introduction to uncertainty quantiﬁcation (uq)....

Introduction to uncertainty quantification (UQ).

With applications.J. Peter ONERA

including contributions of A. Resmini, M. Lazareff, M. Bompard ONERA

ANADE Workshop, Cambridge, September 24th, 2014

Introduction to uncertainty quantification

OUTLINE

Introduction

Probability basics and Monte-Carlo

Orthogonal/orthonormal polynomials

Non-intrusive polynomial chaos methods in 1D

Intrusive polynomial chaos method in 1D

Extension to high dimensions

Examples of application

Conclusion


INTRODUCTION

Example I : drag minimization

• Deterministic minimization of airplane cruise shape : define shape

That minimizes total drag Cd at M=0.82

Satisfying constraints on lift, pitching moment, inner volume...

• Actually, variations of cruise flight Mach number

Waiting for landing slot

Speeding up to cope with pilot maximum flight time

→ Variable Mach number described by D(M)

• Robust definition of airplane cruise shape

Minimize∫Cd(M)D(M)dM , satisfying constraints for all values of M

present in D(M)


INTRODUCTION

Example II : fan design

• Fan operational conditions subject to changes in wind conditions

• Manufacturing subject to tolerances

• Robust design accounts for

variability of external parameters

tolerances for internal parameters

Figure 1: Robust design (from cenaero.be)


INTRODUCTION

Example III : validation process

• Unkown data in experiment

Upwind Mach number not fully controled in wind tunnels dM = 0.001• Unknown physical constant needed in numerical model

Wall roughness constant (milled, brazed, eroded surface...)

• Discrepancy in a computational/experimental validation process !

• Compute the mean and standard deviation of the output of interest due to the uncertain

inputs


INTRODUCTION

Definition of uncertain inputs

• UNCERTAINTY QUANTIFICATION : describe the stochastic behaviour of OUTPUTS of in-

terest due to uncertain INPUTS

• Overview of CFD actual uncertain INPUTS :

Geometrical (manufacturing tolerance)

Operational: flow at boundaries (far field, injection...)

Reference:

/ Proceedings of RTO-MP-AVT-147

Evans T.P., Tattersall P. and Doherty J.J.: Identification and quantifi-

cation of uncertainty sources in aircraft related CFD-computations -

An industrial perspective

• Stochastic behaviour of OUTPUTS. Presently, most often, mean and variance are calcu-

lated


INTRODUCTION

Three issues in (UQ) – 1 terminology

• Lack of agreement on the definition of “error”, “uncertainty”...

• AIAA Guide G-077-1998 Uncertainty is a potential deficiency in any phase are activity

of the modeling process that is due to the lack of knowledge. Error is a recognizable

deficiency in any phase or activity of the modelling process that is not due to the lack of

knowledge

• ASME Guide V& V 20 (in its simpler version adopted for the Lisbon Workshops on CFD un-

certainty). The validation comparison error is defined as the difference between the simu-

lation value and the experimental data value. It is split in numerical, model, input and data

errors (assumed to be independant). Numerical (resp. input, model, data) uncertainty is

a bound of the absolute value of numerical (resp. input, model, data) error


INTRODUCTION

Three issues in (UQ) – 2 (UQ) validation and verification

• (UQ) CFD-based exercise leads to a standard deviation on some outputs

• Compare this standard deviation to the numerical error

Richardson method, GCI...

Pierce et al. Venditti et al. adjoint based formulas for functional outputs

• Compare this standard deviation to the modeling error

Run several (RANS) models

Run better models than (RANS)

• Numerical (UQ) investigation only make sense if standard deviation due to uncertain in-

puts not much smaller than modelling or discretization error


INTRODUCTION

Three issues in (UQ) – 3 lack of shared well-defined problems

• Most often mean and variance of some outputs are computed in (UQ) exercises

as this is feasible with usual methods ?

as this is actually required for applications ?

• from EDF’s specialits point of view, three stochastic criteria of interest:

central dispersion (mean and variance of output)

range (min and max possible values of output)

probability of exceeding a threshold

• Get information from industry in order to define relevant (UQ) exercises

• Share mathematical industrial test cases and mathematical exercices to split the CFD

influence / the one of (UQ) method


INTRODUCTION

Intrusive vs non-intrusive methods

• Non-intrusive methods. No change in the analysis code

Post-processing of deterministic simulations

• Intrusive methods. Changes in the analysis code

Stochastic expansion of state/primitive variables

Galerkin projections

Probably not feasible for large industrial codes


OUTLINE

Introduction







Conclusion


MONTE-CARLO

Basics of probability (1)

A classical introduction to probability basics involves

a sample space Ω (dice values, interval of Mach number values)

events ξ i.e. element of Ω or set of elements of Ω (even dice values,

subinterval of Mach number values) event spaceA, set of subsets of Ω, stable by union, intersection, including

null set ∅ and Ω (also called σ-algebra)

[INPUT] a probability function P on A such that P (Ω) = 1,P (∅) = 0,

plus natural properties for complementary parts and union of disjoint parts

(then denoted D keeping ′′P ′′ for polynomials)

[OUPUT] random variablesX depending of the event ξ (likeCDp orCLp

of an airfoil depending on the far-field Mach number through Navier-Stokes

equations )


MONTE-CARLO


Discrete example : regular 6-face Dice thrown once

events ξ = 1,2,3,4,5 or 6

sample space Ω =1,2,3,4,5,6

probability space Ω = null set plus all discrete sets of these numbers

∅, 1, 2, 3, 4, 5, 6, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 2, 3...1, 2, 3, 4, 5, 6

probability function P (later denoted D) : P (∅) = 0, P (1) =1./6., P (2) = 1./6.,... P (1, 2) = 1./3., P (1, 3) = 1./3 ,

P (1, 4) = 1./3.... P (1, 2, 3, 4, 5, 6) = 1. random variables X , for example dice value to the power three...


