infv∈V

‖K1/2(µ)E∇v‖2

‖v‖2 + ‖v‖2Γ + ‖K1/2ref ∇v‖2

.

44

IntroductionMotivationModel problem

Residual boundBound formFinite dimensional approximations

Stability constantProblem description and SCMLower bound of minimum eigenvalueFinite dimensional approximationsRelationship to complementary variational principle

Spatio-parameter adaptivityAlgorithmRemarks

Numerical results: linear elasticityProblem descriptionUniform refinementAdaptive refinement

SummaryBackup

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