Overview of my (with co-authors) low-rank tensor methods for solving PDEs with uncertain coefficients. Connection with Bayesian Update. Solving a coupled system: stochastic forward and stochastic inverse.

1. Low-rank tensors for PDEs with
uncertain coefficients
Alexander Litvinenko
Center for Uncertainty
Quantification
ntification Logo Lock-up
http://sri-uq.kaust.edu.sa/
Extreme Computing Research Center, KAUST
Alexander Litvinenko Low-rank tensors for PDEs with uncertain coefficients

2. 4*
The structure of the talk
Part I (Stochastic Forward Problem):
1. Motivation
2. Elliptic PDE with uncertain coefficients
3. Discretization and low-rank tensor approximations
Part II (Stochastic Inverse Problem via Bayesian Update):
1. Bayesian update surrogate
2. Examples
Part III (Quantification of uncertainties in aerodynamics)
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3. 4*
My interests and collaborations
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4. 4*
Motivation to do Uncertainty Quantification (UQ)
Motivation: there is an urgent need to quantify and reduce the
uncertainty in multiscale-multiphysics applications.
UQ and its relevance: Nowadays computational predictions are
used in critical engineering decisions. But, how reliable are
these predictions?
Example: Saudi Aramco currently has a simulator,
GigaPOWERS, which runs with 9 billion cells. How sensitive
are these simulations w.r.t. unknown reservoir properties?
My goal is systematic, mathematically founded, develop-
ment of UQ methods and low-rank algorithms relevant for
applications.
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5. 4*
PDE with uncertain coefficient
Consider
− div(κ(x, ω) u(x, ω)) = f(x, ω) in G × Ω, G ⊂ Rd ,
u = 0 on ∂G,
where κ(x, ω) - uncertain diffusion coefficient.
1. Efficient Analysis of High Dimensional Data in Tensor
Formats, Espig, Hackbusch, A.L., Matthies and Zander,
2012.
2. Efficient low-rank approximation of the stochastic
Galerkin matrix in tensor formats, W¨ahnert, Espig, Hack-
busch, A.L., Matthies, 2013.
3. Polynomial Chaos Expansion of random coefficients
and the solution of stochastic partial differential equations
in the Tensor Train format, Dolgov, Litvinenko, Khoromskij,
Matthies, 2016.
0 0.5 1
-20
0
20
40
60
50 realizations of the solution u,
the mean and quantiles
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6. 4*
Canonical and Tucker tensor formats
[Pictures are taken from B. Khoromskij and A. Auer lecture course]
Storage: O(nd ) → O(dRn) and O(Rd + dRn).
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7. 4*
Karhunen Lo´eve and Polynomial Chaos Expansions
Apply both
Truncated Karhunen Lo´eve Expansion (KLE):
κ(x, ω) ≈ κ0(x) + L
j=1 κjgj(x)ξj(θ(ω)), where
θ = θ(ω) = (θ1(ω), θ2(ω), ..., ),
ξj(θ) = 1
κj G (κ(x, ω) − κ0(x)) gj(x)dx.
Truncated Polynomial Chaos Expansion (PCE)
κ(x, ω) ≈ α∈JM,p
κ(α)(x)Hα(θ),
ξj(θ) ≈ α∈JM,p
ξ
(α)
j Hα(θ).
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8. 4*
Discretization of elliptic PDE
Ku = f, where
K:= L
=1 K ⊗ M
µ=1 ∆ µ, K ∈ RN×N, ∆ µ ∈ RRµ×Rµ ,
u:= r
j=1 uj ⊗ M
µ=1 ujµ, uj ∈ RN, ujµ ∈ RRµ ,
f:= R
k=1 fk ⊗ M
µ=1 gkµ, fk ∈ RN and gkµ ∈ RRµ .
(Wahnert, Espig, Hackbusch, Litvinenko, Matthies, 2011)
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9. 4*
Numerical Experiments
2D L-shape domain, N = 557 dofs.
Total stochastic dimension is Mu = Mk + Mf = 20, there are
|JM,p| = 231 PCE coefficients
u =
231
j=1
uj,0 ⊗
20
µ=1
ujµ ∈ R557
⊗
20
µ=1
R3
.
Tensor u has 320 · 557 ≈ 2 · 1012 entries ≈ 16 TB of memory.
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10. 4*
Level sets
Now we compute {ui : ui > b · maxi u},
i := (i1, ..., iM+1)
for b ∈ {0.2, 0.4, 0.6, 0.8}.
The computing time for each b was 10 minutes.
Intermediate ranks of sign(b u ∞1 − u) and of rank(uk )
were less than 24.
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11. 4*
Part II
Part II: Bayesian update
We will speak about Gauss-Markov-Kalman filter for the
Bayesian updating of parameters in a computational model.
Multiple publications with Bojana V. Rosic, Elmar Zander, Oliver Pajonk and H.G. Matthies from TU Braunschweig,
Germany.