MONTE-CARLO


Discrete example : Mach number in [0.81,0.85]

events ξ = a Mach number value in [0.81,0.85]

sample space Ω = [0.81,0.85]

probability space Ω = all subparts of [0.81,0.85]

probability function P to be defined (later denoted D). For example from a

beta distribution with power three :

Pφ(φ) =3532

(1.−φ2)3 φ ∈ [−1, 1] φ = (ξ−0.83)/0.02

P (ξ) =1

0.02Pφ(φ) =

10.02

3532

(1.− (ξ − 0.83

0.02)2)3

random variablesX , aerodynamic pitching moment of a wing with variable

Mach number M∞(“event“ ξ) in the farfield


MONTE-CARLO

Set of probability density functions of β−distributions (on [0,+1] with the α− 1 β − 1

convention for exponants)

Figure 2: Monte-Carlo method with meta-models


MONTE-CARLO


Allows to define probablities for both discrete and continuous spaces

Quantities of interest = functions of the stochastic variables “random variables”

Example = pitching moment of wing J (random variable) depending via NS equations of

uncertain far-field Mach number (uncertain input/event)

Mean – Variance

E(J ) =

ZΩ

J (ξ)D(ξ)dξ

V ar(J ) =

ZΩ

(J (ξ)− E(J ))2D(ξ)dξ

Covariance of two random variables

Cov(J ,G) =

ZΩ

(J (ξ)− E(J ))(G(ξ)− E(G))D(ξ)dξ


MONTE-CARLO

Basics

Monte-Carlo mimics the law of the event in a series of calculations

Reference method for all uncertainty propagation methods

Generation of a sampling (ξ1, ξ2..., ξp..., ξN ) of the p.d.f D(ξ)Example : sampling for normal distribution. ak, bk indep. uni-

formly distributed in [0.,1.] ξk =√−2 ln(ak)cos(2Πbk)

Computation of corresponding flow fields W (ξp), p ∈ [1, N ] Computation of functional outputs J (ξp) = J(W (ξp), X(ξp)) Estimation of following statistical quantities :

E(J ) =

ZJ (ξ)D(ξ)dξ ' JN =

1

N

p=NXp=1

J (ξp)

σ2J = E((J − E(J ))2) ' σ2

JN =1

N − 1

p=NXp=1

(J (ξp)− JN )2

Need to quantify accuracy of estimation


MONTE-CARLO

Accuracy of estimation: mean (1)

Scalar case. variance σJ is known√N JN−E(J )

σJ; N (0, 1) (Normal distribution)

Reminders aboutN (0, 1)

Probability density function (p.d.f.) ofN (0, 1)- DN (x) = 1√2Πe−

x22

Symmetric cumulative distribution function - ΦN (x) = 1√2Π

∫ x−x e

− t22 dt

With ε confidence :

E(J ) ∈ [JN − uε σJ√N , JN + uεσJ√N

] ε = 1√2Π

∫ uε

−uεe−

t22 dt

ε 0.5 0.9 0.95 0.99

uε 0.68 1.65 1.96 2.58

With 99% confidence :

E(J ) ∈ [JN − 2.58 σJ√N, JN + 2.58 σJ√

N] (0.99 = 1√

2Π

∫ 2.58

−2.58

e−t22 dt)


MONTE-CARLO


Scalar case: variance σJ is unknown -√N JN−E(J )

σJN; S(N − 1) (Student dist.)


E(J ) ∈ [JN − uε(N−1)

σJN√N, JN + uε(N−1)

σJN√N

]

uεN as function of ε and N found in tables

Student distribution converges to Normal distribution for large N

Tables for uεN−1

ε N 2 3 20 30 ∞

0.95 12.7 4.30 2.09 2.04 1.96

0.99 63.7 9.93 2.86 2.75 2.58

Figure 3: Value of uε(N−1) for Student distribution S(N − 1) N ≥ 2


MONTE-CARLO


Scalar case: variance σJ is unknown -√N JN−E(J )

σJN; S(N − 1) (Student dist.)


E(J ) ∈ [JN − uε(N−1)

σJN√N, JN + uε(N−1)

σJN√N

]

uεN as function of ε and N found in tables

uεN converges towards uε of normal law for large N

Density function of Student distribution S(N)

. DS(N)(x) =Γ(N+1

2 )

Γ(N2 )√NΠ

(1 + x2

N)−

N+12

Definition of uεN

. ε =

Z uεN

−uεN

DS(N)(t)dt (ε ∈]0, 1.[)


MONTE-CARLO

Accuracy of estimation: variance (1) (skpd)

Scalar case: mean EJ is known

Notations

S2JN = 1

N

∑i=Ni=1 (J (ξp)− EJ )2

Chi-square χ2N probability distribution defined on [0,∞[ with p.d.f. :

Dχ2N

(x) = 1Γ(N/2)2N/2

xN/2−1e−x/2

Chi-square cumulative d.f. :

Φχ2N

(x) =∫ x

0

Dχ2N

(t)dt

Stochastic variable NS2JNσ2J

; χ2N


MONTE-CARLO

Chi-square probabilistic density functions Dχ2N

and

cumulative density functions Φχ2N

(skpd)


MONTE-CARLO


Scalar case: mean E(J ) is known - NS2JNσ2J

; χ2N

With ε = 1− α confidence :

Φ−1χ2N

(α2 ) ≤ N S2JNσ2J≤ Φ−1

χ2N

(1.− α2 )

With ε = 1− α confidence :

σ2J ∈ [N

S2JN

Φ−1χ2N

(1−α2 ), N

S2JN

Φ−1χ2N

(α2 )]

x N 2 20 30

0.005 10.597 39.997 53.672

0.995 0.0100 7.434 13.787

Figure 4: Value of Φ−1χ2N

(x)


MONTE-CARLO

Quality of estimation: variance - 99 % confidence

Application

N = 2⇒ σ2J ∈ [0.189 S2

J2, 200 S2

J2]

N = 20⇒ σ2J ∈ [0.500 S2

J20, 2.69 S2

J20]

N = 30⇒ σ2J ∈ [0.558 S2

J30, 2.17 S2

J30]

N = 100⇒ σ2J ∈ [0.713 S2

J100, 1.49 S2

J100]

Convergence speed of bounds towards 1.

