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
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
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
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
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
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
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
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
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
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
Problem description and SCMLower bound of minimum eigenvalueFinite dimensional approximationsRelationship to complementary variational principle
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
Problem description and SCMLower bound of minimum eigenvalueFinite dimensional approximationsRelationship to complementary variational principle
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
Problem description and SCMLower bound of minimum eigenvalueFinite dimensional approximationsRelationship to complementary variational principle
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
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
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
Numerical results: linear elasticity
Problem descriptionUniform refinementAdaptive refinement
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
Numerical results: linear elasticity
Problem descriptionUniform refinementAdaptive refinement
SetupSCM constructionRB constructionOnline
Numerical results: linear elasticity
Problem descriptionUniform refinementAdaptive refinement
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
Numerical results: linear elasticity
Problem descriptionUniform refinementAdaptive refinement
SetupSCM constructionRB constructionOnline
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
))upper boundlower bound
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
))upper boundlower bound
33
Offline: N = 11
Parameter: µ11 = (0.25, 0.30)Working mesh: N11 = 3906Master mesh: Nmaster,11 = 8400
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
))upper boundlower bound
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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
Numerical results: linear elasticity
Problem descriptionUniform refinementAdaptive refinement
SetupSCM constructionRB constructionOnline
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
Parameter: µ2 = (0.25, 0.70)Working mesh: N2 = 12516Master mesh: Nmaster,2 = 13614
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
Parameter: µ3 = (0.40, 0.70)Working mesh: N3 = 14676Master mesh: Nmaster,3 = 15810
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
Parameter: µ4 = (0.25, 0.30)Working mesh: N4 = 11466Master mesh: Nmaster,4 = 15972
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
Numerical results: linear elasticity
Problem descriptionUniform refinementAdaptive refinement
SetupSCM constructionRB constructionOnline
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
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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