[PPT] - Lecture 2 Error Estimation and Control for Problems with Uncertain PowerPoint Presentation

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Lecture 2 Error Estimation and Control for Problems with Uncertain Coefficients

Serge Prudhomme

D´ epartement de math´ ematiques et de g´ enie industriel Polytechnique Montr´ eal DCSE Fall School 2019 TU Delft, The Netherlands, November 4-8, 2019

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eal) Problems with Uncertain Coefficients November 4-8, 2019 1 / 53

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Outline

Error estimation for PDEs with uncertain coefficients. Adaptive scheme. Numerical examples.

“. . . It is not possible to decide (a) between h or p refinement and (b) whether one should enrich the approximation space Vh or Sh . . . better approaches, yet to be conceived, are consequently needed.”

Spectral Methods for Uncertainty Quantification, Le Maˆ ıtre & Knio 2010

A few words about validation. Application of GOEE to Bayesian Inference. Numerical examples.

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Introduction

Motivation

A(λ; u) = f(λ) → Q(u(λ))

M(λ)=Q(u)

λ

1

λ2 QN(u)

Surrogate model MN M ≈ MN(λ) = Q(uN) Ah(λ; uh) = fh(λ) → Q(uh(λ))

Mh(λ)=Q(uh)

λ

1

λ2 Qh,N(u)

Surrogate model Mh,N Mh ≈ Mh,N(λ) = Q(uh,N)

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Introduction

References

Le Maˆ ıtre et al., 2007, 2010

Polynomial chaos, Stochastic Galerkin, Burger’s equation

Almeida and Oden, 2010

convection-diffusion, sparse grid collocation

Butler, Dawson, and Wildey, 2011

Stochastic Galerkin, PC representation of the discretization error (ignore truncation error)

Butler, Constantine, and Wildey, 2012

Ignore physical discretization error, pseudo-spectral projection, improved linear functional

. . . Bryant, Wildey, Prudhomme, SIAMJUQ, 2015

Pseudo-spectral projection method, adaptivity with respect to quantities of interest

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Model Problem

Model Problem and Discretization

Model Problem: A(λ; u) = f(λ), ∀x ∈ D where uncertain coefficients λ assumes the following representation: λ = λ(ξ), ∀ξ ∈ Ω ∈ Rn (or Ξ) Weak formulation for a given ξ: Find u(ξ) ∈ V such that Bξ(u, v) = Fξ(v) ∀v ∈ V Finite element approximation: Find uh(ξ) ∈ V h ⊂ V such that Bξ(uh, vh) = Fξ(vh) ∀vh ∈ V h

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Model Problem

Surrogate Approximation

Assume that the surrogate approximation of uh is given by: uh,N(x, ξ) =

N

i=0

uh

i (x)Ψi(ξ)

where: uh

i (x) ∈ V h ⊂ V , ∀i = 0, . . . , N.

Ψi(ξ) is a basis function in a finite subpace of L2(Ω). The space L2(Ω) is endowed with the norm: vL2(Ω) =

Ω

(v(ξ))2ρ(ξ)dξ 1/2 where ρ(ξ) is the probability density function of ξ.

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Goal-oriented error estimation

Goal-oriented error estimation (linear case)

Quantity of interest (QoI): Qξ(u) =

D

k(x)u(x, ξ) dx Adjoint problem: Find p(·, ξ) ∈ V such that Bξ(v, p) = Qξ(v) ∀v ∈ V Error representation: Qξ(u) − Qξ(uh) = Fξ(p) − Bξ(uh, p) := Rξ(uh; p)

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Goal-oriented error estimation

Error estimator: Qξ(u) − Qξ(uh) = Rξ(uh; p) ≈ η(ξ) Orthogonality property: If ph ∈ V h then Rξ(uh; ph) = 0 Higher-order approximation of adjoint solution: Compute ˜ p(ξ) ∈ V , V h ⊂ V ⊂ V and η(ξ) = Rξ(uh; ˜ p) Other choices Local interpolation: Rξ(uh; p) ≈ Rξ(uh; ˜ πph − ph) Residual based: Rξ(uh; p) = Bξ(eu, ep) ≈ ηu(ξ)ηp(ξ)