The cumulative distribution of the Chi-Square law ΦN (x) can be ex-

pressed as Φχ2N

(x) = 1Γ(N/2)

∫ x/2

0

tN/2e−tdt =γ(N/2, x/2)

Γ(N/2)(γ

lower incomplete Γ function) Check properties of (the inverse of) Φχ2

N

Check convergence speed of N/Φ−1χ2N

(1− α2 ) and N/Φ−1

χ2N

(α2 )


MONTE-CARLO


Scalar case: mean E(J ) is unknown - Stochastic variable (N − 1)σ2JNσ2J

; χ2N−1

With ε = (1− α) confidence :

σ2J ∈ [(N − 1)

σ2JN

Φ−1χ2N−1

(1−α2 ), (N − 1)

σ2JN

Φ−1χ2N−1

(α2 )]

x N 3 4 20 30

0.005 10.597 12.838 38.582 52.336

0.995 0.0100 0.0717 6.844 13.121

Figure 5: Value of Φ−1χ2N−1

(x)


MONTE-CARLO

Restriction for Monte-Carlo method. Meta-models

Convergence speed of Monte-Carlo for mean value estimation is 1√N

Increasing precision of Monte-Carlo estimation by a factor of 10 requires

multiplying the number of evaluations by a factor of 100

Not usable in the context of complex aeronautics simulation

Cost of one evaluation very high (requires resolution of (RANS) equations)

Aerodynamic function relatively smooth

⇒ replace flow field / aerodynamic function by a model built with a database of flow fields

/ function values


METAMODEL BASED MONTE-CARLO

Use meta-models with Monte-Carlo method

Compute flow

(small) Sampling

and outputsSampling

(optimize inner parameters..)Build meta−model

Monte−Carlo

Meta−model

CFD code

Sampling and m.m. outputs

(large) stochastic sampling

mean, variance,...meta−model based

Evaluate

Compute functional outputs

Meta−model

Figure 6: Monte-Carlo method with meta-models


METAMODEL BASED MONTE-CARLO

Meta-models

Most often, approximation of a function of interest

Meta-models used at ONERA for CFD:

Kriging, Radial Basis Function, Support Vector Regression

Other meta-models of specific interest for UQ:

Adjoint based linear or quadratic Taylor expansion

Discussed in more details in the extension to multi-dimension section

Influence of meta-model accuracy on mean and variance accuracy ?


APPLICATION OF (METAMODEL BASED) MONTE-CARLO

Presentation

Search confidence interval on aerodynamic function CL with uncertainty on AoA

Nominal configuration: NACA0012, M = 0.73, Re = 6M , AoA = 3

elsA(a)

(RANS+(k-w) Wilcox turbulence model) solver (Roe flux+Van Albada lim.)

Figure 7: Mesh

(a) The elsA CFD software: input from research and feedback from industry Mechanics and Industry 14(3) L. Cam-

bier, S. Heib, S. Plot



Distribution of uncertainty

Beta distribution (parameters (2.,2.)) [pdfb(ξ) = 1516

(1− ξ)2(1 + ξ)2] over [-1,1]

p.d.f of angle of attack AoA [pdfa(α) = pdfb(10.(α− 3.))] over [2.9,3.1]

Figure 8: Beta distribution of AoA



Monte-Carlo method: application on CL(1)

0.484

0.486

0.488

0.49

0.492

0.494

0.496

0.498

1 10 100 1000

Estimated mean value90 percent95 percent99 percent

Best MC with MM value

Figure 9: Mean of CL coefficient and confidence interval



Monte-Carlo method: application on CL(2)

0

2e-05

4e-05

6e-05

8e-05

0.0001

0.00012

0.00014

0.00016

0.00018

0.0002

1 10 100 1000

Estimated variance value90 percent95 percent99 percent


Figure 10: Variance of CL coefficient and confidence interval



Monte-Carlo method with meta-models: learning sample

Use learning sample based on Tchebyshev polynomials

2.90 2.92 2.94 2.96 2.98 3.00 3.02 3.04 3.06 3.08 3.10−1

0

1

Figure 11: Tchebychev distribution (11 points)



Monte-Carlo method with meta-models: reconstruction of CL

0.475

0.48

0.485

0.49

0.495

0.5

0.505

0.79 0.792 0.794 0.796 0.798 0.8 0.802 0.804 0.806 0.808 0.81

’sample11.dat’’outputOK.dat’

’outputRBF.dat’’outputSVR.dat’

Figure 12: CL



Monte-Carlo method with meta-models: application on CL(1)

0.484

0.486

0.488

0.49

0.492

0.494

0.496

0.498

1 10 100 1000 10000 100000 1e+06 1e+07

Estimated mean value90 percent95 percent99 percent


Figure 13: Mean of CL coefficient and confidence interval



Monte-Carlo method with meta-models: application on CL(2)

0

2e-05

4e-05

6e-05

8e-05

0.0001

0.00012

0.00014

0.00016

0.00018

0.0002

1 10 100 1000 10000 100000 1e+06 1e+07

Estimated variance value90 percent95 percent99 percent


Figure 14: Variance of CL coefficient and confidence interval


OUTLINE

Introduction







Conclusion


ORTHOGONAL POLYNOMIALS


• Polynomial/spectral expansion and stochastic post-processing

Polynomials orthogonal for the product defined by p.d.f D(ξ) :

