Uncertainty-quantification tasks are often ``many query'' in nature, as they require repeated evaluations of a model that often corresponds to a parameterized system of nonlinear equations (e.g., arising from the spatial discretization of a PDE). To make this task tractable for large-scale models, low-fidelity models (e.g., reduced-order models, coarse-mesh solutions) must be employed. However, such approximations introduce additional error, which may be treated as a source of epistemic uncertainty that must be quantified to ensure rigor in the ultimate UQ result. We present a new approach to quantify the error (i.e., epistemic uncertainty) introduced by these low-fidelity models approximations. The approach (1) engineers features that are informative of the error using concepts related to dual-weighted residuals and rigorous error bounds, and (2) applies machine learning regression techniques (e.g., artificial neural networks, random forests, support vector machines) to construct a statistical model of the error from these features. We consider both (signed) errors in quantities of interest, as well as global state-space error norms. We present several examples to demonstrate the effectiveness of the proposed approach compared to more conventional feature and regression choices. In each of the examples, the predicted errors have a coefficient of determination value of at least 0.998.

1. Kevin Carlberg
Sandia Na(onal Laboratories
SAMSI MUMS Opening Workshop
Duke University
August 21, 2018
Advances in nonlinear model reduction:
least-squares Petrov–Galerkin projection and
machine-learning error models
Kevin Carlberg
iCME *Talk
Stanford University
May 22, 2017
Breaking computaDonal barriers
Using data to enable extreme-scale simulaDons for many-query problems
-8 -6 -4 -2 0 2
-8
-6
-4
-2
0
2
support vector machine
error predicDon
R2
= 0.990
error
ROM h-refinement
e), b) prolongation (fine)
reduced-order model
high-fidelity model
Sandia Na(onal Laboratories is a mul(mission laboratory managed and operated by Na(onal Technology and Engineering Solu(ons of
Sandia, LLC., a wholly owned subsidiary of Honeywell Interna(onal, Inc., for the U.S. Department of Energy’s Na(onal Nuclear Security
Administra(on under contract DE-NA-0003525.
⇡HFM
post(µ|qmeas)
true
prior
⇡HFM
post (µ | qmeas)
⇡
]HFM
post (µ | qmeas)
⇡
]HFM
post (µ | qmeas)

2. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
High-fidelity simulation
2
computa4onal barrier
+Indispensable across science and engineering
- High fidelity: extreme-scale nonlinear dynamical system models
/38
Kevin CarlbergBreaking computa5onal barriers
High-fidelity simulation
computaDonal barrier
๏ uncertainty propagaDon ๏ mulD-objecDve opDmizaDon
+Indispensable across science and engineering
- High fidelity: extreme-scale nonlinear dynamical system models
Many-query problems
๏ Bayesian inference ๏ stochasDc opDmizaDon
Magnetohydrodynamics
courtesy J. Shadid, Sandia
Turbulent reac5ng flows
courtesy J. Chen, Sandia
Antarc5c ice sheet modeling
courtesy R. Tuminaro, Sandia
2
/38High-fidelity simulation
computaDonal barrier
๏ uncertainty propagaDon ๏ mulD-objecDve opDmizaDon
+Indispensable across science and engineering
- High fidelity: extreme-scale nonlinear dynamical system models
Many-query problems
๏ Bayesian inference ๏ stochasDc opDmizaDon
Magnetohydrodynamics
courtesy J. Shadid, Sandia
Turbulent reac5ng flows
courtesy J. Chen, Sandia
Antarc5c ice sheet modeling
courtesy R. Tuminaro, Sandia
/38High-fidelity simulation
computaDonal barrier
๏ uncertainty propagaDon ๏ mulD-objecDve opDmizaDon
+Indispensable across science and engineering
- High fidelity: extreme-scale nonlinear dynamical system models
Many-query problems
๏ Bayesian inference ๏ stochasDc opDmizaDon
Magnetohydrodynamics
courtesy J. Shadid, Sandia
Turbulent reac5ng flows
courtesy J. Chen, Sandia
Antarc5c ice sheet modeling
courtesy R. Tuminaro, Sandia
๏ uncertainty propagaGon ๏ mulG-objecGve opGmizaGon
๏ Bayesian inference ๏ stochasGc opGmizaGon
Many-query problems

3. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
High-fidelity simulation: captive carry /38
Kevin CarlbergBreaking computa5onal barriers 3
High-fidelity simulation: B61 captive carry
3

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Kevin CarlbergAdvances in nonlinear model reduc4on
High-fidelity simulation: captive carry
๏ explore flight
envelope
๏ quanGfy effects of
uncertainGes on store load
๏ robust design of
store and cavity
computa4onal barrier
Many-query problems
+ Validated and predic(ve: matches wind-tunnel experiments to within 5%
- Extreme-scale: 100 million cells, 200,000 Gme steps
- High simula(on costs: 6 weeks, 5000 cores
3

5. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
Approach: exploit simulation data
4
Idea: exploit simula(on data collected at a few points
D
1. Training: Solve ODE for and collect simulaGon data
2. Machine learning: IdenGfy structure in data
3. Reduc(on: Reduce cost of ODE solve for
Many-query problem: solve ODE for µ 2 Dquery
µ 2 Dtraining
µ 2 Dquery Dtraining
ODE:
dx
dt
= f(x; t, µ), x(0, µ) = x0(µ), t 2 [0, Tfinal], µ 2 D

6. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
Model reduction criteria
1. Accuracy: achieves less than 1% error
2. Low cost: achieves at least 100x computaGonal savings
3. Structure preserva;on: preserves important physical properGes
4. Reliability: guaranteed saGsfacGon of any error tolerance (fail safe)
5. Cer;fica;on: quanGfies ROM-induced epistemic uncertainty
5

7. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
Model reduction: previous state of the art
Linear 4me-invariant systems: mature [Antoulas, 2005]
‣ Balanced truncaGon [Moore, 1981; Willcox and Peraire, 2002; Rowley, 2005]
‣ Transfer-funcGon interpolaGon [Bai, 2002; Freund, 2003; Gallivan et al, 2004; Baur et al., 2001]
+ Accurate, reliable, cer(fied: sharp a priori error bounds
+ Inexpensive: pre-assemble operators
+ Structure preserva(on: guaranteed stability
Ellip4c/parabolic PDEs: mature [Prud’Homme et al., 2001; Barrault et al., 2004; Rozza et al., 2008]
‣ Reduced-basis method
+ Accurate, reliable, cer(fied: sharp a priori error bounds, convergence
+ Inexpensive: pre-assemble operators
+ Structure preserva(on: preserve operator properGes
Nonlinear dynamical systems: ineffecGve
‣ Proper orthogonal decomposiGon (POD)–Galerkin [Sirovich, 1987]
- Inaccurate, unreliable: ogen unstable
- Not cer(fied: error bounds grow exponenGally in Gme
- Expensive: projecGon insufficient for speedup
- Structure not preserved: dynamical-system properGes ignored
6

8. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
Our research
Accurate, low-cost, structure-preserving,
reliable, cer;fied nonlinear model reduc;on
‣ accuracy: LSPG projecGon [C., Bou-Mosleh, Farhat, 2011; C., Barone, AnGl, 2017]
‣ low cost: sample mesh [C., Farhat, CorGal, Amsallem, 2013]
‣ low cost: reduce temporal complexity
[C., Ray, van Bloemen Waanders, 2015; C., Brencher, Haasdonk, Barth, 2017; Choi and C., 2017]
‣ structure preserva(on [C., Tuminaro, Boggs, 2015; Peng and C., 2017; C., Choi, Sargsyan, 2017]
‣ reliability: adapGvity [C., 2015]
‣ cer(fica(on: machine learning error models
[Drohmann and C., 2015; Trehan, C., Durlofsky, 2017; Freno and C., 2017]
7

9. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
Our research
8
Accurate, low-cost, structure-preserving,
reliable, cer;fied nonlinear model reduc;on
‣ accuracy: LSPG projecGon [C., Bou-Mosleh, Farhat, 2011*; C., Barone, AnGl, 2017]
‣ low cost: sample mesh [C., Farhat, CorGal, Amsallem, 2013]
‣ low cost: reduce temporal complexity
[C., Ray, van Bloemen Waanders, 2015; C., Brencher, Haasdonk, Barth, 2017; Choi and C., 2017]
‣ structure preserva(on [C., Tuminaro, Boggs, 2015; Peng and C., 2017; C., Choi, Sargsyan, 2017]
‣ reliability: adapGvity [C., 2015]
‣ cer(fica(on: machine learning error models
[Drohmann and C., 2015; Trehan, C., Durlofsky, 2017; Freno and C., 2017]
Collaborators:
‣ Malhew Barone (Sandia)
‣ Harbir AnGl (GMU)
‣ Charbel Farhat (Stanford University)
‣ Julien CorGal (Stanford University)

10. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
1. Training: Solve ODE for and collect simulaGon data
2. Machine learning: IdenGfy structure in data
3. Reduc(on: Reduce the cost of solving ODE for
µ 2 Dtraining
Training simulations: state tensor
9
dx
dt
= f(x; t, µ)ODE:
µ 2 Dquery Dtraining
D
number of
Gme steps T
number of
state variables N

11. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
1. Training: Solve ODE for and collect simulaGon data
2. Machine learning: IdenGfy structure in data
3. Reduc(on: Reduce the cost of solving ODE for
µ 2 Dtraining
Training simulations: state tensor
9
dx
dt
= f(x; t, µ)ODE:
µ 2 Dquery Dtraining
DX =

12. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
1. Training: Solve ODE for and collect simulaGon data
2. Machine learning: IdenGfy structure in data
3. Reduc(on: Reduce the cost of solving ODE for
Tensor decomposition
10
dx
dt
= f(x; t, µ)ODE:
Compute dominant leR singular vectors of mode-1 unfolding
µ 2 Dtraining
µ 2 Dquery Dtraining
X(1) = = U ⌃ VT
X =

13. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
1. Training: Solve ODE for and collect simulaGon data
2. Machine learning: IdenGfy structure in data
3. Reduc(on: Reduce the cost of solving ODE for
Tensor decomposition
10
dx
dt
= f(x; t, µ)ODE:
Compute dominant leR singular vectors of mode-1 unfolding
columns are principal components of the spa(al simula(on data
How to integrate these data with the computa;onal model?
µ 2 Dtraining
µ 2 Dquery Dtraining
X(1) = = U ⌃ VT
X =

14. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
Previous state of the art: POD–Galerkin
1. Training: Solve ODE for and collect simulaGon data
2. Machine learning: IdenGfy structure in data
3. Reduc(on: Reduce the cost of solving ODE for
µ 2 Dtraining
µ 2 Dquery Dtraining
dx
dt
= f(x; t, µ)ODE:
1. Reduce the number of unknowns 2. Reduce the number of equaGons
D
DGalerkin ODE:
dˆx
dt
= T
f( ˆx; t, µ)
dˆx
dt
) = 0
(
(T
(f( ˆx; t, µ)x(t) ⇡ ˜x(t) = ˆx(t)
11

