Sparse data formats and efficient numerical methods for UQ in … · Low-rank approximation of the...
Transcript of Sparse data formats and efficient numerical methods for UQ in … · Low-rank approximation of the...
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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 [email protected]
September 20, 2010
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Outline
Part I. Overview
Modelling of free stream turbulence
Uncertainties in geometry
Part II. Low-rank approximation of the solution
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Outline
Part I. Overview
Modelling of free stream turbulence
Uncertainties in geometry
Part II. Low-rank approximation of the solution
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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).
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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 .
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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
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Outline
Part I. Overview
Modelling of free stream turbulence
Uncertainties in geometry
Part II. Low-rank approximation of the solution
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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
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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.
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Sparse Gauss-Hermite Quadratures
Figure: Sparse Gauss-Hermite grids of order 2 (13 points) and 3 (29points).
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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
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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)
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500 MC realisations of pressure coeff. (cp) in dependence onαi and Mai
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500 MC realisations of skin friction coeff. (cf) in dependence onαi and Mai
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5% and 95% quantiles for cp from 500 MC realisations.
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5% and 95% quantiles for cf from 500 MC realisations.
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0.75 0.8 0.85 0.9 0.950
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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.
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Outline
Part I. Overview
Modelling of free stream turbulence
Uncertainties in geometry
Part II. Low-rank approximation of the solution
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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]
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21 realisations of RAE-2822 airfoil
Covariance function is of Gaussian typecov(ρ) = 10−5exp(−∑2i=1(xi − yi)2/ℓ2i ).
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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
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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)).
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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
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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?
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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)
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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.
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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