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Sampling and Low-Rank Tensor Approximations
Alexander Litvinenko∗
H. G. Matthies∗
∗TU Braunschweig, Brunswick, Germany
http://www.wire.tu-bs.de
$Id: 12_Sydney-MCQMC.tex,v 1.3 2012/02/12 16:52:28 hgm Exp $
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Introduction, aims
Uncertain Input: Variables (α, Ma), geometry of airfoil
Uncertain solution:
1. statistical moments of (v, p, ρ), exceedance probab. P (v ≤ v∗)2. pdf of CL and of CD, position of shock.
Our aims:
1. Low-rank representation of the input data (random fields)
2. Use the deterministic solver as a black box
3. A low-rank format for the solution
4. Postprocessing in the low-rank format
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Example 1:
5% and 95% quantiles for cp from 500 MC realisations.
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Example 2:
5% and 95% quantiles for cf from 500 MC realisations.
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Compression of PCE coefficients
Let RF q(x,θ), θ = (θ1, ..., θM , ...) is approximated:
q(x,θ) =∑β∈J
qβ(x)Hβ(θ) (1)
qβ(x) =1
β!
∫Θ
Hβ(θ)q(x,θ)P(dθ) ≈ 1
β!
nq∑i=1
Hβ(θi)q(x,θi)wi,
where nq - number of quadrature points.
Using low-rank format, obtain
qβ(x) =1
β![q(x,θ1), ..., q(x,θnq)] · [Hβ(θ1)w1, ...,Hβ(θnq)wnq]
T
Denote
cβ :=1
β![Hβ(θ1)w1, ...,Hβ(θnq)wnq]
T ∈ Rnq (2)
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6and approximate the set of realisations in low-rank format:
[q(x,θ1), ..., q(x,θnq)] ≈ ABT .
The matrix of all PCE coefficients will be
RN×|J | 3 [...qβ(x)...] ≈ ABT [...cβ...], β ∈ J . (3)
Put all together, obtain low-rank representation of RS
q(x,θ) =∑β∈J
qβ(x)Hβ(θ) = [...qβ(x)...]HT (θ), (4)
where H(θ) = (..., Hβ(θ), ...).
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Application of response surface
Now, having discretised RS
q(x,θ) ≈ q(θ) = ABT [...cβ...]HT (θ) (5)
Sample RV θ 106 times and then use the obtained sample to compute
• errorbars,
• quantiles,
• cumulative density function.
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Sampling of response surface with residual
Collocation points are θ`, ` = 1..Z.
Algorithm:
1. Compute RS q(x,θ) ≈ ABT [...cβ...]HT (θ) from ` points.
2. (` = `+ 1), evaluate RS in θ`+1 , obtain q(x,θ`+1).
3. Compute residual ‖r(q(x,θ))‖. Only if ‖r‖ is large, solve expensive
determ. problem.
4. Update A, BT , [...cβ...] and go to (2).
If we are lucky, we solve the determ. problem only few times, otherwise
we must solve the determ. problem Z times for all θ1,...,θZ.
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Conjunction of two low-rank matrices
Wk = [q(x,θ1), ..., q(x,θZ)], W′= [q(x,θZ+1), ..., q(x,θZ+m)]
• Suppose Wk = ABT ∈ Rn×Z is given
• Suppose W′ ∈ Rn×m contains new m solution vectors
• Compute C ∈ Rn×k and D ∈ Rm×k such that W′ ≈ CDT .
• Build Anew := [AC] ∈ Rn×2k and
BTnew = blockdiag[BT DT ] ∈ R2k×(Z+m)
• Rank-k truncation of Wnew = AnewBTnew costs
O((n+ Z +m)k2 + k3)
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Mean and variance in the rank-k format
u :=1
Z
Z∑i=1
ui =1
Z
Z∑i=1
A · bi = Ab. (6)
Cost is O(k(Z + n)).
C =1
Z − 1WcW
Tc ≈
1
Z − 1UkΣkΣ
TkU
Tk . (7)
Cost is O(k2(Z + n)).
Lemma: Let ‖W−Wk‖2 ≤ ε, and uk be a rank-k approximation of the
mean u. Then a) ‖u− uk‖ ≤ ε√Z
,
b) ‖C−Ck‖ ≤ 1Z−1ε
2.
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Tensor product structure
Story does not end here as one may choose S =⊗
k Sk,
approximated by SB =⊗K
k=1 SBk, with SBk ⊂ Sk.
Solution represented as a tensor of grade K + 1
in WB,N =(⊗K
k=1 SBk)⊗ UN .
For higher grade tensor product structure, more reduction is possible,
— but that is a story for another talk, here we stay with K = 1.
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Use in MC sampling solution—storage
Inflow and air-foil shape uncertain.
Data compression achieved by updated SVD:
Made from 600 MC Simulations, SVD is updated every 10 samples.
n = 260, 000 Z = 600
Updated SVD: Relative errors, memory requirements:rank k pressure turb. kin. energy memory [MB]
10 1.9e-2 4.0e-3 21
20 1.4e-2 5.9e-3 42
50 5.3e-3 1.5e-4 104
Dense matrix ∈ R260000×600 costs 1250 MB storage.
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Trans-sonic flow with shock with Z = 2600 samples
Relative error for the density mean for rank k = 5, 10, 30, 50.
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Trans-sonic flow with shock with Z = 2600 samples
Relative error for the density variance for rank k = 5, 10, 30, 50.
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Abbildung 1: Comparison of the mean pressures computed with PCE and
with MC. (Left) ∆p := |pPCE137 − pMC|, Case 1 without shock, (Right)
∆p := |pPCE201 − pMC|, Case 9 with shock.
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Errors, Case 1
Abbildung 2: (left) Relative errors in the Frobenius and the maximum
norms for pressure and density. (right) 10 points (α,Ma) were chosen in
the neigbourhood of α = 1.93 and Ma = 0.676.
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Density
The mean density and variance of the density. Case 9, RAE-2822 airfoil.
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Density with preconditioning failed
Density evaluated from two different PCE-based response surfaces (of
order p = 2 and p = 4). Failed.
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Conclusion
• Can compress the set of simulations via SVD.
• The large rank k the more accurate is the approximation and higher
memory requirement.
• PCE produces results which are similar to MC and requires a smaller
number of determ. computations.
• PCE coeffs are computed on sparse GH grid (29 and 201 nodes). For
the mean value there is no difference. The variance on 201 nodes is
better.
• PCE can be used to build response surface/surrogate for statistics.
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Literature
1. Wahnert, A. L. et al., Approximation of the stoch. Galerkin matrix in thelow-rank canonical tensor format , Comp. and Math. with Appl., 2012;
2. Espig, A. L., et al., Efficient low-rank approximation of the stochastic Galerkinmatrix in tensor formats, Springer 2012.
3. H. G. M., E. Zander, Solving stochastic systems with low-rank tensorcompression, Linear Algebra and its Appl., Vol. 436, Issue 10, pp. 3819-3838, 2012.
4. A. L. and H. G. M. Uncertainty Quantification in numerical Aerodynamic vialow-rank Response Surface, PAMM Proc. Appl. Math. Mech., GAMM Darmstadt2012;
5. B. V. Rosic, A. L., O. Pajonk and H. G. M. Sampling-free linear Bayesian updateof polynomial chaos representations, J. of Comp. Phys. 2012;
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