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Kevin CarlbergAdvances in nonlinear model reduc4on
Captive carry
12
V1
‣ Unsteady Navier–Stokes ‣ Re = 6.3 x 106 ‣ M∞ = 0.6
Spa4al discre4za4on
‣ 2nd-order finite volume
‣ DES turbulence model
‣ degrees of freedom1.2 ⇥ 106
Temporal discre4za4on
‣ 2nd-order BDF
‣ Verified Gme step
‣ Gme instances
t = 1.5 ⇥ 10 3
8.3 ⇥ 103

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Kevin CarlbergAdvances in nonlinear model reduc4on
High-fidelity model solution
13
vor(city field
pressure field
Kevin CarlbergAdvances in nonlinear model reduc4on
0
25
50
23
20
17
13

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Kevin CarlbergAdvances in nonlinear model reduc4on
Principal components
x(t) ⇡ ˆx(t)
1
401
21
101
14

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Kevin CarlbergAdvances in nonlinear model reduc4on
Galerkin performance
15
probe
Can we construct a beEer projec;on?
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
Galerkin: dim 368
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
Galerkin: dim 204
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
high-fidelity:
dim 1.2x106
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
Galerkin: dim 564
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
pressure at probe
1.6
2.0
2.4
2.8
Gme
0 2 4 6 8 10 12
- Galerkin projec(on fails regardless of basis dimension

19. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
Galerkin: time-continuous optimality
ODE Galerkin ODE
dˆx
dt
= T
f( ˆx; t, µ)
+ Time-con(nuous Galerkin solu(on: opGmal in the minimum-residual sense:
dˆx
dt
= T
f( ˆx; t, µ)
- Time-discrete Galerkin solu(on: not generally opGmal in any sense
dx
dt
= f(x; t) f( ˆx; t)
O∆E Galerkin O∆E
rn
(x) := ↵0x t 0f(x; tn
) +
kX
j=1
↵j xn j
t
kX
j=1
j f(xn j
; tn j
)
rn
(xn
) = 0, n = 1, ... , Nn = 1, ... , T T
rn
( ˆxn
) = 0, n = 1, ... , Nn = 1, ... , T
16
r(v, x; t) := v f(x; t)
dˆx
dt
(x, t) = argmin
v2range( )
kr(v, x; t)k2

20. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
Residual minimization and time discretization
ODE
residual
minimiza(on
(me
discre(za(on
dx
dt
= f(x; t)
Galerkin ODE
dˆx
dt
(x, t) = argmin
v2range( )
kr(v, x; t)k2
, n
(ˆxn
)T
rn
( ˆxn
) = 0ˆxn
= argmin
v2range( )
kArn
(v)k2
n
(ˆxn
) := AT
A(↵0I t 0
@f
@x
( ˆxn
; t))
Least-squares Petrov–Galerkin (LSPG) projec(on
residual
minimiza(on
LSPG O∆E
ˆxn
= argmin
v2range( )
kArn
(v)k2
[C., Bou-Mosleh, Farhat, 2011]
n = 1, ... , T
(me
discre(za(on
O∆E
rn
(xn
) = 0
n = 1, ... , T
Galerkin O∆E
T
rn
( ˆxn
) = 0
n = 1, ... , T
17

21. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
Discrete-time error bound
18
Theorem [C., Barone, AnGl, 2017]
If the following condiGons hold:
1. is Lipschitz conGnuous with Lipschitz constant
2. The Gme step is small enough such that ,
3. A backward differenGaGon formula (BDF) Gme integrator is used,
4. LSPG employs , then
f(·; t) 
0 < h := |↵0| | 0| tt
+ LSPG sequen(ally minimizes the error bound
A = I
kxn
ˆxn
Gk2 
1
h
krn
G( ˆxn
G)k2+
1
h
kX
`=1
|↵`|kxn `
ˆxn `
G k2
kxn
ˆxn
LSPGk2 
1
h
min
ˆv
krn
LSPG( ˆv)k2+
1
h
kX
`=1
|↵`|kxn `
ˆxn `
LSPGk2

22. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
LSPG performance
19
probe
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
Galerkin: dim 368
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
Galerkin: dim 204
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
high-fidelity:
dim 1.2x106
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
Galerkin: dim 564
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
LSPG: dim 368
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
LSPG: dim 204FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3LSPG: dim 564
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
pressure at probe
1.6
2.0
2.4
2.8
Gme
0 2 4 6 8 10 12
+ LSPG is far more accurate than Galerkin

23. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
Our research
20
Accurate, low-cost, structure-preserving,
reliable, cer;fied nonlinear model reduc;on
‣ accuracy: LSPG projecGon [C., Bou-Mosleh, Farhat, 2011; C., Barone, AnGl, 2017]
‣ low cost: sample mesh [C., Farhat, CorGal, Amsallem, 2013*]
‣ low cost: reduce temporal complexity
[C., Ray, van Bloemen Waanders, 2015; C., Brencher, Haasdonk, Barth, 2017; Choi and C., 2017]
‣ structure preserva(on [C., Tuminaro, Boggs, 2015; Peng and C., 2017; C., Choi, Sargsyan, 2017]
‣ reliability: adapGvity [C., 2015]
‣ cer(fica(on: machine learning error models
[Drohmann and C., 2015; Trehan, C., Durlofsky, 2017; Freno and C., 2017]

24. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
Wall-time problem
21
probe
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
Galerkin: dim 368
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
Galerkin: dim 204
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
high-fidelity:
dim 1.2x106
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
Galerkin: dim 564
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
LSPG: dim 368
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
LSPG: dim 204FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3LSPG: dim 564
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
pressure at probe
1.6
2.0
2.4
2.8
Gme
0 2 4 6 8 10 12
‣ High-fidelity simula(on: 1 hour, 48 cores
‣ Fastest LSPG simula(on: 1.3 hours, 48 cores
Why does this occur?
Can we fix it?

25. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
1. Training: collect residual tensor while solving ODE for
2. Machine learning: compute residual PCA and sampling matrix
3. Reduc4on: compute regression approximaGon
Cost reduction by gappy PCA [Everson and Sirovich, 1995]
minimize
ˆv
k A rn
( ˆv)k2
k2
Can we select to make this less expensive?A
ˆv)k2
rn
⇡ ˜rn
= r(P r)+
Prn
r P
0 1 2 3 4 5 6 7 8 9 10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 1 2 3 4 5 6 7 8 9 10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
minimize
ˆv
k
k2
rn
(
rn
(
rn
(
rn
(
rn
(˜rn
0 1 2 3 4 5 6 7 8 9 10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 1 2 3 4 5 6 7 8 9 10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
index
value
r
˜rn
rn
Prn
Rijk
µ 2 Dtraining
22

26. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
1. Training: collect residual tensor while solving ODE for
2. Machine learning: compute residual PCA and sampling matrix
3. Reduc4on: compute regression approximaGon
Cost reduction by gappy PCA [Everson and Sirovich, 1995]
minimize
ˆv
k A rn
( ˆv)k2
k2
Can we select to make this less expensive?A
rn
( ˆv)k2 + Only a few elements
of d must be computedrn
rn
⇡ ˜rn
= r(P r)+
Prn
r P
(P r)+
P
0 1 2 3 4 5 6 7 8 9 10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 1 2 3 4 5 6 7 8 9 10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
minimize
ˆv
k
k2
rn
(
rn
(
rn
(
rn
(
rn
(
0 1 2 3 4 5 6 7 8 9 10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 1 2 3 4 5 6 7 8 9 10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
index
value
minimize| {z }
A
r
˜rn
rn
Prn
Rijk
µ 2 Dtraining
22

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Kevin CarlbergAdvances in nonlinear model reduc4on
Sample mesh [C., Farhat, Cortial, Amsallem, 2013]
vor(city field pressure field
LSPG ROM with
32 min, 2 cores
+ 229x savings in core–hours
+ < 1% error in (me-averaged drag
+ HPC on a laptop
sample
mesh
minimize
ˆv
k(P r)+
Prn
( ˆv)k2
A = (P r)+
P
Prn
|{z}
A
high-fidelity
5 hours, 48 cores
Implemented in three computa;onal-mechanics codes at Sandia
23

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Kevin CarlbergAdvances in nonlinear model reduc4on
Ahmed body [Ahmed, Ramm, Faitin, 1984]
24
V1
‣ Unsteady Navier–Stokes ‣ Re = 4.3 x 106 ‣ M∞ = 0.175
Spa4al discre4za4on
‣ 2nd-order finite volume
‣ DES turbulence model
‣ degrees of freedom
Temporal discre4za4on
‣ 2nd-order BDF
‣ Time step
‣ Gme instances
t = 8 ⇥ 10 5
s
1.7 ⇥ 107
1.3 ⇥ 103

29. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
Ahmed body results [C., Farhat, Cortial, Amsallem, 2013]
25
pressure
field
+ 438x savings in core–hours
+ HPC on a laptop
sample
mesh
high-fidelity model
13 hours, 512 cores
LSPG ROM with A = (P r)+
P
4 hours, 4 cores
+ Largest nonlinear dynamical system on which ROM has ever had success

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Kevin CarlbergAdvances in nonlinear model reduc4on
Our research
26
Accurate, low-cost, structure-preserving,
reliable, cer;fied nonlinear model reduc;on
‣ accuracy: LSPG projecGon [C., Bou-Mosleh, Farhat, 2011; C., Barone, AnGl, 2017]
‣ low cost: sample mesh [C., Farhat, CorGal, Amsallem, 2013]
‣ low cost: reduce temporal complexity
[C., Ray, van Bloemen Waanders, 2015; C., Brencher, Haasdonk, Barth, 2017; Choi and C., 2017]
‣ structure preserva(on [C., Tuminaro, Boggs, 2015; Peng and C., 2017; C., Choi, Sargsyan, 2017]
‣ reliability: adapGvity [C., 2015]
‣ cer(fica(on: machine learning error models
[Drohmann and C., 2015; Trehan, C., Durlofsky, 2017; Freno and C., 2017]

31. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
Our research
27
Accurate, low-cost, structure-preserving,
reliable, cer;fied nonlinear model reduc;on
‣ accuracy: LSPG projecGon [C., Bou-Mosleh, Farhat, 2011; C., Barone, AnGl, 2017]
‣ low cost: sample mesh [C., Farhat, CorGal, Amsallem, 2013]
‣ low cost: reduce temporal complexity
[C., Ray, van Bloemen Waanders, 2015; C., Brencher, Haasdonk, Barth, 2017; Choi and C., 2017]
‣ structure preserva(on [C., Tuminaro, Boggs, 2015; Peng and C., 2017; C., Choi, Sargsyan, 2017]
‣ reliability: adapGvity [C., 2015]
‣ cer(fica(on: machine learning error models
[Drohmann and C., 2015; Trehan, C., Durlofsky, 2017; Freno and C., 2017]

32. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
Our research
28
Accurate, low-cost, structure-preserving,
reliable, cer;fied nonlinear model reduc;on
‣ accuracy: LSPG projecGon [C., Bou-Mosleh, Farhat, 2011; C., Barone, AnGl, 2017]
‣ low cost: sample mesh [C., Farhat, CorGal, Amsallem, 2013]
‣ low cost: reduce temporal complexity
[C., Ray, van Bloemen Waanders, 2015; C., Brencher, Haasdonk, Barth, 2017; Choi and C., 2017]
‣ structure preserva(on [C., Tuminaro, Boggs, 2015; Peng and C., 2017; C., Choi, Sargsyan, 2017]
‣ reliability: adapGvity [C., 2015]
‣ cer(fica(on: machine learning error models
[Drohmann and C., 2015; Trehan, C., Durlofsky, 2017; Freno and C., 2017]

33. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
Our research
Accurate, low-cost, structure-preserving,
reliable, cer;fied nonlinear model reduc;on
‣ accuracy: LSPG projecGon [C., Bou-Mosleh, Farhat, 2011; C., Barone, AnGl, 2017]
‣ low cost: sample mesh [C., Farhat, CorGal, Amsallem, 2013]
‣ low cost: reduce temporal complexity
[C., Ray, van Bloemen Waanders, 2015; C., Brencher, Haasdonk, Barth, 2017; Choi and C., 2017]
‣ structure preserva(on [C., Tuminaro, Boggs, 2015; Peng and C., 2017; C. and Choi, 2017]
‣ reliability: adapGvity [C., 2015]
‣ cer(fica(on: machine learning error models
[Drohmann and C., 2015; Trehan, C., Durlofsky, 2017; Freno and C., 2018]
29
Collaborators:
‣ MarGn Drohmann (formerly Sandia)
‣ Wayne Uy (Cornell University)
‣ Fei Lu (Johns Hopkins University)
‣ Malhias Morzfeld (U of Arizona)
‣ Brian Freno (Sandia)

34. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
Surrogate modeling in UQ
30
outputsinputs µ surrogate model qsurr
‣ surrogate noise model:
‣ surrogate likelihood:
- inconsistent with HFM noise model
qmeas = qsurr(µ) + "
⇡surr(qmeas | µ) = ⇡"(qmeas qsurr(µ))
⇡"(·)
‣ high-fidelity-model (HFM) noise model:
‣ measurement noise has probability distribuGon
‣ HFM likelihood:
"
outputsinputs µ high-fidelity model qHFM
qmeas = qHFM(µ) + "
⇡HFM(qmeas | µ) = ⇡"(qmeas qHFM(µ))

35. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
Surrogate modeling in UQ
31
qHFM(µ) = qsurr(µ) + (µ)
‣ HFM noise model:
‣ HFM likelihood: ⇡HFM(qmeas | µ) = ⇡"(qmeas qHFM(µ))
= ⇡"(qmeas qsurr(µ) (µ))
qmeas = qHFM(µ) + "
= qsurr(µ) + (µ) + "
+ equivalent to HFM formulaGon
+ not pracGcal: the (determinisGc) error is generally unknown(µ)
How can we account for the error in a manner that is
consistent and prac;cal?
(µ)

36. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
Surrogate modeling in UQ
32
qHFM(µ) = qsurr(µ) + (µ)
Approach: sta(s(cal model for the error that models its uncertainty˜(µ)
˜qHFM(µ)
| {z }
stochastic
= qsurr(µ)
| {z }
deterministic
+ ˜(µ)
|{z}
stochastic
‣ staGsGcal HFM noise model: qmeas = ˜qHFM(µ) + "
= qsurr(µ) + ˜(µ) + "
+ consistent with HFM noise model
+ pracGcal if the staGsGcal error model is computable
⇡]HFM
(qmeas | µ) = ⇡"+˜(qmeas qsurr(µ))‣ stochasGc HFM likelihood:
˜
Desired proper4es in sta4s4cal error model
1. cheaply computable: similar cost to evaluaGng the surrogate
2. low variance: introduces lille epistemic uncertainty
3. generalizable: correctly models the error
˜(µ)
How can we construct a sta;s;cal error model for reduced-order models?