1Becker & Rannacher 2001, Oden & Prudhomme, 2001

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Goal-oriented error estimation

Case with Uncertain Parameters

Goal is to estimate:

Qξ
u
− Qξ
uh,N
L2(Ω)

Adjoint solution: p ≈ ˜ pM(x, ξ) =

M

i=0

˜ pi(x)Ψi(ξ) where: ˜ pi(x) ∈ V , i = 0, . . . , M, with V h ⊂ V ⊂ V . Ψi(ξ) are the basis functions in a finite subpace of L2(Ω). However, we can choose M = N.

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Goal-oriented error estimation

Error Estimate

η =

Rξ(uh,N, ˜

pN)

L2(Ω)

where: Rξ(uh,N, v) = Fξ(v) − Bξ(uh,N, v) Such an estimate requires a large number of residual evaluations for the computation of the L2 norm. Instead we compute: Rξ(uh,N, ˜ pN) ≈

M

i=0
Rξ(uh,N, ˜

pN)

iΨi(ξ) := E(ξ)

and consider estimate of total approximation error as: η ≈ E(ξ)L2(Ω)

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Goal-oriented error estimation

Error decomposition

Decomposition: Qξ

u
− Qξ
uh,N

= Qξ

u
− Qξ
uh
error due to

physical discretization

+ Qξ

uh

− Qξ

uh,N
error due to approx

in parameter space

Bound on total error:

Qξ
u
− Qξ
uh,N
L2(Ω) ≤
Qξ
u
− Qξ
uh
L2(Ω)

+

Qξ
uh

− Qξ

uh,N
L2(Ω)
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Goal-oriented error estimation

Estimations of error constributions

Physical space discretization error: Qξ

u
− Qξ
uh

≈ Rξ

uh; ˜

p

But:

Rξ

uh; ˜

p

≈

N

i=0
Rξ
uh; ˜

p

i

Ψi(ξ) := ED(ξ) with M = N. With the same expansion order for uh,N, ˜ pN, and ED, the coefficients of all three responses can be computed simultaneously. Then, by quadrature with m points:

Qξ
u
− Qξ
uh
L2(Ω) ≈

m

k=1
ED(ξk)
2ωk

1/2 Parameter space discretization error: Qξ

uh

− Qξ

uh,N

≈ E(ξ) − ED(ξ) := EΩ(ξ) ⇒

EΩ(ξ)
L2(Ω)
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Adaptive Strategy

Adaptivity Strategy

if

ED
L2(Ω) >
EΩ
L2(Ω)

Refine physical approximation space V h (h ← h

2 )

else

Refine random approximation space SN (N ← N + 1)

end for a given physical mesh, refine approximation in Ω to the level of physical discretization error use error indicator to guide h refinement in parameter space anisotropic p-refinement in higher dimensions

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non-Intrusive approach

Pseudo-spectral projection method

Model Problem: A(λ; u) = f(λ), ∀x ∈ D where uncertain coefficients λ assumes the following representation: λ = λ(ξ), ∀ξ ∈ Ω ∈ Rn (or Ξ) Non-intrusive approach (“pseudo-spectral projection method”): u(x, ξ) ≈

N

k=0

uk(x)Ψk(ξ) ≈

N

k=0

um

k (x)Ψk(ξ) := uN(x, ξ)

where uk(x) :=

Ω

u(x, ξ)Ψk(ξ) ρ(ξ) dξ ≈

m(N)

j=1

u(x, ξj) Ψk(ξj) wj := um

k (x)

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non-Intrusive approach

Model Problem and Discretization

Gaussian quadrature: Select quadrature rule {ξj, wj}m(N)

j=1

according to probability density ρ. Parameterized discrete solution (the surrogate/reduced model): Solve for uh(x, ξj). Then: uh,m

k

(x) =

m(N)

j=1

uh(x, ξj) Ψk(ξj) wj and uh,N(x, ξ) =

N

k=0

uh,m

k

(x)Ψk(ξ)

