Sparse data formats and efficient numericalmethods for UQ in numerical aerodynamics

NASPDE-2010 Workshop, Freiberg,Alexander Litvinenko, Hermann G. Matthies

Institut für Wissenschaftliches Rechnen,TU Braunschweiglitvinen@tu-bs.de

September 20, 2010

1

Outline

Part I. Overview

Modelling of free stream turbulence

Uncertainties in geometry

Part II. Low-rank approximation of the solution

2

Outline

Part I. Overview

Modelling of free stream turbulence

Uncertainties in geometry

Part II. Low-rank approximation of the solution

3

Problem setup

Stationar Navier-Stokes equation:

v · ∇v − 1Re

∇2v + ∇p = g, and ∇ · v = 0.

+ b.c. and Wilcox-k-w turbulence modeldomain: RAE-2822 airfoil with some area around

Solver: TAU has more than 300 parameters! (developed inDLR)Many of them are or can be uncertain!The first task: Classify uncertain parameters (p. 25).

4

Overview of uncertainties

Uncertain Input:

1. Variables α and Ma

2. Geometry of airfoil

3. Parameters of a turbulence model

Uncertain solution:

1. mean value and variance of v

2. exceedance probabilities P(v > v∗)

3. probability density functions of v .

5

Our Aims

1. Sparse (low-rank) representation of the input data (randomfields)

2. The whole computation process must be done in areasonable time

3. Use the deterministic solver as a black box

4. A sparse (low-rank) data format for the solution

6

Outline

Part I. Overview

Modelling of free stream turbulence

Uncertainties in geometry

Part II. Low-rank approximation of the solution

7

Modelling of uncertainties in free stream turbulence

α

v

v

u

u’

α’v1

2

Random vectors v1(θ) and v2(θ) model free stream turbulence 8

v1 =Iuθ1√

2and v2 =

Iuθ2√2

.

where θ1 and θ2 two Gaussian random variables.

Let θ :=√

θ21 + θ22 and β := arctg

v2v1

Then

α′

= arctgsin α + z sin βcos α − z cos β , where z :=

Iθ√2

and the new Mach number

Ma′

= Ma

√

1 +I2θ2

2−√

2Iθ cos(β + α).

where Ma′

= u′

us, us speed of sound.

9

Sparse Gauss-Hermite Quadratures

Figure: Sparse Gauss-Hermite grids of order 2 (13 points) and 3 (29points).

10

α(θ1, θ2), Ma(θ1, θ2), where θ1, θ2 have Gaussiandistributions

Statistics obtained on sparse Gauss-Hermite grid with 137points.Input uncertainties in α and Ma

mean st. dev. σ σ/meanα 2.8 0.2 0.071Ma 0.73 0.0026 0.0036

results in uncertainties in the solution lift CL and drag CD

CL 0.85 0.0373 0.044CD 0.0187 0.0031 0.163

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Table: Comparison of results obtained by a sparse Gauss-Hermitegrid (n grid points) with 17000 MC simulations.

n 137 381 645 MC,17000

σCL/CL 0.044 0.042 0.042 0.045σCD/CD 0.163 0.159 0.16 0.159|CL − CL0|/CL 7.6e-4 1.3e-3 1.6e-3 4.2e-4|CD−CD0|/CD 1.7e-2 1.5e-2 1.4e-2 2.1e-2

12

Assume that α, Ma have Gaussian distributions:

2.6

2.8

3

3.2

0.7

0.71

0.72

0.73

0.74

0.75

0.76

0.8

0.82

0.84

0.86

0.88

0.9

0.92

alpha

Cl(alpha, Ma), I=0.005, RAE−2822.Wilcox

Ma

Cl(a

lph

a, M

a)

2.6

2.8

3

3.2

0.71

0.72

0.73

0.74

0.75

0.76

0.01

0.015

0.02

0.025

0.03

0.035

alpha

CD(alpha, Ma), I=0.005, RAE−2822.Wilcox

Ma

CD

(alp

ha

, M

a)

13

500 MC realisations of pressure coeff. (cp) in dependence onαi and Mai

14

500 MC realisations of skin friction coeff. (cf) in dependence onαi and Mai

15

5% and 95% quantiles for cp from 500 MC realisations.

16

5% and 95% quantiles for cf from 500 MC realisations.

17

0.75 0.8 0.85 0.9 0.950

5

10

15

20

25Lift: Comparison of densities

0.005 0.01 0.015 0.02 0.025 0.03 0.0350

50

100

150Drag: Comparison of densities

0.75 0.8 0.85 0.9 0.950

0.2

0.4

0.6

0.8

1Lift: Comparison of distributions

0.005 0.01 0.015 0.02 0.025 0.03 0.0350

0.2

0.4

0.6

0.8

1Drag: Comparison of distributions

sgh13

sgh29

MC

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Figure: Intervals [mean − 3σ, mean + 3σ], σ standard deviation, ineach point of RAE2822 airfoil for the pressure, density, cp and cf.Build for 645 points of sparse Gauss-Hermite grid.