37. /38
Kevin CarlbergAdvances in nonlinear model reduc4on
Approximate-solution surrogate models
33
r(x(µ); µ) = 0<latexit 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MEw+7Cx12QPr1p1/Qt/3WvtqyIYi4cOypeStCHFReNuytFFPmG3a84UUTxT44xXwoGusCPvMJlOD1rRV3lejREiA7aNJFUyu7HIsMLyugU0X2nzNt2rghCe7NHhClbntbIftJLQ+jCvR6cCyV53+GzMMVqQOG1K7yTcpynhnuEUGOmP3rKiiYYRdmGdwK0QZXZReeueQc3iaaxSMZ20pKjG86T+RM6313jWjTD3Ki4K+2Q52dWqb/JIuxTo7NTT1p5/jFpxkzQ6txCsmRmQvjMuCNBZRG+ASRjdnjOHdOKtk5u7iOCVzB8dmRZwy9oXtLcJpQ1fWjuXcktX2z/Wg4v2NEdkkCrBWYnY1UXHtPlBjgiAFOwcmosZ4CCSbTYKwPpc0Ap/gIGPbLDH8WD3moSdZMZWt52opzoEliw6qw/QlIG9DhKOAhKOvBJmMf6g200fdDL5gLsmeODtkFmlSOM5ysjFo0i/SfKQ7nJOk7Hf5TVunrTGTdrscumsaDG6LvOU3gNp3lzyWyLOlZ0Debq3MmXK424ZdKfmdDma0NOcFgtrEkyhMcCVy21YgivJWbbrZuhGWbjnkxkCLEWASB6ktZAzAFmOAGwGY1l8AcZXrftuigKRk693eCBF5RU9zVvOc+PoAIjn5490bP+eWAja5weTMdZuQOR0OB7y6LsAxJEsOyjBMaw/SAK8gODJrGVQqaItKE0hiqFESOmgClAIKF4tAdDvWOHGFu0S/A8g/ihLUBI8SzCWDGRIaQpRDCXhEuS388jwQu+BQalWSC2CsXGazOjbNXjA8pltogTw/SBsI40y99whWEKd2KbSEqYoHcJqTZfJvCrzDI2vUCngeGaXt4fc1oLHb3UkiFSOfYTfAakguTPGO1j3RTagUiim5ODe8dvX1Ndg98IAWZ1EkVvoNK/D3S5QptIjpbPCQLmRqPDKHscwvDMIAtwNEnGdkUi0W7ZMFHcNJBbcbh5z97ttbMdbhNMb+wInTtA2STkMKicam6dF4lxwZ+tFtjoFk9Q+D/tOS+IxUBgs2X/0Rajy02xVajN7hT7Z3/nGTkd9+mEJS6t2XuIx3WlxgtsYrLa76YoMLHe1g8MJnjtIoi6ZzEdWfUIOp2AY9psMOPS/QDFxQ4x3xZBnUJDuY6nHfsmXj7YdCXtxllHjidO8SzEhWK8B8ewVSbjZwJ0DIPZrHpyNHOhNAzrMuAoTwvpPEzWc6GMazcR/hMfSBLA6cI3mpgT5G50uGSB3F+G2XquyPG0YOq3jY2/BZzNod9fmq1ABubeZ3KYY5EINMngixUTZTY7bEGxAlAi0B++BIRMBVKuUuKpWVZaBqjWzHaxf6CBOSI6qaXuVzV+3dATvNSZxxqYzMb2/upScAEEZZTxmsz58liGtihkz5ZVXMl9pabZIVoiiqtp8DlW1iYDyISSGzzdR//Wc77ywImQBRLEb47+tYVNVe7NT1ELIaSag38C8T3hUdjtFqDu4HaFDFzB9C747tdImYPjWnofsA0ZhS6LH1iKfUBcPfNcLTpyh7BIeY3AiOyoRpuoHdwAR4nRGSIh5aMMUURVxnZBdC7wSq2iQa+IsluWpJPvCwVScSFIuuxhL3AFpap0+8p08kqeOWB3y79DwTkXwt5mEBTKUIZM98ontYiXbS/Dnfyi67YtzP13PqqmRjchdZFAT2AVJbEYkRWjkXtH6a8XZDZY2BNRfx2VC3rp8uly8xR2FXOV+I+5Bxh3AMqLsSvHEWIhRgDK4wm1FDtsTGSJbLXGq2qlz1UsCZLXJJ1BjkolyhBGnAJjno/R1YHtkuLLCwsRrlcT3IRtlgng3t+RpUI3smpm7HziKS3MJTXZWwN5IB30TyzYNQTQb8dTe9zxx3+4r9v20Ka8kUReDAjJI0YqeOijm4EEhgdU8LtL+OBp4wQ23xX57lzgg4hna5Skg+DQMmjBHAjjd4serifM+53UwjDtbwHEYvzJ2FsJV1DsIafGw+n6Wm6lQTnwnvLoduhIHq9pGEaDLcJmqXOcVZuTnDyoUl+M52wFgL9arY/BKMvlVAz5QOa+kTCaIBShZ+rHHfUWvyU8QFGmCSv1NJ3xXrQNJ4NN4MZp0IbCWCSTw269OE4VlOK66lR5w2S3kaZ5e0/eokMOJE6/JEyeEHRCnXMkACJ7oAALvYP4yc3qaRwhVGyy5FasdHbpaGNho8QipziGHnV+JczcCUcUuicm+dOmNaGVCjMsIH4CmbrNLCpyBJm6GS7COQduToW6fXPPofL4laxnrLVUpQ8xFXhGwGYzvZsNjhut9qffJUl/aGinFYXdj3UTfrSmTPEsmme62H4fvbToSZK6+HAGhhZfE2csvR0rodmmeLnnf+YVtt1N/wn4b+FgeKuCxM+jdkxqI/DoFldloQh2pchffCgqLIwsesCZcz7AubIU+IBTdUTlnShME29X+i/PuStqKxa15dpn0+JLFA8kNHfKX9xonYq9u2zfmbdQD109MsuswgdzK98lnr/01o5SX5it0vVwn9h42VGO6+65jzx6vo86hKhHtahn8qK1g8hiPsLfdBHtLxV37CK9HI/jfJshTtjbXyrjVPKdghR1tF55bQte+kjF43uqU2lezvbVDrir4tvFneAaeidNO+rW2545KDeuqQh00hNos1beK6s