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Numerical results

Example 1: Smooth response surface in 2D

Convection-diffusion problem in 2D: −∇ · (2∇u) + 10 sin π

2 ξ1

10 cos (πξ2)
· ∇u = f(ξ)

in D = (0, 1)2 u = 0

n ∂D

Loading f is chosen such that, with ξ1, ξ2 ∼ U(0, 1): u(x, y, ξ) = 400

ξ1(x − x2)e− 20

ξ1 (x−ξ1)2

ξ2(y − y2)e− 20

ξ2 (y−ξ2)2

Quantity of interest: Q(u(·, ξ)) = 1 4 1

0.5

1

0.5

u(x, y, ξ) dxdy ≈

D

q(x, y) u(x, y, ξ) dxdy

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Numerical results

Example 1: Effectivity indices

EΩL2

Ω

EDL2

Ω

EL2

Ω

EL2

Ω

Q(u)−Q(uh,p,N )L2

Ω

5.12427e-01 3.28574e-03 4.34727e-01 .851 1.79962e-01 3.48349e-03 1.95149e-01 .782 5.23817e-02 6.59002e-03 4.25596e-02 .921 2.30547e-02 3.77558e-03 2.85842e-02 .949 6.17006e-03 5.77325e-03 8.41438e-03 .998 2.21929e-03 4.48790e-03 7.25161e-03 .987 2.20458e-03 3.98680e-04 2.80610e-03 .984 7.00606e-04 4.31703e-04 9.24221e-04 .990 3.58282e-04 4.13817e-04 8.06397e-04 1.01 3.58118e-04 1.47612e-04 5.17592e-04 1.03 1.38497e-04 1.49756e-04 2.71081e-04 1.11 8.78811e-05 2.61145e-05 1.06502e-04 1.02 5.10997e-05 2.59334e-05 7.73100e-05 1.00 1.34534e-05 2.59640e-05 3.87553e-05 .985 1.33674e-05 1.22096e-05 2.57607e-05 .981

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Numerical results

Example 2: Response surface with discontinuity

Convection-diffusion model in 2D: −2∆u + sin( 3π

2 ξ1)

4⌊ξ2 − ξ1⌋

· ∇u = f(ξ)

in D = (0, 1)2 u = 0

n ∂D

Loading f is chosen so that u(x, y, ξ) = 10 sin 3π 2 ξ1

4⌊ξ2 − ξ1⌋
· (x − x2)(y − y2)

where ⌊ξ2 − ξ1⌋ =

ξ1 ≤ ξ2

−1 ξ1 > ξ2 with ξ1, ξ2 ∼ U(0, 1).

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Numerical results

Example 2: Response surface with discontinuity

0.2 0.4 0.6 0.8 1 0.5 1 −1 −0.5 0.5 1

ξ1 ξ2 bx(ξ)

0.2 0.4 0.6 0.8 1 0.5 1 −4 −2 2 4

ξ1 ξ2 by(ξ)

Dependence of the convection velocity over parameter space in x and y direction.

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Numerical results

Example 2: Response surface with discontinuity

Q(u(·, ξ)) = u 1 3, 1 3, ξ

≈
D

q(x, y) u(x, y, ξ) dxdy q(x, y) = 100 π exp

− 100(x − 1/3)2 − 100(y − 1/3)2

0.2 0.4 0.6 0.8 1 0.5 1 −2 −1 1 2 ξ1 ξ2

True response for QoI over parameter space.

