parametric manifold models for validation; veriﬁcation for...

Parametric Manifold Modelsfor Validation;

Verification forParametric Manifold Models

Anthony T PateraMassachusetts Institute of Technology

NRC Symposium on VVUQWashington, DC28 March 2012

Acknowledgements Collaborators

P Huynh* D Knezevic* M Dihlmann

J Eftang E Rønquist*

S Vallaghe K Urban

T Tonn

Also S Boyaval, A-L Gerner, M Grepl, B Haasdonk, J Hesthaven, KC Hoang,C Le Bris, T Leurent, GR Liu, Y Maday, C Nguyen, J Penn, A Quarteroni,G Rozza, K Steih, K Veroy, K Willcox, and M Yano.

NRC Symposium on VVUQ AT Patera 2

Acknowledgements Sponsors

AFOSR/Office of Secretary of Defense

Office of Naval Research

MIT-Singapore International Design Center

Deshpande Center for Technological Innovation


Acknowledgements AFOSR/OSD MURI(F Fahroo)

Multi-Scale Fusion of Information UncertaintyQuantification and Management

in Large-Scale Simulations

Brown: K Karniadakis, J Hesthaven, B RozovskyCaltech: T HouCornell: N Zabaras

MIT: A Patera, K Willcox


FrequentistValidation Framework

with“Exact”† Model

†Exact in the sense of verification, not validation.


“Exact” Model

Given µ ∈ D ⊂ RP , µ: parameter P -vector

umodel( · ;µ) ∈ X(Ω) ,

for Ω a domain in Rd.

Note the field umodel(x;µ) may derive from aparametrized partial differential equation.†

†In an example we also consider time-dependent fields.


Measurements of Physical System

Given measurement functionals,

`measi : X → R, 1 ≤ i ≤ m ,

assume experiments yield outputs

Y expi = `meas

i (uphy)︸︷︷︸Y

phyi

+ ε(ωi), 1 ≤ i ≤ m ,

for ε(ω) a zero-mean Gaussian process.

Note uphy(x) serves to “interpret” Y phyi .


Validation Hypothesis

For given µ ∈ D, δ ≥ 0

H(µ):

For all µ′ ∈ Nδ(µ) ,

‖Y phy − Y model(µ′)‖ ≤ δ .

Here Y modeli (µ′) = `meas

i (umodel( · ;µ′)), 1 ≤ i ≤ m;Nδ(µ) a neighborhood in D about µ.

Note: H(µ) FALSE, ∀µ ∈ D ⇒ “unmodeled physics.”


Statistical Test

For any µ ∈ D,

REJECT H(µ) vs. ACCEPT H(µ)

if and only if

Fε(Y exp, Y model(µ)) > Bδ(µ) ;

require PrErrorType I ≤ 1− γ.†

† H(µ) TRUE: PrREJECT H(µ) ≤ 1− γ.


FrequentistValidation Framework

withApproximate Model


Approximate Model

Given µ ∈ D ⊂ RP , µ: parameter P -vector

umodel( · ;µ) ∈ X(Ω) ⊂ X(Ω) ,

where u(µ) ≈ umodel(µ).

Introduce Y modeli (µ) = `meas

i (u( · ;µ)), 1 ≤ i ≤ m.


Measurements of Physical System

Given measurement functionals,

`measi : X → R, 1 ≤ i ≤ m ,

assume experiments yield outputs

Y expi = `meas

i (uphy)︸︷︷︸Y

phyi

+ ε(ωi), 1 ≤ i ≤ m ,

for ε(ω) a zero-mean Gaussian process.

Note uphy(x) is an “interpretation” of Y phyi .


Validation Hypothesis

For given µ ∈ D, δ ≥ 0

H(µ):

For all µ′ ∈ Nδ(µ) ,

‖Y phy − Y model(µ′)‖ ≤ δ .

Here Y modeli (µ′) = `meas

i (umodel( · ;µ′)), 1 ≤ i ≤ m;Nδ(µ) a neighborhood in D about µ.

Note: H(µ) FALSE, ∀µ ∈ D ⇒ “unmodeled physics.”


Statistical Test

For any µ ∈ D,


if and only if

Fε(Y exp, Y model(µ)) > Bδ(µ)Verification

; (≥ Bδ(µ))




Statistical Test

For many µ ∈ D,


if and only if

Fε(Y exp, Y model(µ)) > Bδ(µ)Verification

; (≥ Bδ(µ))




Parametric Manifold Models

Approximation Space


Parametric ManifoldMh

Introduce uh(µ) ∈ Xh ⊂ X: a highly accuratefinite element (FE) surrogate for umodel(µ).

. ..Greedy Samplingsnapshots

uuhh((˜µµ11))uuhh((˜µµ22))

uuhh((˜µµNN ))

XXhh

MMhh ≡≡ uuhh((µµ)) µµ ∈∈ DD

Approximation Space: XD = spanuh(µn), 1 ≤ n ≤ N.NRC Symposium on VVUQ AT Patera 17


Approximation Space

Reduced Basis (RB)


Galerkin Projection

Given weak form (parametrized PDE),

µ ∈ D → A(w, v;µ), ∀ w, v in X(Ω) ,†

uRB ∈ XD satisfies

A(uRB, v;µ) = 0, ∀ v ∈ XD .

Advantage: N = dim(XD) dim(Xh) .

†Note umodel(µ) ∈ X: A(umodel(µ), v;µ) = 0, ∀ v ∈ X;uh(µ) ∈ Xh: A(uh(µ), v;µ) = 0, ∀ v ∈ Xh.


A Posteriori Error Estimation

For any µ ∈ D,

‖uRB(µ)− umodel(µ)‖X≤ ‖uRB(µ)− uh(µ)‖X + ‖uh(µ)− umodel(µ)‖X≈ ‖uRB(µ)− uh(µ)‖X ; OFFLINE-ONLINE

provide rigorous error bound

‖uRB(µ)− uh(µ)‖ ≤ ∆hRB(µ) ,

where ∆hRB(µ) ≡ Ch

stability(µ) ‖A(uRB, · ;µ)‖(Xh)′ .


OFFLINE-ONLINE Computational Strategy

OFFLINE Construct XD (once): expensive.

ONLINE Evaluate µ→ uRB, ∆hRB: inexpensive.

In the many-query (or real-time) context, amortizeOFFLINE effort over many ONLINE evaluations.

Note: ONLINE cost independent of dim(Xh),hence choose Xh conservatively rich.



Approximation Space

Reduced Basis (RB)

Empirical Interpolation (EI)


Collocation

OFFLINE: Given XD, identifyquasi optimal interpolation points xk;associated characteristic functions Vk ∈ XD;

for 1 ≤ k ≤ N .

ONLINE: For µ ∈ D, form uEI(µ) ∈ XD as

uEI(x;µ) =N∑k=1

uh(xk;µ)Vk(x) .†

Advantage: Ω ⊂ Rd replaced by xkk=1,...,N .

†In practice, EI applied to functions of approximations of uh(µ).


Parameter Manifold Models

Approximation Space

Reduced Basis (RB)

Empirical Interpolation (EI)

Dirty Laundry


Issues

Difficulty Palliative

Stability Constant, Chstability Successive Constraint

P 1: µ ∈ D ⊂ RP Domain Decomposition in Ω

D Large: µ ∈ D ⊂ RP Domain Decomposition in D

Nonpolynomial Nonlinearity Empirical Interpolation

Long-Time Integration Space-Time Inf-Sup


Numerical Example


Cast of Characters

“Exact” Model:

(µ1, µ2) ∈ D → umodel(t, x;µ) ∈ X

FE Surrogate: dim(Xh) ≈ 3,000

(µ1, µ2) ∈ D → uh(t, x;µ) ∈ Xh

RB Approximate Model: dim(XD) ≤ 40 †

(µ1, µ2) ∈ D → uRB(t, x;µ) ∈ XD, ∆hRB(t, µ)

Physical System: uphy(t, x)

†RB 100× faster (ONLINE) than FE.


Case I: Model

1100

1100

ΩΩ

ΓΓinin

zero

flux

zero

flux

ΓΓ00

ΓΓ∞∞

ΩΩmeme

00.5.500.5.5

ΩΩ11µµ11 ΩΩ00

µµ00 == 11ΩΩ22µµ22

∂∂uummoodeldel

∂∂nn== 11

YY momodedellii ((µµ)) ==

11

——ΩΩmeme —— ΩΩmeme

uumomodedell((ttii,, xx;;µµ)) ddxx

∂∂uummoodeldel

∂∂tt−−∇∇ ·· ((µµkk∇∇uumomodeldel)) == 00 in ΩΩkk

Parametrization: µ ≡ (µ1, µ2) ∈ [0.2, 5]2 ≡ D.


Case I: Physical System(fully modeled physics)

1100

1100

ΩΩ

ΓΓininze

ro fl

uxze

ro fl

ux

ΓΓ00

ΓΓ∞∞ΩΩmeme

00.5.500.5.5

ΩΩ11ΩΩ00ΩΩ22

∂∂uuphphyy

∂∂nn== 11

αα11 == 44αα22 == 11 αα00 == 11

∂∂uuphphyy

∂∂tt−−∇∇ ·· ((ααkk∇∇uuphphyy)) == 00 inin ΩΩkk

YY expexpii ==

11

——ΩΩmmee —— ΩΩmeme

uuphphyy((ttii,, xx)) ddxx

++ ((ttii;;ωωii))unknown

Gaussian process

uphy(t, x) = umodel(t, x;µ∗ ≡ (4, 1)(α1,α2)

) †.†In practice, umodel(t, x;µ∗) replaced by uh(t, x;µ∗).


Case I: Validation(noise: ≈ 1%, uncorrelated)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.51

1.52

2.53

3.54

4.55

µµ∗∗

µµ11

µµ22

PrPr REJECT HH ((µµ∗∗ ≤≤ 00..0505))reject HH when HH is true

REJECT

ACCEPT


Case I: Effort

Experiment (synthetic):

m = 10 repeats× 100 measurements in time

= 1,000 data in total.

