Uncertainty Quantification, Bayesian Update surrogate, data assimilation, low-rank techniques, numerical aerodynamics, INLA, sparse matrices

1. Uncertainty Quantification and related areas -
An overview
Alexander Litvinenko,
talk given at Department of Mathematical Sciences,
Durham University
Bayesian Computational Statistics & Modeling, KAUST
https://bayescomp.kaust.edu.sa/
Stochastic Numerics Group
http://sri-uq.kaust.edu.sa/
Alexander Litvinenko, talk given at Department of Mathematical Sciences, Durham UniversityUncertainty Quantification and related areas - An overview

2. 4*
The structure of the talk
1. Part I. Introduction to UQ
2. Part II. Low-rank tensors for representation of
big/high-dimensional data
3. Part III. Inverse Problem via Bayesian Update
4. Part IV. R-INLA and advance numerics for spatio-temporal
statistics
5. Part V. High Performance Computing, parallel algorithms
13
13
17
17
14
14 17
13
17
14 15
13 13
17 29
13 48
15
13 13
13 13
15 13
13
13 16
23
8 8
13 15
28 29
8
8 15
8 15
8 15
19
18 18
61
57
23
17 17
17 17
23 35
57 60
61 117
17
17 17
17 17
14 14
14
7 7
14 14
34
21 14
17 14
28 28
10
10 13
17 17
17 17
11 11
17
11 11
69
40
17 11
17 11
36 28
69 68
10
10 11
9 9
10 11
9
9 12
14 14
21 21
14
14
11
11 11
42
14
11 11
11 11
14 22
38 36
12
12 13
12 12
10 10
12
10 10
23
12 10
10 10
15 15
13
10 10
15 15
69
97
49
28
16 15
12 12
21 21
48 48
83 132
48 91
16
12 12
13 12
8 8
13
8 8
26
13 8
13 8
22 21
13
13 13
9 9
13 13
9
9 13
49
26
9 12
9 13
26 22
49 48
12
12 14
12 14
12 14
15
9 9
18 18
26
15 15
14 14
26 35
15
14 14
15 14
15 14
16
16 19
97
68
29
16 18
16 18
29 35
65 64
97 132
18
18 18
15 15
18 18
15
15
14
7 7
33
15 16
15 17
32 32
16
16 17
14 14
16 17
14
14 18
64
33
11 11
14 18
31 31
72 65
11
11
8
8 14
11 18
11 13
18
13 13
33
18 13
15 13
33 31
20
15 15
19 15
18 15
19
18 18
53
87
136
64
35
19 18
14 14
35 35
64 66
82 128
61 90
33 62
8
8 13
14 14
17 14
18 14
17
17 18
29
17 18
10 10
35 35
19
10 10
13 10
19 10
13
13
10
10 14
70
28
13 15
13 13
29 37
56 56
15
13 13
15 13
15 13
19
19
10
10 15
23
11 11
12 12
28 33
11
11 12
11 12
11 12
18
15 15
115
66
23
18 15
18 15
23 30
49 49
121 121
18
18 18
12 12
18 18
12
12 18
22
11 11
11 11
27 27
11
11 11
11 11
10 10
17
10 10
62
22
17 10
17 10
21 21
59 49
13
10 10
18 18
10 10
11 11
10
10 11
27
10 11
10 11
32 21
12
12 15
12 13
12 15
13
13 19
88
115
62
27
13 19
13 14
27 32
62 59
115 121
61 90
10
10 11
14 14
21 14
12 12
14
10 10
12 12
29
14 12
15 12
35 35
14
14 15
11 11
14 15
11
11
8
8 16
69
29
11 18
11 23
28 28
62 62
18
18
8
8 15
15 15
13 13
15
13 13
29
15 13
13 13
33 28
16
13 13
16 13
15 13
18
15 15
135
62
29
18 15
18 15
22 22
69 62
101 101
10
10 11
19 19
15 15
7 7
15
7 7
40
15 7
15 7
40 22
19
19
9
9 13
18 18
19 22
18
18
11
10 10
11 11
62
31
18 20
11 11
31 31
39 39
20
11 11
19 11
12 11
19
12 12
26
12 12
14 12
13 13
12
12 14
13 13
2