< Pl, Pm >=∫Pl(ξ)Pm(ξ)D(ξ)dξ = δlm

• Basic references

Orthogonal polynomials – Abramowitz and Stegun: Handbook of Mathe-

matical functions. (1972). Chapter 22 Spectral expansions – J. P. Boyd: Chebyshef and Fourier spectral methods

(2001)• Expand fields on this basis

In non-intrusive PC method – fields of interest

In intrusive PC method – state/primitive variables



Orthonormal polynomials (1/3)

• Families of polynomials

Normal distribution Dn(ξ) = 1√2Πe−

ξ2

2 on R → Hermitte polynomials

Gamma distributionDg(ξ) = exp(−ξ) on R+ → Laguerre polynomials

Uniform distribution Du(ξ) = 0.5 on [−1, 1]→ Legendre polynomials

Chebyshev distribution Dcf (ξ) = 1/Π/√

1− ξ2 on [−1, 1] →Chebyshev (first-kind) polynomials

Chebyshev distribution Dcs(ξ) =√

1− ξ2 on [−1, 1]→ Chebyshev

(second-kind) polynomials Beta distribut. Dβ(ξ) = (1− ξ)α(1 + ξ)β/

∫ 1

−1(1− u)α(1 + u)βdu

α > −1. , β > −1. on [−1,+1] → Jacobi polynomials (incl. Cheby-

shev polynomials) Non-usual probabilistic density functions, Dl(ξ) computed by Gram-

Schmidt orthogonalisation process.




• Example : 1D problem. Hermite polynomials for normal law

Probability density function (p.d.f.) of Dn(ξ) = 1√2Πe−

ξ2

2

• Family of orthonormal polynomials for < f, g >=R +∞−∞ f(ξ)g(ξ)Dn(ξ)dξ

PH0(ξ) = 1 PH1(ξ) = ξ

PH2(ξ) = ξ2 − 1 PH3(ξ) = ξ3 − 3ξ PH4(ξ) = ξ4 − 6ξ2 + 3

• Normalization of Hermite polynomials PHj(ξ) = 1√j!(2Π)(1/4)

PHj(ξ)

• Orthonormality relation for PHj :

< PHj , PHk >=∫ +∞−∞ PHj(ξ)PHk(ξ)Dn(ξ)dξ = δjk




• Example : 1D problem on [-1,1]. First-kind Chebyshef polynomials

Probability density function (p.d.f.) of Dcf (ξ) = 1Π

1√1−ξ2

• Family of orthonormal polynomials for < f, g >=R 1

−1f(t)g(t)Dcf (t)dt

T 0(ξ) = 1 T 1(ξ) = ξ

T 2(ξ) = 2ξ2 − 1 T 3(ξ) = 4ξ3 − 3ξ T 4(ξ) = 8ξ4 − 8ξ2 + 1

• Normalization of Chebyshev polynomials T0 = T 0 ... Tn =√

2 Tn (n ≥ 1)

• Orthonormality relation for T j :

< Tj , Tk >=∫ +∞−∞ Tj(ξ)Tk(ξ)Dcf (ξ)dξ = δjk

• specific property Tn(cos(θ)) = cos(nθ) (hence ||Tn||∞ ≤ 1. )


OUTLINE

Introduction







Conclusion


GENERALIZED POLYNOMIAL CHAOS METHOD (N.-I.)

Basics

• Polynomial/spectral expansion and stochastic post-processing

Polynomials orthogonal for the product defined by p.d.f D(ξ) :

< Pl, Pm >=∫Pl(ξ)Pm(ξ)D(ξ)dξ = δlm

Expand field of in interest (entire flow field, flow field at the wall...). i is the

discrete space variable (mesh index) Spectral expansion :

W (i, ξ) ' PCW (i, ξ) =∑l=M−1l=0 Cl(i)Pl(ξ)

Stochastic post-processing for PCW instead of W

• References [Wiener 1938][Ghanem et al. 1991]



Stochastic post-processing

• Expansion of flow field on structured mesh depending on stochastic variable ξ


• Stochastic post-processing (mean and variance)

done for the expansion PCW instead of W

straightforward evaluation of mean value

E(PCW (i, ξ)) =∫ ∑l=M−1

l=0 Cl(i)Pl(ξ)D(ξ)dξ = C0(i)E(PCW (i, ξ)) =< PCW (i, ξ), 1 >= C0(i)

straightforward evaluation of variance

E((PCW (i, ξ)− C0(i))2) =∫ (∑l=M−1

l=1 Cl(i)Pl(ξ))2

D(ξ)dξ

E((PCW (i, ξ)− C0(i))2) =∑l=M−1l=1 Cl(i)2



Accuracy of truncated spectral extension (1/3)

• General idea : the highest the regularity of the function of interest

→ the fastest the convergence of the truncated spectral expansion

• Discussed for Chebyshef first-kind polynomials

• Classical expension of a continuous fonction over [-1,1]

f(ξ) =∑∞n=0 CnTn(ξ)

• Definition of coefficients

C0 = 1Π

∫ +1

−11√

1−ξ2f(ξ)T o(ξ)dξ

Cn = 2Π

∫ +1

−11√

1−ξ2f(ξ)Tn(ξ)dξ (n > 0)

• Convergence speed of truncated series f(ξ) =PNn=0 CnTn(ξ) ?




• Cos Fourier series of g periodic over [−Π,+Π] : g(θ) =P∞n=0 Ancos(nθ)

A0 = 12Π

R +Π

−Πf(θ)dθ An = 1

Π

R +1

−1g(θ)cos(nθ)dθ

• Integration by part of An for a Ck function

An = − 1nΠ

∫ +1

−1g′(θ)sin(nθ)dθ

An = − 1n2Π

∫ +1

−1g(2)(θ)cos(nθ)dθ

......................

An = 1nkΠ

∫ +1

−1g(k)(θ)cos(nθ + kΠ/2)dθ

• Upper bound of the coefficients for a Ck function

|An| ≤ 1nkΠ

∫ +1

−1|g(k)(θ)|dθ ≤ 2

nk||g(k)||∞

• Upper bound of the residual

|g(θ)−∑Nn=0Ancos(nθ)| ≤

∑∞N+1 |An| ≤ ...