Adaptive scheme: If EΩiL2

Ωi > 0.75 maxj

EΩj
L2

Ωj

split Ωi into 2n new elements by bisection in each stochastic direction end

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Numerical results

Example 2: Response surface with discontinuity

h-refinement of parameter space

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ξ1 ξ2 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ξ1 ξ2 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ξ1 ξ2 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ξ1 ξ2 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ξ1 ξ2 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ξ1 ξ2

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Numerical results

Example 2: Response surface with discontinuity

Adaptive hΩ refinement

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Total error ξ1 ξ2 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Parameter space mesh ξ1 ξ2

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Numerical results

Example 2: Response surface with discontinuity

p-refinement of parameter space

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Total error ξ1 ξ2 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5

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Numerical results

Example 2: Convergence of true error

100 101 102 103 104 Number of Evaluations 10−2 10−1 100

||E||L2(Ξ)

−1

2

−1

4

||ED||L2(Ξ) = O(10−4) Uniform h-refinement Adaptive h-refinement p-refinement

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Numerical results

Example 3: Flow at low Reynolds numbers

Navier-Stokes equations: −ν∆u + u · ∇u + ∇p = 0 ∇ · u = 0 u = uin, x ∈ Γin u = 0, x ∈ Γw ∪ Γcyl σ · n = 0, x ∈ Γo Parameterization of uncertainty:    ν = ξ1 uin,x = ξ2 3 32(4 − y2)

−2 −1 1 2 3 4 5 6 −2 −1 1 2

Γw Γw Γi Γo Γcyl

QoI: Qξ (u) = ux (x0, ξ) Let ξ1 ∼ U(0.01, 0.1), ξ2 ∼ U(1, 3) s.t. Re = ξ2 8ξ1 ∈ [1.25, 37.5]

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Numerical results

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Numerical results

Example 3: Flow at low Reynolds numbers

−2 −1 1 2 3 4 5 6 −2 −1 1 2

−2 −1 1 2 3 4 5 6 −2 −1.5 −1 −0.5 0.5 1 1.5 2 −2 −1 1 2 3 4 5 6 −2 −1.5 −1 −0.5 0.5 1 1.5 2 −2 −1 1 2 3 4 5 6 −2 −1.5 −1 −0.5 0.5 1 1.5 2

Sequence of physical meshes obtained from adaptive refinement procedure.

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Numerical results

Example 3: Flow at low Reynolds numbers

0.02 0.04 0.06 0.08 0.1 1 2 3 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ξ1 ξ2 Q(u(⋅,ξ))

Approximate response surface for QoI computed at the final stage of adaptive refinement.

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Numerical results

Example 3: Flow at low Reynolds numbers

104 105 106 107 Total Degrees of Freedom 10−4 10−3 10−2 10−1

||E||L2(Ξ) ||Q||L2(Ξ)

Ω and Ξ Refinement Adaptive Refinement Ξ Refinement Ω Refinement

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Numerical results

Statistical quantities of interest (sQoI)

Which features of Q(u) are we interested in? Moments: S(u) = Q(u) S(u) = Var [Q(u)] Probability of failure: S(u) = P [Q(u) > tol] =

Ω

1{Q(u)>tol}ρ(ξ) dξ If S is nonlinear, e.g. variance ES = Var [Q(u)] − Var

Q(uh,N)
= Var
Q(u) − Q(uh,N)
≈ Var
EQ
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Numerical results

Statistical quantities of interest (sQoI)

P ({Qξ(u) > tol}) =

Ω

χ{Qξ(u)>tol}ρ(ξ)dξ ≈

Ω

K (Qξ(u))ρ(ξ)dξ := S(u)

−2 −1 1 2 Qξ(u)-TOL 0.0 0.5 1.0

χ K

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Conclusions

Concluding remarks and future work

Error representation for the total error in surrogate models and contributions from each approximation space. Development of adaptive refinement strategies based on error decomposition. Extension to statistical QoI (sQoI), such as probabilities of failure.

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Validation

A few words about validation

Questions pertaining to validation

How good is the model? Modeling is more of an art. “A model is a simplification or an approximation of reality and hence will not reflect all of reality.” Burnham and Anderson, 2002

If one wants to be philophical. . .