3. Part I. Introduction into UQ
3

4. 4*
Motivation to do Uncertainty Quantification (UQ)
Motivation: there is an urgent need to quantify and reduce the
uncertainty in multiscale-multiphysics applications.
Nowadays computational predictions
are used in critical engineering de-
cisions. But, how reliable are these
predictions?
Example: Saudi Aramco currently
has a simulator, TeraPOWERS, which
runs trillion-cell simulation. How
sensitive are these simulations w.r.t.
unknown reservoir properties?
My goal is development of UQ
methods and low-rank algo-
rithms relevant for applications. (pictures are taken from internet)
4

5. 4*
Uncertainty Quantification
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, Litvinenko., Matthies, Zander,
2012.
2. Efficient low-rank approx. of the stoch. Galerkin ma-
trix in tensor formats, W¨ahnert, Espig, Hackbusch, A.L.,
Matthies, 2013.
3. PCE of random coefficients and the solution of stochas-
tic PDEs in the Tensor Train format, Dolgov, Litvinenko,
Khoromskij, Matthies, 2016.
4. Application of H-matrices for computing the KL expan-
sion, Khoromskij, Litvinenko, Matthies Computing 84 (1-2),
49-67, 2009
0 0.5 1
-20
0
20
40
60
50 realizations of the solution u,
the mean and quantiles
Related work by R. Scheichl, Chr. Schwab, A. Teckentrup, F. Nobile, D. Kressner,...
5

6. 4*
Discretisation techniques
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.
1. S. Dolgov, B.N. Khoromskij, A. Litvinenko, H.G. Matthies, Computation of the Response Surface in the
Tensor Train data format, arXiv preprint arXiv:1406.2816, 2014
2. S. Dolgov, B.N. Khoromskij, A. Litvinenko, H.G. Matthies, Polynomial Chaos Expansion of Random Co-
efficients and the Solution of Stochastic Partial Differential Equations in the Tensor Train Format, IAM/ASA
J. Uncertainty Quantification 3 (1), 1109-1135, 2015
6

7. 4*
Generalized Polynomial Chaos Expansions (gPCE)
Decompose ξj(θ) from above:
ξj(θ) ≈
α∈JM,p
ξ
(α)
j Hα(θ),
Hα(θ) could be multivariate Hermite polynomi-
als, α = (α1, ..., αM).
And, combining with KLE above, obtain
κ(x, ω) ≈ κ0(x) +
L
j=1
κjgj(x)
α∈JM,p
ξ
(α)
j Hα(θ)
Need a compact representation of a sum above. → TENSORS!
Can also apply gPCE directly
κ(x, ω) ≈
α∈JM,p
κ(α)
(x)Hα(θ).
7

8. Part II. Low-rank tensor approximations
of big data
Goal: To provide fast and cheap numerical techniques for
working with big and high-dimensional data.
How ?: By reducing linear algebra cost from O(nd ) to
O(drn).
8

9. 4*
Curse of dimensionality
Assume we have nd data. Our aim is to reduce
storage/complexity from O(nd ) to O(dn).
For n = 100 and d = 10, then just to store one needs
8 · 10010 ≈ 8 · 1020 = 8 · 108 TB. If we assume that a modern
computer compares 107 numbers per second, then the total
time for comparison 1020 elements will be 1013 seconds or
≈ 3 ∗ 105 years. In some chemical applications we had n = 100
and d = 800.
how to compute the mean ?
how to compute maxima and minima ?
how to compute level sets, i.e. all elements from an interval
[a, b] ?
how to compute the number of elements in an interval
[a, b] ?
9

10. 4*
Canonical (CP) and Tucker tensor formats
Tensor of order d is a multidimensional array over a d-tuple
index set I = I1 × · · · × Id ,
A = [ai1...id
: i ∈ I ] ∈ RI
, I = {1, ..., n }, = 1, .., d.
Storage: O(nd ) → O(dRn) and O(Rd + dRn).
A. Litvinenko, D. Keyes, V. Khoromskaia, B.N. Khoromskij, H.G. Matthies, Tucker Tensor analysis of Matern
functions in spatial statistics, preprint arXiv:1711.06874, 2017
10

11. 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
, gkµ ∈ RRµ
.
Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats, W¨ahnert, Espig,
Hackbusch, Litvinenko, Matthies, 2013.
We analyzed tensor ranks (compression properties) of the
stochastic Galerkin operator K.
11