... ≤ 2||g(k)||∞∫∞N

1ukdu ≤ 2||g(k)||∞

(k−1)Nk−1




• Chebyshev expansion of a Ck function f(ξ) over [-1,1]

f(ξ) =∑∞k=0AkT k(ξ)

• Change variable: ξ = cos(θ), g = f cos is Ck over [−Π,Π]

g(θ) = f(cosθ) =∑∞k=0AkT k(cos(θ))

g(θ) =∑∞k=0Akcos(k θ) is a Fourier series !!!

• All results concerning Fourier series have a counterpart for Tchebyshev expansions

Express ||g(k)||∞ from ||f (k)||∞ ... ||f ′||∞Express the upper bound of |Ak|, use ||T k||∞ = 1.Find the upper bound of truncation error of spectral expansion of f

• Extension to f(i, ξ) =P∞k=0 Ak(i)T k(ξ)



Coefficient computations (1/4)



Straightforward stochastic post-processing for mean and variance (pre-

sented before) Accuracy of ideal PCW depending on degree and regularity - theory of

spectral expansions (just discussed)

• Computation of Cl coefficients ?

• Accuracy of actual PCW expansion ? (articles on this topic ?)



Coefficients computation - Gaussian quadrature (2/4)



From orthonormality property Cl(i) =< PCW,Pl >

Under regularity assumptions Cl(i) =< W,Pl >

• Gaussian quadrature

Cl(i) =< PCW,Pl >=< W,Pl >=∫W (i, ξ)Pl(ξ)D(ξ)dξ

Computation by Gaussian quadrature associated to D with G points.∫f(ξ)D(ξ)dξ '

∑k=Gk=1 wkf(ξk)

(wk,ξk) depend on D(ξ) Exact quadrature if f(ξ) is a polynomial of degree≤ (2G− 1).

/ Polynomial chaos : Cl(i) exact if W (i, ξ)Pl(ξ) polynomial of

ξ of degree lower than/equal to (2G− 1).



Coefficients computation - collocation (3/4)

• Other way : collocation - least-square collocation

Identify W (i, ξl) and PCW (i, ξl) for M values of ξ. “Collocation”

W (i, ξk) =∑l=M−1l=0 Cl(i)Pl(ξk) k ∈ 1,M solved for Cl(i)

Solve least-square problem for more values of W (i, ξl) than coefficients

in the expansion

• Less general accuracy results than for Gauss quadrature



Coefficients computation - collocation (4/4)

• Theorical result for Chebyshev series (see Boyd pages 95-96). Expansion of a Ck func-

tion f(ξ) over [-1,1] f(ξ) =P∞k=0 AkTk(ξ)

Collocation at xk = −cos( kΠM−1 ) k ∈ [0,M−1] (Chebyshev extrema)

obtain PM−1(ξ) =∑M−1k=0 AkT k(ξ)

Collocation at xk = −cos( (2k+1)Π2M ) k ∈ [0,M−1] (Chebyshev roots)

compute QM−1(ξ) =∑M−1k=0 AkT k(ξ)

Upper bounds of truncation error

|f(ξ)−∑Nn=0 AnTn(ξ)| ≤ 2

∑∞N+1 |An| ≤ ...

|f(ξ)−∑Nn=0 AnTn(ξ)| ≤ 2

∑∞N+1 |An| ≤ ...

Factor TWO w.r.t. truncation of exact series


PROBABILISTIC COLLOCATION (N.-I.)

Basics (1/2)

• Another approach for non-intrusive polynomial chaos based on Lagrangian polynomial

expansion. [Tatang 1995] [Xiu et al. 2005] [Loeven et al. 2007] for compressible CFD

• Presented for 1D problem ( i is the discrete space variable – mesh index)

W (i, ξ) ' SCW (i, ξ) =∑l=Nl=1 Wl(i)Hl(ξ)

Hl(ξ) = Πm=Nm=1,m<>l

(ξ − ξm)(ξl − ξm)

( polynomial of degree (N − 1) )

• Note that actually SCW (i, ξl) = Wl(i).→ no coefficient computation, defineWl(i) =

W (i, ξl)

• Definition of (ξ1, ξ2, ..., ξN ) (most often, not absolutely necessary)

N points of the N-point quadrature associated to D(ξ)∫f(ξ)D(ξ)dξ '

∑m=Nm=1 ωmf(ξm)



Basics (2/2)

• Expansion of flow field depending on stochastic variable ξ


done for the expansion SCW instead of W


E(SCW (i, ξ)) =< SCW (i, ξ), 1 >=∑m=Nm=1 ωmWm(i)


E((SCW (i, ξ)− E(SCW (i)))2) =∑m=Nm=1 ωmWm(i)2

−(∑m=Nm=1 ωmWm(i))2

Both exact for SCW as N -point Gaussian quadrature is exact for polyno-

mial of degree up to 2N − 1



Link between polynomial chaos and probabilistic collocation

• Two polynomial expansions

• Case when the methods merge

same degree (M − 1) for both expansions

coefficents of chaos expansion computed by collocation

same set of M collocation points

polynomials of degree (M − 1) having same values in M distinct abscis-

sas → SCW = PCW (expressed in two different basis)

exact evaluation of mean and variance for and SCW and PCW

same evaluation of mean and variance by the two methods


OUTLINE

Introduction







Conclusion


INTRUSIVE POLYNOMIAL CHAOS

References

• A stochastic projection method for fluid flow. I Basic formulation. Le Maître OP, Knio OM, Najm, HN,

Ghanem RG. JCP 173 (2001)

• A stochastic projection method for fluid flow. II Random process. Le Maître OP, Reagan MT, Najm

HN, Ghanem RG, Knio OM. JCP 181 (2002)

• Intrusive polynomial chaos method for the compressible Navier Stokes equations. Chris Lacor.