What is a good model? “All models are wrong, but some are useful” George Box, 1976 To answer those questions, one must have an objective in mind. Quantities of interest: Specific objectives that can be expressed as the target outputs

f a model, defined as functionals
f the solutions:

Q(u) =

Ω

k(x)u(x)dx

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Validation

DECISION KNOWLEDGE Errors Discretization MODELS COMPUTATIONAL Errors Observational VERIFICATION VALIDATION PHYSICAL REALITIES THE UNIVERSE

f

Modeling Errors MODELS THEORY / MATHEMATICAL OBSERVATIONS

Oden, Moser, and Ghattas, SIAM News, (Nov. 2010) Oden and Prudhomme, IJNME (Sept. 2010)

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Validation

Definitions:

Verification:

The process of determining the accuracy with which a computational model can produce results deliverable by the mathematical model on which it is based. ⇒ Code and Solution Verification

Validation:

The process of determining the accuracy with which a model can predict observed physical events (or the important features of a physical reality).

P. Roache (2009): “The process of determining the degree to which a model (and its

associated data) is an accurate representation of the real world from the perspective of the intended uses of the model”.

Uncertainty Quantification:

The process of determining the degree of uncertainty in the prediction of the QoI. Typically, the degree of uncertainty is given by the probability distribution for the QoI.

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Validation

Control of Errors

Errors are all a matter of comparison! Code Verification: Using the method of “manufactured solutions”, for example, we can easily compare the computed solution with the manufactured solution. Solution Verification: In this case, the solution of the problem is unknown and one can use convergence (uniform or adaptive methods) to assess the accuracy of the approximate solutions. Calibration Process: Comparison of observable data with model estimates of the observables. Validation Process: The main idea behind validation is to know whether a model can be used for prediction purposes. ⇒ What should we compare in this case?

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Validation Process

1. Calibration/Paremeter Identification:

Identification of values of parameters of a model designed to bring model results into agreement with measurements.

2. Validation:

The process of determining the accuracy with which a model can predict observed physical events (or the important features of a physical reality).

3. Prediction:

The forecast of an event (a predicted event cannot be measured or

bserved, for then it ceases to be a prediction).
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Validation Process

The Prediction Pyramid for validating a single model

f QoI

Prediction QP

If model predicts QoI within prescribed accuracy the model is said to be not invalidated

SV Prediction Scenario Validation Scenario(s) SC Calibration Scenarios SP

Experimental Data DC

n calibration

scenarios Experimental Data

n validation

scenario(s) DV (As simple as possible, numerous) (more complex than calibration) No Data Available

The Validation Pyramid

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Validation Process

Classical Approach for Validation

Solve Forward Problem

n

Validation Scenario M O ( , γ ) <

?

Yes No

c Dv

Solve Calibration

θ

c

Parameter(s) with Model Problem Inverse

n

Scenario Unknown Parameter(s)

θ

Calibration Validation Dv Oc Observables Estimate

c

D Data

v

D Data Calibrated Model with Model not Invalid Increased Confidence Model is Invalid

M

Ocal, Dval
≤ γtol (?)
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Validation Process

Proposed Validation Process (2009)

M Q ( , γ ) <

?

Yes No

c Qv

Solve Calibration Validation Solve

θ

c

θv

Model with Parameter(s) Parameter(s) with Parameter(s)

θ

with Model Problem Inverse

n

Scenario Problem Inverse

n

Scenario Same Model Unknown Parameter(s)

θ

Calibration Validation Scenario Prediction

n

Problem Forward Solving the QoI Estimate Quantification Uncertainty Sensitivity/ Data D Data D

c v

Q Qv

c

Model with Re−Calibrated Calibrated Model not Invalid Increased Confidence Model is Invalid

M

Qcal, Qval
≤ γtol (?)
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Efficient Bayesian inference - model selection for RANS

Bayesian inference for RANS using surrogate models

Objective The main objective here is to develop a methodology based on response surface models and goal-oriented error estimation for efficient and reliable parameter estimation in turbulence modeling. Efficient Bayesian inference: Approximate response surface models can be used to reduce the computational cost of the process. Ma and Zabaras, 2009. Marzouk et al., 2014; Le Maˆ ıtre et al., etc. etc. UQ for RANS models: Uncertainty in RANS model parameters is a known issue in the turbulence community, but quantifying the effect of this uncertainty is seldom analyzed in the CFD literature. Cheung et al., 2011. Oliver and Moser, 2011. Prudhomme and Bryant, AMSES, 2015.

