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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 to uncertainty quantification
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 to uncertainty quantification
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 to uncertainty quantification
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 to uncertainty quantification
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 to uncertainty quantification
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 to uncertainty quantification
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 to uncertainty quantification
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 to uncertainty quantification
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
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 to uncertainty quantification
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 )
Introduction to uncertainty quantification
MONTE-CARLO
Basics of probability (2)
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...
Introduction to uncertainty quantification
MONTE-CARLO
Basics of probability (3)
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
Introduction to uncertainty quantification
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
Introduction to uncertainty quantification
MONTE-CARLO
Basics of probability (4)
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
Introduction to uncertainty quantification
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
Introduction to uncertainty quantification
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)
Introduction to uncertainty quantification
MONTE-CARLO
Accuracy of estimation: mean (2)
Scalar case: variance σJ is unknown -√N JN−E(J )
σJN; S(N − 1) (Student dist.)
With ε confidence :
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
Introduction to uncertainty quantification
MONTE-CARLO
Accuracy of estimation: mean (3)
Scalar case: variance σJ is unknown -√N JN−E(J )
σJN; S(N − 1) (Student dist.)
With ε confidence :
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.[)
Introduction to uncertainty quantification
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
Introduction to uncertainty quantification
MONTE-CARLO
Chi-square probabilistic density functions Dχ2N
and
cumulative density functions Φχ2N
(skpd)
Introduction to uncertainty quantification
MONTE-CARLO
Accuracy of estimation: variance (2) (skpd)
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)
Introduction to uncertainty quantification
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 )
Introduction to uncertainty quantification
MONTE-CARLO
Accuracy of estimation: variance (3) (skpd)
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)
Introduction to uncertainty quantification
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
Introduction to uncertainty quantification
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
Introduction to uncertainty quantification
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 ?
Introduction to uncertainty quantification
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
Introduction to uncertainty quantification
APPLICATION OF (METAMODEL BASED) MONTE-CARLO
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
Introduction to uncertainty quantification
APPLICATION OF (METAMODEL BASED) MONTE-CARLO
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
Introduction to uncertainty quantification
APPLICATION OF (METAMODEL BASED) MONTE-CARLO
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
Best MC with MM value
Figure 10: Variance of CL coefficient and confidence interval
Introduction to uncertainty quantification
APPLICATION OF (METAMODEL BASED) MONTE-CARLO
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)
Introduction to uncertainty quantification
APPLICATION OF (METAMODEL BASED) MONTE-CARLO
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
Introduction to uncertainty quantification
APPLICATION OF (METAMODEL BASED) MONTE-CARLO
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
Best MC with MM value
Figure 13: Mean of CL coefficient and confidence interval
Introduction to uncertainty quantification
APPLICATION OF (METAMODEL BASED) MONTE-CARLO
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
Best MC with MM value
Figure 14: Variance of CL coefficient and confidence interval
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 to uncertainty quantification
ORTHOGONAL POLYNOMIALS
Orthogonal/orthonormal 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
Introduction to uncertainty quantification
ORTHOGONAL POLYNOMIALS
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.
Introduction to uncertainty quantification
ORTHOGONAL POLYNOMIALS
Orthonormal polynomials (2/3)
• 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
Introduction to uncertainty quantification
ORTHOGONAL POLYNOMIALS
Orthonormal polynomials (3/3)
• 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. )
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 to uncertainty quantification
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]
Introduction to uncertainty quantification
GENERALIZED POLYNOMIAL CHAOS METHOD (N.-I.)
Stochastic post-processing
• Expansion of flow field on structured mesh depending on stochastic variable ξ
W (i, ξ) ' PCW (i, ξ) =∑l=M−1l=0 Cl(i)Pl(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
Introduction to uncertainty quantification
GENERALIZED POLYNOMIAL CHAOS METHOD (N.-I.)
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) ?
Introduction to uncertainty quantification
GENERALIZED POLYNOMIAL CHAOS METHOD (N.-I.)
Accuracy of truncated spectral extension (2/3)
• 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
Introduction to uncertainty quantification
GENERALIZED POLYNOMIAL CHAOS METHOD (N.-I.)
Accuracy of truncated spectral extension (3/3)
• 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)
Introduction to uncertainty quantification
GENERALIZED POLYNOMIAL CHAOS METHOD (N.-I.)
Coefficient computations (1/4)
• Expansion of flow field on structured mesh depending on stochastic variable ξ
W (i, ξ) ' PCW (i, ξ) =∑l=M−1l=0 Cl(i)Pl(ξ)
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 ?)
Introduction to uncertainty quantification
GENERALIZED POLYNOMIAL CHAOS METHOD (N.-I.)
Coefficients computation - Gaussian quadrature (2/4)
• Expansion of flow field on structured mesh depending on stochastic variable ξ
W (i, ξ) ' PCW (i, ξ) =∑l=M−1l=0 Cl(i)Pl(ξ)
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).