Slides presented at ONERA Scientific Day on error approximation and uncertainty quantification.

3/12/2010 (available on ONERA’s web site)

• Comparison and intrusive and non-intrusive polynomial chaos methods for CFD applications in aero-

nautics. Onorato G, Loeven GJA, Ghobarnias G, Bijl H, Lacor C. Proceedings of ECCOMAS CFD 2010

• [Application] Assessment of intrusive and non-intrusive non-deterministic CFD methodologies

based on polynomial chaos expansions. Dinsecu C, Smirnov S, Hirsch C, Lacor C. IJESMS 2 (2010)



Principle

• Expand primitive variables (not state-variables ! unless you get ratio of polynomial ex-

pansions) on PC basisU(x, t, ξ) =

∑l=M−1l=0 Ul(x, t)Pl(ξ)

• Input expansions in stochastic partial differential equations

• Do a Galerkin projection for all Pl (l ∈ [0,M − 1]) :

Multiply all equations by Pl(ξ) Apply the mean (in other words, multiply all equations by

Pl(ξ)×D(ξ) sum over ξ domain)

• Discretize resulting system of coupled partial differential equations

• Stochastic post-processing



Example 1 (1/3)

• Steady incompressible flow. Continuity equation

• One uncertain parameter (angle of attack or...) following Dcf (ξ) = 1Π

1√1−ξ2

• Order 2 expansion of primitive variables

U(x, ξ) =∑l=2l=0 Ul(x)Tl(ξ) p(x, ξ) =

∑l=2l=0 pl(x)Tl(ξ)

• reminder

T0(ξ) = 1 T1(ξ) =√

2ξ T2(ξ) =√

2(2ξ2 − 1)

• Continuity equation

div (U(x, ξ)) = div(∑l=2

l=0 Ul(x)Tl(ξ))

div (U(x, ξ)) =∑l=2l=0 div(Ul(x))Tl(ξ)



Example 1 (2/3)

• Steady incompressible flow. Continuity equation. One uncertain parameter (angle of

attack or...) following Dcf (ξ) = 1Π

1√1−ξ2


div (U(x, ξ)) =∑l=2l=0 div(Ul(x))Tl(ξ) = 0

• Galerkin projection for T0(ξ),T1(ξ) and T2(ξ) (For T1 for example)

div (U(x, ξ))T1(ξ)Dcf (ξ) =Pl=2l=0 div(Ul(x))Tl(ξ)T1(ξ)Dcf (ξ) = 0

A =R 1

−1div (U(x, ξ)))T1(ξ)Dcf (ξ) = 0



Example 1 (3/3)

• Steady incompressible flow. Continuity equation. One uncertain parameter (angle of

attack or...) following Dcf (ξ) = 1Π

1√1−ξ2


div (U(x, ξ)) =∑l=2l=0 div(Ul(x))Tl(ξ) = 0

• Galerkin projection for T0(ξ),T1(ξ) and T2(ξ) (For T1 for example)

A =R 1

−1

Pl=2l=0 div(Ul(x))Tl(ξ)T1(ξ)Dcf (ξ)dξ = 0

A =Pl=2l=0 div(Ul(x))

R 1

−1Tl(ξ)T1(ξ)Dcf (ξ)dξ = div(U1(x)) = 0

Using orthonormality property : div(U1(x)) = 0 (of course div(U0(x))and div(U2(x)) are also proved to be equal to zero using corresponding

Galerkin projection)



Example 2 (1/3)

• Unsteady incompressible flow. Euler equations. Momentum equation. One uncertain

parameter following Dcf (ξ) = 1Π

1√1−ξ2

• Order 2 expansion of primitive variables (→ final number of equations multiplied by three)

U(x, t, ξ) =∑l=2l=0 Ul(x, t)Tl(ξ) p(x, t, ξ) =

∑l=2l=0 pl(x, t)Tl(ξ)

• Momentum equation∂U∂t + (U.∇)U +∇p = 0

∂∂t

(Pl=2l=0 Ul(x, t)Tl(ξ))+

Pl=2l=0

Pn=2n=0(Ul(x, t).∇)Un(x, t)Tl(ξ)Tn(ξ)+

∇Pl=2l=0 pl(x, t)Tl(ξ) = 0



Example 2 (2/3)

• Steady incompressible flow. Euler equations. Momentum equation. One uncertain pa-

rameter following Dcf (ξ) = 1Π

1√1−ξ2

• Momentum equation∂∂t (

Pl=2l=0 Ul(x, t)Tl(ξ))

+Pl=2l=0

Pn=2n=0(Ul(x, t).∇)Un(x, t)Tl(ξ)Tn(ξ)

+∇Pl=2l=0 pl(x, t)Tl(ξ) = 0

• Multiplying by Tk ∀k ∈ [0, 2] and taking the expected value for Dcf∂∂t (

Pl=2l=0 Ul(x, t) < TlTk >

+Pl=2l=0

Pn=2n=0(Ul(x, t).∇)Un(x, t) < TlTnTk >

+∇Pl=2l=0 pl(x, t) < TlTk >= 0



Example 2 (3/3)

• Steady incompressible flow. Euler equations. Momentum equation. One uncertain pa-

rameter following Dcf (ξ) = 1Π

1√1−ξ2

• Momentum equation∂∂t (

Pl=2l=0 Ul(x, t) < TlTk >)

+Pl=2l=0

Pn=2n=0(Ul(x, t).∇)Un(x) < TlTnTk >

+∇Pl=2l=0 pl(x, t) < TlTk >= 0

• Using orthogonality property∂∂tUk(x, t) +

Pl,n=2l,n=0(Ul(x, t).∇)Un(x, t) < TlTnTk >

+∇pk(x, t) = 0

• Set of p.d.e for (p0, p1, p2) (U0, U1, U2). Coupling through momentum equation



Stochastic post-processing

• Expand primitive variables on PC basis

U(x, t, ξ) =∑l=M−1l=0 Ul(x, t)Pl(ξ)

• Straightforward post-processing at continuous level

E(U(x, t, ξ)) = U0(x, t)E((U(x, t, ξ)− U0(x, t))2) =

∑l=M−1l=1 Ul(x, t)2

• Actual stochastic post-processing at discrete level

Use corresponding formulas at discrete level

Derive mean and variance of (linearized) outputs of interest



Compressible Euler equations

• Including several technical details

Pseudo spectral approach for cubic terms

Truncation of higher order terms

Three applications

• Feasibility for large codes ?