19

Outline

Part I. Overview

Modelling of free stream turbulence

Uncertainties in geometry

Part II. Low-rank approximation of the solution

20

Uncertainties in geometry

Random boundary perturbations:∂Dε(ω) = {x + εκ(x , ω)n(x) : x ∈ ∂D}.where κ(x , ω) is a random field.

How to generate geometry with uncertainties ?Algorithm:

1. Assume cov. function cov(x , y) for random field κ(x , ω)given

2. Compute Cij := cov(xi , xj) for all grid points (in a sparseformat!)

3. Solve eigenproblem Cφi = λiφi4. Then κ(x , ω) ≈ ∑mi=1

√λiφiξi(ω), where ξi(ω) are

uncorrelated random variables.

Sparse approximation of dense matrix C is done in [Khoromskij,Litvinenko, Matthies, 2009]

21

21 realisations of RAE-2822 airfoil

Covariance function is of Gaussian typecov(ρ) = 10−5exp(−∑2i=1(xi − yi)2/ℓ2i ).

22

Uncertainties in geometry

mean st. dev. σ σ/meanCL 0.8552 0.0049 0.0058CD 0.0183 0.00012 0.0065

PCE of order 1 with 3 random variables and sparseGauss-Hermite grid wite 25 points were used.

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Outline

Part I. Overview

Modelling of free stream turbulence

Uncertainties in geometry

Part II. Low-rank approximation of the solution

24

Low-rank approximation of the solution

Let W := [v1, v2, ..., vZ ], where vi are solution vectors.Given tSVD W̃ = ŨΣ̃Ṽ T = ABT ≈ W .How to compute tSVD of [W̃ , vZ+1, ..., vZ+K ] with a linearcomplexity ? (M. Brand, 2006)

v =1Z

Z∑

i=1

vi =1Z

Z∑

i=1

A · bi = Ab, (1)

C =1

Z − 1WcWTc =

1Z − 1UΣV

T VΣT UT =1

Z − 1UΣΣT UT .

(2)Diagonal of C can be computed with the complexityO(k2(Z + n)).

25

Decay of eigenvalues

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−20

−15

−10

−5

0

5

log, #eigenvalues

log

, va

lue

s

pressuredensitycpcf

Figure: Decay (in log-scales) of 100 largest eigenvalues of solutionmatrices: [pressure], [density], [cf], [cp] ∈ R512×645 on the surface ofRAE-2822 airfoil. 26

Relative errors of rank-k approximations

k press. density tke ev xv zv memory, MB10 1.9e-2 1.9e-2 4.0e-3 1.4e-3 1.1e-2 1.3e-2 2120 1.4e-2 1.3e-2 5.9e-3 4.1e-4 9.7e-3 1.1e-2 4250 5.3e-3 5.1e-3 1.5e-4 7.7e-5 3.4e-3 4.8e-3 104

Table: each matrix ∈ R260000×600. Dense matrix format costs 1.25 GB.

Conclusion: already with a small rank a good accuracy can beachieved. Why?

27

Comparison of computing times

rank k Update time, sec. SVD time, sec.10 107 153720 150 208450 228 8236

Table: Computing times of rank-k approximations of W ∈ R260000×600.

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Pressure, density, turb. kinetic energa, eddy viscosity, velosityin x and z directions.

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Conclusion

◮ Two different ways of modelling uncertainties in α, Ma.◮ Uncertainties in the input data α, Ma and in the geometry

strongly influence on the drag CD and weakly on the lift CL.◮ Results obtained with the sparse GH grid are very similar

to the MC results, but require much smaller computingtime.

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Future work

1. Bayesian update of uncertain input parameters

2. Adaptive refinement in stochastic space (number of PCEterms, which ones)

31

Literature

1. A.Litvinenko, H. G. Matthies, Sparse Data Representationof Random Fields, PAMM, 2009.

2. B.N. Khoromskij, A.Litvinenko, H. G. Matthies, Applicationof hierarchical matrices for computing the Karhunen-Loèveexpansion, Springer, Computing, 84:49-67, 2009.

3. B.N. Khoromskij, A.Litvinenko, Data Sparse Computationof the Karhunen-Loève Expansion, AIP ConferenceProceedings, 1048-1, pp. 311-314, 2008.

4. H. G. Matthies, Uncertainty Quantification with StochasticFinite Elements, Encyclopedia of ComputationalMechanics, Wiley, 2007.

32

Acknowledgement

Project MUNA under the framework of the GermanLuftfahrtforschungsprogramm funded by the Ministry ofEconomics (BMWA).

Elmar Zander:A Malab/Octave toolbox for stochastic Galerkin methods(KLE, PCE, sparse grids, tensors, many examples etc)

http://ezander.github.com/sglib/

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Part I. OverviewModelling of free stream turbulenceUncertainties in geometryPart II. Low-rank approximation of the solution