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Efficient Bayesian inference - model selection for RANS

Fully developed incompressible channel flow

Mean flow equations: u = U + u′      DUi Dt = −1 ρ ∂P ∂xi + ∂ ∂xj

ν ∂Ui

∂xj − u′

iu′ j

∇ · U = 0

Eddy viscosity model of Reynolds stress (closure model): u′

iu′ j = −νT

∂Ui ∂xj + ∂Uj ∂xi

Channel equations: assuming homogeneous turbulence in x

∂ ∂y

(ν + νT )∂U

∂y

= 1,

y ∈ (0, H)

1Durbin and Petterson Reif, 2001; Pope, 2000

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Efficient Bayesian inference - model selection for RANS

Spalart-Allmaras (SA) model

Eddy viscosity is given by: νT = ˜ νfv1, fv1 = (˜ ν/ν)3/((˜ ν/ν)3 + cv13), where ˜ ν is governed by the transport equation D˜ ν Dt = P˜

ν(κ, cb1) − ε˜ ν(κ, cb1, σSA, cw2)

+ 1 σSA ∂ ∂xj

(ν + ˜

ν) ∂˜ ν ∂xj

+ cb2

∂˜ ν ∂xj ∂˜ ν ∂xj

with

P˜

ν = production term

D˜

ν = wall destruction term

Parameter Values κ 0.41 cb2 0.622 cb1 0.1355 cv1 7.1 σSA 2/3 cw2 0.3

1Allmaras, Johnson, and Spalart, 2012; Oliver and Darmofal, 2009

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Forward model: Find U and ˜ ν such that

       1 = ∂ ∂y

(ν + νT (cv1))∂U

∂y

0 = P˜

ν(κ, cb1) − ε˜ ν(κ, cb1, σSA, cw2) +

1 σSA ∂ ∂y

(ν + ˜

ν)∂˜ ν ∂y

+ cb2

∂˜ ν ∂y 2

Boundary conditions (H = half-height of channel): U(0) = 0, ∂yU(H) = 0, ˜ ν(0) = 0, ∂y˜ ν(H) = 0 Weak formulation: Find (U, ˜ ν) ∈ V s.t. B((U, ˜ ν); (V, µ)) = F(V, µ), ∀(V, µ) ∈ V Quantity of interest and adjoint problem: Find (Z, ζ) ∈ V s.t. B′((U, ˜ ν); (Z, ζ), (V, µ)) = Q(V, µ) = H

0 V dy

∀(V, µ) ∈ V

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Adaptive Response Surface

Uniform distributions for all parameters in (0.5, 1.5) × nominal value QoI scenario is at Reτ = 5000, only adapt in parameter space. Adapted expansion order (after 17 iterations): κ cb1 σSA cb2 cv1 cw2 6 3 3 1 2 2

100 101 102 103 104 Number of evaluations 10−3 10−2 10−1 100 101 102 ||E||L2(Ξ) adaptive uniform

10 20 30 40 50 60 70 80 Q 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 surrogate full model

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Bayesian inference

Bayes rule: p(ξ|q) = L(ξ|q) p(ξ) p(q) where:        q ∈ Rn = Calibration data L(ξ|q) = [p(q|ξ)] = Likelihood p(ξ) = Prior p(ξ|q) = Posterior The denominator in Bayes theorem acts as a normalization constant and can be expressed as: p(q) =

Ξ

p(q|ξ)p(ξ)dξ

1Calvetti and Somersalo, 2007; Jaynes, 2003; Kaipio and Somersalo, 2005

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Efficient Bayesian inference - model selection for RANS