OUTLINE

Introduction







Conclusion


EXTENSION TO HIGH DIMENSIONS

Basics of multidimensional probability (1)

• Several R values stochastic variables

a sample space Ω ∈ Rd

events i.e. element of Ω or set of elements of Ω elementary event ξ Rd valued vector

event spaceA, set of subsets of Ω, stable by union, intersection, including

null set ∅ and Ω (also called σ-algebra)

For the sake of simplicity Ω ∈ R2

Define, cumulative distribution of (ξ1, ξ2), probabilty density function,

marginal distributions, independance, covariance




• Cumulative density function

Φ(u1, u2) = P (ξ1 ≤ u1 ∩ ξ2 ≤ u2)

• Probabilistic density function

Dξ(ξ1, ξ2) =∂2Φξ∂x∂y

(ξ1, ξ2)

probability of ξ ∈ [ξ1 − dx/2, ξ1 + dx/2][ξ2 − dy/2, ξ2 + dy/2] is Dξ(ξ1, ξ2)dxdy




• Marginal distribution w.r.t. the two variables

Φ1(ξ1) =

Z ξ1

−∞

„Z +∞

−∞Dξ(x, y)dy

«dx

Φ2(ξ2) =

Z ξ2

−∞

„Z +∞

−∞Dξ(x, y)dx

«dy

• Marginal probabilistic density function

D1(ξ1) =

Z +∞

−∞Dξ(ξ1, y)dy

D2(ξ2) =

Z +∞

−∞Dξ(x, ξ2)dx




• Independant ξ1 and ξ2 = one variable provides no information about the other :

φξ(ξ1, ξ2) = Φ1(ξ1)Φ2(ξ2)

Dξ(ξ1, ξ2) = D1(ξ1)D2(ξ2)

• Null covariance is ensured for independant stochastic variable. Null covariance is (by far)

a less stronger property than independance of stochatic variables

• Example : independant variables ξ1 ξ2 on [−1,+1]2:

Dξ(ξ1, ξ2) = Dξ(ξ1, ξ2) =35

32(1.− ξ2

1)3 × 15

16(1.− ξ2

2)2




• Example : independant variables ξ1 ξ2 on [−1,+1]2:

Dξ(ξ1, ξ2) = Dξ(ξ1, ξ2) =35

32(1.− ξ2

1)3 × 15

16(1.− ξ2

2)2

• Example = dependant variables ξ1 ξ2 on [−1,+1]2 :

Daξ (ξ1, ξ2) = Dξ(ξ1, ξ2) = 3/4 (1− 1/4 (ξ1 − ξ2)2)

Dbξ(ξ1, ξ2) = Dξ(ξ1, ξ2) = 3/4 (1− 1/4 (ξ1 + ξ2)2)

• Exercise = calculate all previous quantities :

Daξ1(ξ1) = 3/4((5/6− 1/2ξ2

1) Daξ2(ξ2) = 3/4(5/6− 1/2ξ2

2)

Dbξ1(ξ1) = 3/4((5/6− 1/2ξ2

1) Dbξ2(ξ2) = 3/4(5/6− 1/2ξ2

2)

Ea(ξ1) = 0 Ea(ξ2) = 0 Eb(ξ1) = 0 Eb(ξ2) = 0

cova(ξ1, ξ2) = 1/6 covb(ξ1, ξ2) = −1/6



Changes from 1D to dimension d

• Several uncertainties considered simultaneously

ξ is a vector ξ = (ξ1, ξ2.., ξd)Caution : subscript = components <> exponent = number in a sampling

• What does not change:

Monte-Carlo method

• What does not change very much:

Monte-Carlo plus metamodel

Polynomial chaos method for independant uncertain ξk and tensorial basis

of polynomials. Size of polynomial basis (collocation grid) Gd

• What does change:

Polynomial chaos method for dependant uncertain ξk or degree-M multi-

variate polynomials



Monte-Carlo method Rd

• Basic Monte-Carlo

No difference between R and Rd

• Monte-Carlo plus Kriging, RBF, SVM/SVR...

Number of inner parameters of the meta-model prop. to d

Cost of the definition of an accurate meta-model increasing with d

• Monte-Carlo plus adjoint-based first or second order Taylor expansion

First order : solve one adjoint equation to compute first-order derivative

Second order : solve d direct equations plus one adjoint equation to com-

pute first- and second-order derivatives



Independant uncertain variables. Tensorial-product polynomial

basis (1/2)

• D(ξ) = D1(ξ1)D2(ξ2)

• For 2D problem (i generic grid index)

W (i, ξ) ' PCW (i, ξ) =∑l1=M,l2=Ml1=0,l2=0 Cl1,l2(i)Pl1(ξ1)Ql2(ξ2)

P orthonormal polynomials associated to D1(ξ1)Q orthonormal polynomials associated to D2(ξ2)

• Calculate Cl1,l2(i) by collocation

• Calculate Cl1,l2(i) by quadrature using :

Cl1,l2(i) =< PCW,Pl1Ql2 >=

ZPCW (i, ξ)Pl1(ξ1)Ql2(ξ2)D1(ξ1)D2(ξ2)dξ1dξ2

Cl1,l2(i) '< W,Pl1Ql2 >=

ZW (i, ξ)Pl1(ξ1)Ql2(ξ2)D1(ξ1)D2(ξ2)dξ1dξ2

using a G×G grid based on Gaussian quadrature points



Independant uncertain variables. Tensorial-product polynomial

basis (2/2)

• Accuracy issue. How close is W (i, ξ)Pl1(ξ1)Ql2(ξ2) from a (2G − 1) in ξ1 times a

(2G− 1) polynomial in ξ2 ?