12. 4*
Numerical computation of NLBU
Look for ϕ such that q(ξ) = ϕ(z(ξ)), z(ξ) = y(ξ) + ε(ω):
ϕ ≈ ˜ϕ =
α∈Jp
ϕαΦα(z(ξ))
and minimize q(ξ) − ˜ϕ(z(ξ)) 2
L2
, where Φα are polynomials
(e.g. Hermite, Laguerre, Chebyshev or something else).
Taking derivatives with respect to ϕα:
∂
∂ϕα
q(ξ) − ˜ϕ(z(ξ)), q(ξ) − ˜ϕ(z(ξ)) = 0 ∀α ∈ Jp
Inserting representation for ˜ϕ, solve linear system for ϕα.
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13. 4*
Numerical computation of NLBU
Finally, the assimilated parameter qa will be
qa = qf + ˜ϕ(ˆy) − ˜ϕ(z), (1)
z(ξ) = y(ξ) + ε(ω),
˜ϕ = β∈Jp
ϕβΦβ(z(ξ))
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14. 4*
Example: 1D elliptic PDE with uncertain coeffs
− · (κ(x, ξ) u(x, ξ)) = f(x, ξ), x ∈ [0, 1]
+ Dirichlet random b.c. g(0, ξ) and g(1, ξ).
3 measurements: u(0.3) = 22, s.d. 0.2, x(0.5) = 28, s.d. 0.3,
x(0.8) = 18, s.d. 0.3.
κ(x, ξ): N = 100 dofs, M = 5, number of KLE terms 35, beta distribution for κ, Gaussian covκ, cov.
length 0.1, multi-variate Hermite polynomial of order pκ = 2;
RHS f(x, ξ): Mf = 5, number of KLE terms 40, beta distribution for κ, exponential covf , cov. length 0.03,
multi-variate Hermite polynomial of order pf = 2;
b.c. g(x, ξ): Mg = 2, number of KLE terms 2, normal distribution for g, Gaussian covg , cov. length 10,
multi-variate Hermite polynomial of order pg = 1;
pφ = 3 and pu = 3
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15. 4*
Example: Updating of the parameter
0 0.5 1
0
0.5
1
1.5
0 0.5 1
0
0.5
1
1.5
Figure: Original and updated parameter κ.
Collaboration with Y. Marzouk, MIT, and TU Braunschweig. We
try to build an equivalent of KLD for PCE expansion.
Collaborate with H. Najm, Sandia Lab. We try to compare our
technique with his advanced MCMC technique for chemical
combustion eqn.
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16. 4*
Example: updating of the solution u
0 0.5 1
-20
0
20
40
60
0 0.5 1
-20
0
20
40
60
0 0.5 1
-20
0
20
40
60
0 0.5 1
-20
0
20
40
60
0 0.5 1
-20
0
20
40
60
Figure: Original and updated solutions, mean value plus/minus 1,2,3
standard deviations. Number of available measurements {0, 1, 2, 3, 5}
[graphics are built in the stochastic Galerkin library sglib, written by E. Zander in TU Braunschweig]
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17. 4*
Part III: My contribution to MUNA

18. 4*
Example: uncertainties in free stream turbulence
α
v
v
u
u’
α’
v1
2
Random vectors v1(θ) and v2(θ) model free stream turbulence
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19. 4*
Example: 3sigma intervals
Figure: 3σ interval, σ standard deviation, in each point of RAE2822
airfoil for the pressure (cp) and friction (cf) coefficients.
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20. 4*
Mean and variance of density, tke, xv, zv, pressure
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21. 4*
Domain decomposition
Application of domain decomposition and Hierarchical matrices
for solving multi-scale problems.
(a)macroscopic scale (b)microscopic scale (c)molecular scale
Ω
v
T
repeated cells
v
. . . .
.
.
.
.
.
.
.
.
.
.
.
.
mean value
hH
TH Th
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22. 4*
Conclusion
Introduced
Low-rank tensor methods to solve elliptic PDEs with
uncertain coefficients,
Post-processing in low-rank tensor format, computing level
sets
Bayesian update surrogate ϕ (as a linear, quadratic,...
approximation)
Quantification of uncertainties in Numerical Aerodynamics
Domain decomposition and Hierarchical matrices for
multiscale problems
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23. 4*
Thank you
Thank you!
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24. 4*
My experience since 2002
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25. 4*
Literature
1. PCE of random coefficients and the solution of stochastic partial
differential equations in the Tensor Train format, S. Dolgov, B. N.
Khoromskij, A. Litvinenko, H. G. Matthies, 2015/3/11,
arXiv:1503.03210
2. Efficient analysis of high dimensional data in tensor formats, M.
Espig, W. Hackbusch, A. Litvinenko, H.G. Matthies, E. Zander Sparse
Grids and Applications, 31-56, 40, 2013
3. Application of hierarchical matrices for computing the
Karhunen-Loeve expansion, B.N. Khoromskij, A. Litvinenko, H.G.
Matthies, Computing 84 (1-2), 49-67, 31, 2009
4. Efficient low-rank approximation of the stochastic Galerkin matrix
in tensor formats, M. Espig, W. Hackbusch, A. Litvinenko, H.G.
Matthies, P. Waehnert, Comp. & Math. with Appl. 67 (4), 818-829,
2012
5. Numerical Methods for Uncertainty Quantification and Bayesian
Update in Aerodynamics, A. Litvinenko, H. G. Matthies, Book
”Management and Minimisation of Uncertainties and Errors in
Numerical Aerodynamics”, pp 265-282, 2013
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26. 4*
Literature
1. A. Litvinenko and H. G. Matthies, Inverse problems and
uncertainty quantification
http://arxiv.org/abs/1312.5048, 2013
2. L. Giraldi, A. Litvinenko, D. Liu, H. G. Matthies, A. Nouy, To
be or not to be intrusive? The solution of parametric and
stochastic equations - the ”plain vanilla” Galerkin case,
http://arxiv.org/abs/1309.1617, 2013
3. O. Pajonk, B. V. Rosic, A. Litvinenko, and H. G. Matthies, A
Deterministic Filter for Non-Gaussian Bayesian Estimation,
Physica D: Nonlinear Phenomena, Vol. 241(7), pp.
775-788, 2012.
4. B. V. Rosic, A. Litvinenko, O. Pajonk and H. G. Matthies,
Sampling Free Linear Bayesian Update of Polynomial
Chaos Representations, J. of Comput. Physics, Vol.
231(17), 2012 , pp 5761-5787
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