Bayesian inference

Model selection: Suppose that we have a set of models M = {M1, M2, . . . , Mn} Model plausibility quantifies the relative probability that model Mi actually generated the observed data: p(Mi|q, M) ∝ E(Mi|q, M) p(Mi|M) Likelihood E(Mi|q, M) measures the probability of recovering data q given the model Mi: E(Mi|q, M) := p(q|Mi, M) =

Ξ

p(q|ξ, Mi, M)p(ξ|Mi, M) dξ Model evidence of Mi is used to compare models relative to one another and identify the model that is most likely capable of reproducing the data: log(E)

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Efficient Bayesian inference - model selection for RANS

Calibration data: Data is obtained from direct numerical simulation (DNS) 1 Mean velocity measurements taken at Reτ = 944 and Reτ = 2003 Uncertainty models: Three multiplicative error models, with z = y/H: u+ (z; ξ) − U +(x; ξ) = ǫ(z; ξ)U +(z; ξ)

independent homogeneous covariance correlated homogeneous covariance correlated inhomogeneous covariance

Reynolds stress model T +(z; ξ) −

u′

iu′ j

+ (z; ξ) = ǫ(z; ξ)

1Del Alamo et al., 2004; Hoyas and Jim´

enez, 2006

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Efficient Bayesian inference - model selection for RANS

Numerical Results

Independent homogeneous covariance:

ǫ(z)ǫ(z′)
= σ2δ(z − z′)

0.85 0.9 0.95 1 1.05 5 10 15 20 25

κ

0.7 0.8 0.9 1 1.1 5 10 15

cv1

0.8 1 1.2 1.4 1.6 1.8 1 2 3 4 5

σ Solid - full model, dashed - surrogate

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Efficient Bayesian inference - model selection for RANS

Numerical Results

Correlated homogeneous covariance:

ǫ(z)ǫ(z′)
= σ2 exp
−1/2(z − z′)2

l2

0.85

0.9 0.95 1 1.05 5 10 15 20

κ

0.7 0.8 0.9 1 1.1 2 4 6 8 10 12

cv1

0.5 1 1.5 2 1 2 3 4

σ Solid - full model, dashed - surrogate

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Efficient Bayesian inference - model selection for RANS

Numerical Results

Correlated inhomogeneous covariance:

ǫ(z)ǫ(z′)
= σ2
2l(z)l(z′)

l2(z) + l2(z′) 1/2 exp

−

(z − z′)2 l2(z) + l2(z′)

0.8

1 1.2 1.4 1.6 2 4 6 8

κ

0.8 1 1.2 1.4 1.6 1 2 3 4

cv1

2 4 6 8 0.5 1 1.5

σ Solid - full model, dashed - surrogate

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Numerical Results

Reynolds stress uncertainty:

ǫ(z)ǫ(z′)
= kin(z, z′) + kout(z, z′)

where kin models the error near the walls and kout far from the walls.

0.9 1 1.1 1.2 1.3 5 10 15 20 25

κ

0.8 1 1.2 1.4 1.6 2 4 6 8

cv1

2 4 6 8 0.5 1 1.5

σin

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Efficient Bayesian inference - model selection for RANS

Numerical Results: Model selection

Model evidence (log(E)):

Surrogate Full model Independent homogeneous

1.457

8.862 Correlated homogeneous 1.963 8.045 Correlated inhomogeneous 164.9 164.0 Reynolds stress 164.8 169.0

Relative runtimes (in seconds):

Surrogate Full model Independent homogeneous 130 1720 Correlated homogeneous 162 1906 Correlated inhomogeneous 151 1735 Reynolds stress 147 1743 Cumulative 590 7104

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SLIDE 54

Conclusions

Concluding remarks and future work

We need to think about errors and error control in computational science and engineering. We need quantitative methods to assess the reliability of our predictions (with respect to a given goal). Formulation of the problem should be done with respect to the goal

f the computation, not necessarily by minimizing the energy of the

system.

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