• Stochastic post-processing done for the expansion PCW instead of W


E(PCW (i, ξ)) =< PCW (i, ξ), 1 >= C0,0(i)


E((PCW (i, ξ)− E(PCW (i)))2) =∑l1=M,l2=Ml1=0,l2=0 (l1,l2)<>(0,0) Cl1,l2(i)2



Independant uncertain variables. Tensorial-grid probabilistic

collocation (1/2)

• D(ξ) = D1(ξ1)D2(ξ2)

• Lagrangian polynomial expansion

• For 2D problem

W (i, ξ) ' SCW (i, ξ) =∑l1=N,l2=Nl1=1,l2=1 Wl1,l2(i)Hl1(ξ1)Hl2(ξ2)

Hl(ξj) = Πm=Nm=1,m<>l

(ξj − ξj,m)(ξj,l − ξj,m)

(j = 1 or 2)

• Note that actually SCW (i, ξl1 , ξl2) = Wl1,l2(i).→ no coefficient computation, define

Wl1,l2(i) = W (i, ξl1 , ξl2)

• Definition of (ξj,1, ξj,2, ..., ξj,N ) (j = 1 or 2) (most often, not absolutely necessary)

N points of the N-point quadrature associated to Dj(ξj)

• Mean and variance



Independant uncertain variables. Tensorial-grid probabilistic

collocation (2/2)


done for the expansion SCW instead of W


E(SCW (i, ξ)) =< SCW (i, ξ), 1 >=∑l1=N,l2=Nl1=1,l2=1 ω1,l1ω2,l2Wl1,l2(i)


E((SCW (i, ξ)− E(SCW (i)))2) =∑l1=N,l2=Nl1=1,l2=1 ω1,l1ω2,l2Wl1,l2(i)2

−(∑l1=N,l2=Nl1=1,l2=1 ω1,l1ω2,l2Wl1,l2(i))2



Dependant uncertain variables. True degree M polynomials in Rd

• Number of terms for a degree M polynomial in dimension d(M+d)!M !d!

• Number of continuous equations for intrusive approach

Multiplied by (M+d)!M !d! instead of (M + 1)

• Computation of coefficients for non-intrusive approach

Collocation : from at least (M+d)!M !d! flow computations

Exact quadrature for degree-M polynomial [Smolyak 1963]...

Define clever enrichment...

Huge interest the community. Huge literature...

...Attend (UQ) course level two


OUTLINE

Introduction







Conclusion


Some outputs of NODESIM-CFD

M. Lazareff

• NODESIM-CFD = EU project 2007-2009

• Some important outputs of NODESIM-CFD

Split (UQ) issues and CFD analysis issues. Test (UQ) propagation method

on well defined CFD test cases AND mathematical models

Compare standard deviation due to uncertainty to numerical errors and

modelization errors

Need for accurate definition of the input p.d.f.

Actually, most often non intrusive methods for complex test cases


RAE2822 profile

Configuration characteristics

• RAE2822 airfoil Re = 6.5106,M = 0.734, α = 2.79o

• Classical test case of high-Re transonic CFD

• Complex flow with upper and lower shocks

• Strong dependance on turbulence model and mesh density


RAE2822 profile, k − ω model, Re = 6.5106 around M = 0.734, α = 2.79o

TPS CL response surface with SST correction

medium CFD grid, low-density collocation (blue spheres)


RAE2822 profile, k − ω model, Re = 6.5106 around M = 0.734, α = 2.79o

TPS CL response surface without (transparent) and with (opaque, lower) SST correction

medium CFD grid, low-density collocation (white/blue spheres)


NASA Rotor37 compressor

Configuration characteristics

• Isolated rotor blade. Transonic flow. Subsonic outlet.

• Uncertainty propagation exercises

β distribution (bounded support)

uncertainty on outlet static-pressure and tip-clearance

uncertainty quantification for several points of characteristic curve (several

outlet static pressure <> several mass flows) polynomial chaos (gaussian quadrature) applied to functional outputs

• good quadrature convergence is observed

computed order-4 moments almost identical to order-3 ones

• use order-3 Jacobi polynomials (associated to β distributions)

only 4 CFD points needed for near-complete characterisation



Barplots for uncertainties on

outlet pressure (left) and tip clearance (right) variations



Tip clearance : order-1 (mean) moments for global quantities



Tip clearance : order-2 (stdev) moments for global quantities



Application of intrusive polynomial chaos method

Assessment of intrusive and non-intrusive non-deterministic CFD methodologies based

on polynomial chaos expansions. [Dinsecu C, Smirnov S, Hirsch C, Lacor C. IJESMS 2 (2010)]

Comparison between intrusive PC and probabilistic collocation

NASA Rotor 37 (Compressor map)

Uncertain parameters : outlet static pressure, height of tip clearance

Output parameters : local static pressure (middle gridline, mid-span), pres-

sure ration, isentropic efficiency Very consistent results for mean and standard deviation


OUTLINE

Introduction







Conclusion


Conclusion

• Uncertainty quantification

more and more interest and projects (EU, RTO . . . )

get definition of industry relevant problems

“robust design” optimization process

• Methods

1D definition

multi-D, sparsity, clever enrichment...

• Challenges

Get relevant p.d.f. for uncertain parameters

Deal efficiently with large number of uncertain parameters

Deal efficiently with geometrical uncertainties

introduction to uncertainty quantiﬁcation (uq)....

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