Computation: 10,000 ONLINE evaluations

µ→ Y model(µ), ∆hRB(µ), Bδ(µ)

⇒ ACCEPT or REJECT H(µ).


Case II: Model

1100

1100

ΩΩ

ΓΓinin

zero

flux

zero

flux

ΓΓ00

ΓΓ∞∞

ΩΩmeme

00.5.500.5.5

ΩΩ11µµ11 ΩΩ00

µµ00 == 11ΩΩ22µµ22

∂∂uummoodeldel

∂∂nn== 11

YY momodedellii ((µµ)) ==

11

——ΩΩmeme —— ΩΩmeme

uumomodedell((ttii,, xx;;µµ)) ddxx

∂∂uummoodeldel

∂∂tt−−∇∇ ·· ((µµkk∇∇uumomodeldel)) == 00 in ΩΩkk

Parametrization: µ ≡ (µ1, µ2) ∈ [0.2, 5]2 ≡ D.


Case II: Physical System(unmodeled physics: crack)

1100

1100

ΩΩ

ΓΓinin

zero

flux

zero

flux

ΓΓ00

ΓΓ∞∞

ΩΩmeme

00.5.500.5.5

ΩΩ11ΩΩ00ΩΩ22

∂∂uuphphyy

∂∂nn== 11

αα11 == 44αα22 == 11 αα00 == 11

∂∂uuphphyy

∂∂tt−−∇∇ ·· ((ααkk∇∇uuphphyy)) == 00 inin ΩΩkk

YY expexpii ==

11

——ΩΩmmee —— ΩΩmeme

uuphphyy((ttii,, xx)) ddxx

++ ((ttii;;ωωii))unknown

Gaussian process

ΓΓcracracckk

uphy(t, x) 6= umodel(t, x;µ) for any µ ∈ D.NRC Symposium on VVUQ AT Patera 33

Case II: Validation(noise: ≈ 1%, uncorrelated)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.51

1.52

2.53

3.54

4.55

µµ11

µµ22

REJECT

ACCEPT

µµ∗∗“ ”

Crack Length = 1: crack conflated withlower conductivity medium.


Case II: Validation(noise: ≈ 1%, uncorrelated)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.51

1.52

2.53

3.54

4.55

REJECT

µµ∗∗“ ”

µµ11

µµ22

Crack Length = 1.5: unmodeled physics detected.


AFOSR/OSD MURI

Multi-Scale Fusion of Information UncertaintyQuantification and Management

in Large-Scale Simulations

Brown: K Karniadakis, J Hesthaven, B RozovskyCaltech: T HouCornell: N Zabaras

MIT: A Patera, K Willcox


MURI Architecture

while (model insufficient)

propose/update model: µPDE, SPDE;

develop Reduced Order Model (ROM);ISOMAP, ANOVA, RB/EI, MEPC, . . .

verify ROM;

validate model⇐ Fε (data, ROM);

end

design under uncertainty⇐ ACCEPTED H(µ);


MURI Domains & Applications

Thermofluids

Acoustics (for Noise Mitigation . . . )

Electromagnetics (for Stealth . . . )

(Multiscale) Materials


References


Frequentist Validation

V Chew, Confidence, prediction, and tolerance regions for the multivariatenormal distribution. J Am Stat Ass 61(315): 605–617, 1966.

O Balci, R Sargent, Validation of simulation models via simultaneousconfidence intervals. Am J Math Manage Sci 4: 375–406, 1984.

J McFarland, S Mahadevan, Multivariate significance testing and modelcalibration under uncertainty. Comput Methods Appl Mech Engrg 197(29– 32):2467–2479, 2008.

DBP Huynh, DJ Knezevic, AT Patera, Certified reduced basis model validation:A frequentistic uncertainty framework. Comput Methods Appl Mech Engrg201–204:13–24, 2012.



AK Noor, Recent advances in reduction methods for nonlinear problems.Comput Struct 13:31–44, 1981.

BO Almroth, P Stern, FA Brogan, Automatic choice of global shape functions instructural analysis. AIAA J 16:525–528, 1978.

TA Porsching, Estimation of the error in the reduced basis method solution ofnonlinear equations. Math Comput 45(172):487–496, 1985.

L Machiels, Y Maday, IB Oliveira, AT Patera, DV Rovas, Output Bounds forReduced-Basis Approximations of Symmetric Positive Definite EigenvalueProblems. CR Acad Sci Paris Series I 331:153–158, 2000.

G Rozza, DBP Huynh, AT Patera, Reduced Basis Approximation and APosteriori Error Estimation for Affinely Parametrized Elliptic Coercive PartialDifferential Equations — Application to Transport and Continuum Mechanics.Arch Comput Methods Eng 15(3):229–275, 2008.

M Barrault, Y Maday, NC Nguyen, AT Patera, An ‘Empirical Interpolation’Method: Application to Efficient Reduced-Basis Discretization of PartialDifferential Equations. CR Acad Sci Paris Series I 339:667–672, 2004.


ATP Research Group:

augustine.mit.edu

AFOSR/OSD/MURI:

www.dam.brown.edu/AFOSR MURI 2009.htm


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