8h6pJDJuQgj8sKneXsbxkw54t7igqa94XvcIM2YTT2tjBLo5EP72vqKnWjUryAnpHAWy00jcV/LLGB0I/OchQJYavUaUEr3cphYwStRCDDHUW04P5rYJCRjYucBaR4H0aF3zgSslzBxSitSV45SFudmh3mF2w7c9jzRSVeIIhZDRNyYMbnMDXxy3zYAqD8cIz+Omm23gvDbB0yAf6JitDbsgHD/eaSgksR+TWze5D0zB0BmXNzZWfHdGypJPuoicEPUc03qNk/ZsARhiTyJjOoWKW/o35PBXDAwl6UE0M0/cFQHTjqIe0e5W6Rcx8uWPKW7T4hrjAwH1IaD4OSyKNzmVDHsJoj/VXUYxjslsuLQG2S2y8N+Y/5mEYc5FFUM41FA+h/3YQxXrFNWOrDF/YVgKM3T8+wWPJfN4EzyUeYSQEidxzGf6PM7PdkK0m8upUdYDXSl8YK0jmOR4aW2R0XQnuc1RDLvKriEsWM/tR2wvHKkRyyYamFXVn8zPcc0Yva0KDH9BeoXMQVPcAM/rKIEDSLK3V+7MQFkfflVuRiJbNnbE8P8XjOUuwWUKMF3ayQWbHrjLWtVyib1XN3K1x7L/FZyYZh7RGfsugpdCcOwJ42QVCBbskP4/QvG/rBCQPUuPqMn295bxdNFhfVPK9F/PC9E/inEt5l0zydqP2x97hP0zYHotlpO6HSrg2JsYWeBIVq75T1vQDTBNriXwTF8+1GIfqlF2SLfRkUKyITUvBpAGaLPlWi3Zv4pBWxJkXx7K0y7PJ8oMFcLeJbGhC94ijynRyKED8NRgQAr1UgV30qvFzE2RA/28Vfumfi6FUhAOh6rxhzHacWTLz5D1Z+jcYUnKgiWvqO9Io4jDHf/9kbURGTi80NEOTr9iw5ZKCkv7+I9G2tppzhwWPnm8e9ncff9Hb29v+w35o4PfbK0Xvf7Pd2vvnDt1/3dnb3LDjdEeE4nE7+ZyH4G+Y5Zv9rrHb3j/vf9nb29/+IoSUyXZ9FkuLhfhovU/pTW1Px+fKzjT3715ndD+ePdvZ2d/Z+2N/407/zX27+7dq/rP3b2tba3tq3a39ae7nWX3u3Nl77z7X/Wvvvtf/59O2ni0+TT1MG/fWvOOf3a8a/T/n/AZHNsGc=</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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MEw+7Cx12QPr1p1/Qt/3WvtqyIYi4cOypeStCHFReNuytFFPmG3a84UUTxT44xXwoGusCPvMJlOD1rRV3lejREiA7aNJFUyu7HIsMLyugU0X2nzNt2rghCe7NHhClbntbIftJLQ+jCvR6cCyV53+GzMMVqQOG1K7yTcpynhnuEUGOmP3rKiiYYRdmGdwK0QZXZReeueQc3iaaxSMZ20pKjG86T+RM6313jWjTD3Ki4K+2Q52dWqb/JIuxTo7NTT1p5/jFpxkzQ6txCsmRmQvjMuCNBZRG+ASRjdnjOHdOKtk5u7iOCVzB8dmRZwy9oXtLcJpQ1fWjuXcktX2z/Wg4v2NEdkkCrBWYnY1UXHtPlBjgiAFOwcmosZ4CCSbTYKwPpc0Ap/gIGPbLDH8WD3moSdZMZWt52opzoEliw6qw/QlIG9DhKOAhKOvBJmMf6g200fdDL5gLsmeODtkFmlSOM5ysjFo0i/SfKQ7nJOk7Hf5TVunrTGTdrscumsaDG6LvOU3gNp3lzyWyLOlZ0Debq3MmXK424ZdKfmdDma0NOcFgtrEkyhMcCVy21YgivJWbbrZuhGWbjnkxkCLEWASB6ktZAzAFmOAGwGY1l8AcZXrftuigKRk693eCBF5RU9zVvOc+PoAIjn5490bP+eWAja5weTMdZuQOR0OB7y6LsAxJEsOyjBMaw/SAK8gODJrGVQqaItKE0hiqFESOmgClAIKF4tAdDvWOHGFu0S/A8g/ihLUBI8SzCWDGRIaQpRDCXhEuS388jwQu+BQalWSC2CsXGazOjbNXjA8pltogTw/SBsI40y99whWEKd2KbSEqYoHcJqTZfJvCrzDI2vUCngeGaXt4fc1oLHb3UkiFSOfYTfAakguTPGO1j3RTagUiim5ODe8dvX1Ndg98IAWZ1EkVvoNK/D3S5QptIjpbPCQLmRqPDKHscwvDMIAtwNEnGdkUi0W7ZMFHcNJBbcbh5z97ttbMdbhNMb+wInTtA2STkMKicam6dF4lxwZ+tFtjoFk9Q+D/tOS+IxUBgs2X/0Rajy02xVajN7hT7Z3/nGTkd9+mEJS6t2XuIx3WlxgtsYrLa76YoMLHe1g8MJnjtIoi6ZzEdWfUIOp2AY9psMOPS/QDFxQ4x3xZBnUJDuY6nHfsmXj7YdCXtxllHjidO8SzEhWK8B8ewVSbjZwJ0DIPZrHpyNHOhNAzrMuAoTwvpPEzWc6GMazcR/hMfSBLA6cI3mpgT5G50uGSB3F+G2XquyPG0YOq3jY2/BZzNod9fmq1ABubeZ3KYY5EINMngixUTZTY7bEGxAlAi0B++BIRMBVKuUuKpWVZaBqjWzHaxf6CBOSI6qaXuVzV+3dATvNSZxxqYzMb2/upScAEEZZTxmsz58liGtihkz5ZVXMl9pabZIVoiiqtp8DlW1iYDyISSGzzdR//Wc77ywImQBRLEb47+tYVNVe7NT1ELIaSag38C8T3hUdjtFqDu4HaFDFzB9C747tdImYPjWnofsA0ZhS6LH1iKfUBcPfNcLTpyh7BIeY3AiOyoRpuoHdwAR4nRGSIh5aMMUURVxnZBdC7wSq2iQa+IsluWpJPvCwVScSFIuuxhL3AFpap0+8p08kqeOWB3y79DwTkXwt5mEBTKUIZM98ontYiXbS/Dnfyi67YtzP13PqqmRjchdZFAT2AVJbEYkRWjkXtH6a8XZDZY2BNRfx2VC3rp8uly8xR2FXOV+I+5Bxh3AMqLsSvHEWIhRgDK4wm1FDtsTGSJbLXGq2qlz1UsCZLXJJ1BjkolyhBGnAJjno/R1YHtkuLLCwsRrlcT3IRtlgng3t+RpUI3smpm7HziKS3MJTXZWwN5IB30TyzYNQTQb8dTe9zxx3+4r9v20Ka8kUReDAjJI0YqeOijm4EEhgdU8LtL+OBp4wQ23xX57lzgg4hna5Skg+DQMmjBHAjjd4serifM+53UwjDtbwHEYvzJ2FsJV1DsIafGw+n6Wm6lQTnwnvLoduhIHq9pGEaDLcJmqXOcVZuTnDyoUl+M52wFgL9arY/BKMvlVAz5QOa+kTCaIBShZ+rHHfUWvyU8QFGmCSv1NJ3xXrQNJ4NN4MZp0IbCWCSTw269OE4VlOK66lR5w2S3kaZ5e0/eokMOJE6/JEyeEHRCnXMkACJ7oAALvYP4yc3qaRwhVGyy5FasdHbpaGNho8QipziGHnV+JczcCUcUuicm+dOmNaGVCjMsIH4CmbrNLCpyBJm6GS7COQduToW6fXPPofL4laxnrLVUpQ8xFXhGwGYzvZsNjhut9qffJUl/aGinFYXdj3UTfrSmTPEsmme62H4fvbToSZK6+HAGhhZfE2csvR0rodmmeLnnf+YVtt1N/wn4b+FgeKuCxM+jdkxqI/DoFldloQh2pchffCgqLIwsesCZcz7AubIU+IBTdUTlnShME29X+i/PuStqKxa15dpn0+JLFA8kNHfKX9xonYq9u2zfmbdQD109MsuswgdzK98lnr/01o5SX5it0vVwn9h42VGO6+65jzx6vo86hKhHtahn8qK1g8hiPsLfdBHtLxV37CK9HI/jfJshTtjbXyrjVPKdghR1tF55bQte+kjF43uqU2lezvbVDrir4tvFneAaeidNO+rW2545KDeuqQh00hNos1beK6s8h6pJDJuQgj8sKneXsbxkw54t7igqa94XvcIM2YTT2tjBLo5EP72vqKnWjUryAnpHAWy00jcV/LLGB0I/OchQJYavUaUEr3cphYwStRCDDHUW04P5rYJCRjYucBaR4H0aF3zgSslzBxSitSV45SFudmh3mF2w7c9jzRSVeIIhZDRNyYMbnMDXxy3zYAqD8cIz+Omm23gvDbB0yAf6JitDbsgHD/eaSgksR+TWze5D0zB0BmXNzZWfHdGypJPuoicEPUc03qNk/ZsARhiTyJjOoWKW/o35PBXDAwl6UE0M0/cFQHTjqIe0e5W6Rcx8uWPKW7T4hrjAwH1IaD4OSyKNzmVDHsJoj/VXUYxjslsuLQG2S2y8N+Y/5mEYc5FFUM41FA+h/3YQxXrFNWOrDF/YVgKM3T8+wWPJfN4EzyUeYSQEidxzGf6PM7PdkK0m8upUdYDXSl8YK0jmOR4aW2R0XQnuc1RDLvKriEsWM/tR2wvHKkRyyYamFXVn8zPcc0Yva0KDH9BeoXMQVPcAM/rKIEDSLK3V+7MQFkfflVuRiJbNnbE8P8XjOUuwWUKMF3ayQWbHrjLWtVyib1XN3K1x7L/FZyYZh7RGfsugpdCcOwJ42QVCBbskP4/QvG/rBCQPUuPqMn295bxdNFhfVPK9F/PC9E/inEt5l0zydqP2x97hP0zYHotlpO6HSrg2JsYWeBIVq75T1vQDTBNriXwTF8+1GIfqlF2SLfRkUKyITUvBpAGaLPlWi3Zv4pBWxJkXx7K0y7PJ8oMFcLeJbGhC94ijynRyKED8NRgQAr1UgV30qvFzE2RA/28Vfumfi6FUhAOh6rxhzHacWTLz5D1Z+jcYUnKgiWvqO9Io4jDHf/9kbURGTi80NEOTr9iw5ZKCkv7+I9G2tppzhwWPnm8e9ncff9Hb29v+w35o4PfbK0Xvf7Pd2vvnDt1/3dnb3LDjdEeE4nE7+ZyH4G+Y5Zv9rrHb3j/vf9nb29/+IoSUyXZ9FkuLhfhovU/pTW1Px+fKzjT3715ndD+ePdvZ2d/Z+2N/407/zX27+7dq/rP3b2tba3tq3a39ae7nWX3u3Nl77z7X/Wvvvtf/59O2ni0+TT1MG/fWvOOf3a8a/T/n/AZHNsGc=</latexit>
High-fidelity model
‣ governing equaGons:
‣ quanGty of interest:
Types of approximate solu4ons
‣ Reduced-order model:
‣ Low-fidelity model:
‣ Inexact solu(on: compute such that
rLF(xLF; µ) = 0, ˜x = p(xLF)<latexit 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</latexit><latexit 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
T
r( ˆx; µ) = 0, ˜x = ˆx
x(k)
, k = 1, ... , K
kr(x(K)
; µ) = 0k2  ✏, ˜x = x(K)
Approximate-solu4on surrogate model
‣ approximate soluGon:
‣ quanGty of interest:
˜x(µ) ⇡ x(µ)
qsurr(µ) := q(˜x(µ))
qHFM(µ) := q(x(µ))
What methods exist for quan;fying the error ?(µ) := qHFM(µ) qsurr(µ)