Introduction to uncertainty quantification
GENERALIZED POLYNOMIAL CHAOS METHOD (N.-I.)
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
Introduction to uncertainty quantification
GENERALIZED POLYNOMIAL CHAOS METHOD (N.-I.)
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
Introduction to uncertainty quantification
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)
Introduction to uncertainty quantification
PROBABILISTIC COLLOCATION (N.-I.)
Basics (2/2)
• Expansion of flow field depending on stochastic variable ξ
• Stochastic post-processing (mean and variance)
done for the expansion SCW instead of W
straightforward evaluation of mean value
E(SCW (i, ξ)) =< SCW (i, ξ), 1 >=∑m=Nm=1 ωmWm(i)
straightforward evaluation of variance
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
Introduction to uncertainty quantification
PROBABILISTIC COLLOCATION (N.-I.)
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
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 to uncertainty quantification
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)
Introduction to uncertainty quantification
INTRUSIVE POLYNOMIAL CHAOS
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
Introduction to uncertainty quantification
INTRUSIVE POLYNOMIAL CHAOS
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)
Introduction to uncertainty quantification
INTRUSIVE POLYNOMIAL CHAOS
Example 1 (2/3)
• Steady incompressible flow. Continuity equation. One uncertain parameter (angle of
attack or...) following Dcf (ξ) = 1Π
1√1−ξ2
• Continuity equation
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
Introduction to uncertainty quantification
INTRUSIVE POLYNOMIAL CHAOS
Example 1 (3/3)
• Steady incompressible flow. Continuity equation. One uncertain parameter (angle of
attack or...) following Dcf (ξ) = 1Π
1√1−ξ2
• Continuity equation
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)
Introduction to uncertainty quantification
INTRUSIVE POLYNOMIAL CHAOS
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
Introduction to uncertainty quantification
INTRUSIVE POLYNOMIAL CHAOS
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
Introduction to uncertainty quantification
INTRUSIVE POLYNOMIAL CHAOS
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
Introduction to uncertainty quantification
INTRUSIVE POLYNOMIAL CHAOS
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
Introduction to uncertainty quantification
INTRUSIVE POLYNOMIAL CHAOS
Compressible Euler equations
• Including several technical details
Pseudo spectral approach for cubic terms
Truncation of higher order terms
Three applications
• Feasibility for large codes ?
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 to uncertainty quantification
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
Introduction to uncertainty quantification
EXTENSION TO HIGH DIMENSIONS
Basics of multidimensional probability (2)
• 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
Introduction to uncertainty quantification
EXTENSION TO HIGH DIMENSIONS
Basics of multidimensional probability (3)
• 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
Introduction to uncertainty quantification
EXTENSION TO HIGH DIMENSIONS
Basics of multidimensional probability (4)
• 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
Introduction to uncertainty quantification
EXTENSION TO HIGH DIMENSIONS
Basics of multidimensional probability (5)
• 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
Introduction to uncertainty quantification
EXTENSION TO HIGH DIMENSIONS
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
Introduction to uncertainty quantification
EXTENSION TO HIGH DIMENSIONS
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
Introduction to uncertainty quantification
EXTENSION TO HIGH DIMENSIONS
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
Introduction to uncertainty quantification
EXTENSION TO HIGH DIMENSIONS
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
straightforward evaluation of mean value
E(PCW (i, ξ)) =< PCW (i, ξ), 1 >= C0,0(i)
straightforward evaluation of variance
E((PCW (i, ξ)− E(PCW (i)))2) =∑l1=M,l2=Ml1=0,l2=0 (l1,l2)<>(0,0) Cl1,l2(i)2
Introduction to uncertainty quantification
EXTENSION TO HIGH DIMENSIONS
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
Introduction to uncertainty quantification
EXTENSION TO HIGH DIMENSIONS
Independant uncertain variables. Tensorial-grid probabilistic
collocation (2/2)
• Stochastic post-processing (mean and variance)
done for the expansion SCW instead of W
straightforward evaluation of mean value
E(SCW (i, ξ)) =< SCW (i, ξ), 1 >=∑l1=N,l2=Nl1=1,l2=1 ω1,l1ω2,l2Wl1,l2(i)
straightforward evaluation of variance
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
Introduction to uncertainty quantification
EXTENSION TO HIGH DIMENSIONS
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
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 to uncertainty quantification
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
Introduction to uncertainty quantification
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
Introduction to uncertainty quantification
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)
Introduction to uncertainty quantification
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)
Introduction to uncertainty quantification
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
Introduction to uncertainty quantification
NASA Rotor37 compressor
Barplots for uncertainties on
outlet pressure (left) and tip clearance (right) variations
Introduction to uncertainty quantification
NASA Rotor37 compressor
Tip clearance : order-1 (mean) moments for global quantities
Introduction to uncertainty quantification
NASA Rotor37 compressor
Tip clearance : order-2 (stdev) moments for global quantities
Introduction to uncertainty quantification
NASA Rotor37 compressor
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
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 to uncertainty quantification
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