Stochastic spectral likelihood embedding for the calibration of heat transfer models

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Stochastic spectral likelihood embedding for the calibration of heat transfer models

Author(s):

Wagner, Paul-Remo; Marelli, Stefano; Sudret, Bruno Publication Date:

2021

Permanent Link:

https://doi.org/10.3929/ethz-b-000477805

Rights / License:

In Copyright - Non-Commercial Use Permitted

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Stochastic spectral likelihood embedding for the calibration of heat transfer models

WCCM-ECCOMAS 2020

P.-R. Wagner, S. Marelli, B. Sudret

Chair of Risk, Safety and Uncertainty Quantification | ETH Zürich

Motivation

Heat transfer model for timber structures under fire:

• _Restrict the spread of fire;

• _Simulate time-dependent heat evolution in structural components with component additive method (CAM);

• _{Based on} temperature-dependent parameters ( λ(T ) , c(T ) and ρ(T ) );

• _Determine λ(T ) , c(T ) and ρ(T ) with measurements of time-dependent heat evolution.

Inverse problem: Bayesian model calibration

framework

Outline

Bayesian model calibration

Stochastic spectral likelihood embedding Heat transfer problem

Conclusions

Framework

Consider a computational model M with input parameters X ∼ π(x) and measurements Y , the Bayesian inverse problem reads:

π(x|Y) = L(x; Y)π(x)

Z ^where Z =

Z

D_X

L(x; Y)π(x)dx

with:

• L : D

→ R

^: likelihood function (measure of how well the model fits the data)

• π(x|Y) : posterior density function

Outline

Bayesian model calibration

Stochastic spectral likelihood embedding Heat transfer problem

Conclusions

Stochastic spectral embedding

For Y = L (X ) with E

Y

< ∞:

Y ≈ X

k∈K

1

(X)R

(X), where R

(X)

= X

α∈A^k

a

Ψ

(X ) ≈ R

.

Sequential partitioning approach

• Static experimental design X

• Algorithm:

1. initialize K = {(0, 1)}

2. for terminal domains k ∈ T ⊆ K do:

2.1 Partition domain D

in half: D

2.2 Expand R

to R

2.3 Add k

to K

3. Stop if fewer than N

points in domain

[ Marelli et al. (2020) ]

Stochastic spectral embedding - adaptive enrichment

Y ≈ X

k∈K

1

(X)R

(X), where R

(X)

= X

α∈A^k

a

Ψ

(X ) ≈ R

.

Sequential partitioning approach

• Adaptive experimental design enrichment

• Algorithm:

1. initialize K = {(0, 1)}

2. for refinement domain k = arg max

{E

} do:

2.1 Partition domain D

in half: D

2.2

X 2.3 Expand R

to R

2.4 Add k

to K

3. Stop if computational budget exhausted

[ Wagner et al. (2020) ]

Stochastic spectral embedding - the details

Refinement domain selection Refine domain

k = arg max

k∈T

{E

} where

E

= V

E

with domain-wise probability mass V

and leave-one-out error E

Partitioning strategy Split along direction

d = arg max

i∈{1,···,M}

E

where

E

= Var

R

(X

)

− Var

R

(X

)

.

Stochastic spectral likelihood embedding

After expanding the likelihood with SSLE as L(X) ≈ P

k∈K

1

(X)R

(X), the full posterior distribution or the following quantities of interest can be computed analytically:

Post-processing a

Z = E [L(X)] ≈ X

k∈K

V

a

π(x|Y) ≈ π(x) Z

X

k∈K

1

(x)R

(x)

E [h(X)|Y] ≈ 1 Z

X

k∈K

V

· X

α∈A^k

a

b

after h(x) ≈ X

α∈A^k

b

Ψ

(x)

[ Nagel et al. (2016), Wagner et al. (2020) ]

Outline

Bayesian model calibration

Stochastic spectral likelihood embedding Heat transfer problem

Conclusions

Heat transfer problem

• Model as 1D heat transfer problem under standard ISO-fire curve

• N independent time-dependent temperature measurements at interface and discrete times

Y = {T

, · · · ,T

}

where T

= {T

, · · · , T

} stores measurements at t

= 401 time steps

• Computational forward model solves the heat equation (FE-method) and returns temperature at discrete time steps

• For present study: polynomial chaos expansion of

FE-model

Heat transfer problem - parameterization

Parameter Physical Meaning Range Unit

X

Start of second key process [300, 800]

C

X

Main effect of second key process [0.1, 1] -

(relative between X

and 850

C)

X

λ(180

C) and λ(X

) [0.1, 0.25]

/

X

λ(1200

C) [0.1, 1.2]

/

X

c(140

C) [1.4 · 10

, 6.5 · 10

]

/

X

c(X

+ (850

C − X

) · X

) [1 · 10

, 8 · 10

]

/

Heat transfer problem - Bayesian calibration

• Likelihood with X

= (X

, . . . , X

) and N = 5 measurement time series:

L(X; Y) =

N

Y

i=1

N (T

|M(X), Σ),

where Σ = σ

· I

and σ = 100

C

• Prior distributions:

π(X) =

6

Y

i=1

U(X

, X

)

Posterior moments and correlations

SSLE ( 10

L evaluations) vs. MCMC ( 10

L evaluations)

Moments, SSLE vs. MCMC

E[X_i|Y]

p

Var [X_i|Y]

X₁ 534597 116128

X₂ 0.5930.627 0.2630.25 X₃ 1.45·10⁻⁴

1.58·10⁻⁴

2.11·10⁻⁵ 2.15·10⁻⁵ X₄ 8.28·10⁻⁴

8.63·10⁻⁴

2.49·10⁻⁴ 2.3·10⁻⁴ X₅ 2.99·10⁴

3.22·10⁴

5.4·10³5·10³

X6 1.6·10⁴ 1.76·10⁴

1.05·10⁴ 1.15·10⁴

Correlation

Posterior marginals

SSLE ( 10

L evaluations) vs. MCMC ( 10

L evaluations)

Posterior mean predictions

Outline

Bayesian model calibration

Stochastic spectral likelihood embedding Heat transfer problem

Conclusions

Conclusion

• Bayesian inversion is a powerful tool for model calibration ;

• Stochastic spectral likelihood embedding aims at avoiding any MCMC sampling, by expressing the likelihood function as a sum of local, residual expansions;

• Cost of method can be optimized by allowing non equal splitting and rotation.

Conclusion

Chair of Risk, Safety & Uncertainty Quantification www.rsuq.ethz.ch

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