12. 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.
Instead we store only
231 · (557 + 20 · 3) ≈ 144000 entries
≈ 1.14 MB.
12

13. 4*
How to compute the variance in CP format
Let u ∈ Rr and
˜u := u − u
d
µ=1
1
nµ
1 =
r+1
j=1
d
µ=1
˜ujµ ∈ Rr+1, (1)
then the variance var(u) of u can be computed as follows
var(u) =
˜u, ˜u
d
µ=1 nµ
=
1
d
µ=1 nµ


r+1
i=1
d
µ=1
˜uiµ

 ,


r+1
j=1
d
ν=1
˜ujν


=
r+1
i=1
r+1
j=1
d
µ=1
1
nµ
˜uiµ, ˜ujµ .
Numerical cost is O (r + 1)2 · d
µ=1 nµ .

14. 4*
Level sets
Now we compute level sets
{ui : ui > b · max
i
u},
i := (i1, ..., iM+1)
for b ∈ {0.2, 0.4, 0.6, 0.8}.
The computing time for each b was 10 minutes.
Key words: u is tensor, Newton methods, inverse u−1, sign(u),
frequency, characteristic function.
13

15. 4*
Computation of level sets and frequency
1. To compute level sets and frequencies we need
characteristic function.
2. To compute characteristic function we need sign(u) function.
3. To compute sign(u) function we need u−1.
Proposition
Let I ⊂ R, u ∈ T , and χI(u) its characteristic. We have
LI(u) = χI(u) u
and rank(LI(u)) ≤ rank(χI(u)) rank(u).
The frequency FI(u) ∈ N of u respect to I is
FI(u) = χI(u), 1 ,
where 1 = d
µ=1
˜1µ, ˜1µ := (1, . . . , 1)T ∈ Rnµ .

16. Part III. Inverse Problem via Bayesian
Update
15

17. 4*
Developing of cheap Bayesian update surrogate
1. H.G. Matthies, E. Zander, B.V. Rosic, A. Litvinenko, Parameter estimation via conditional expectation:
a Bayesian inversion, Advanced modeling and simulation in engineering sciences 3 (1), 24, 2016
2. Inverse Problems in a Bayesian Setting, H.G. Matthies, E. Zander, O. Pajonk, B. Rosic, A. Litvinenko.
Computational Methods for Solids and Fluids Multiscale Analysis, ISSN: 1871-3033, 2016
Related work by A. Stuart, Chr. Schwab, A. El Sheikh, Y. Marzouk, H. Najm, O. Ernst
10 0 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x
20 0 20
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
y
0 10 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z
xf
xa
yf
ya
zf
za

18. 4*
Numerical computation of Bayesian Update surrogate
Notation: ˆy – measurements from engineers, y(ξ) – forecast
from the simulator, ε(ω) – the Gaussian noise.
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 known
polynomials (e.g. Hermite).
Taking derivatives with respect to ϕα:
∂
∂ϕα
q(ξ) − ˜ϕ(z(ξ)), q(ξ) − ˜ϕ(z(ξ)) = 0 ∀α ∈ Jp
16

19. 4*
Numerical computation of NLBU
∂
∂ϕα
E

q2
(ξ) − 2
β∈J
qϕβΦβ(z) +
β,γ∈J
ϕβϕγΦβ(z)Φγ(z)


= 2E

−qΦα(z) +
β∈J
ϕβΦβ(z)Φα(z)


= 2


β∈J
E [Φβ(z)Φα(z)] ϕβ − E [qΦα(z)]

 = 0 ∀α ∈ J .
17

20. 4*
Numerical computation of NLBU
Now, rewriting the last sum in a matrix form, obtain the linear
system of equations (=: A) to compute coefficients ϕβ:



... ... ...
... E [Φα(z(ξ))Φβ(z(ξ))]
...
... ... ...







...
ϕβ
...



 =




...
E [q(ξ)Φα(z(ξ))]
...



 ,
where α, β ∈ J , A is of size |J | × |J |.
A. Litvinenko, H.G. Matthies, Inverse problems and uncertainty quantification, arXiv preprint arXiv:1312.5048, 2013
18

21. 4*
Numerical computation of NLBU
Finally, the assimilated parameter qa will be
qa = qf + ˜ϕ(ˆy) − ˜ϕ(z), (2)
z(ξ) = y(ξ) + ε(ω),
˜ϕ = β∈Jp
ϕβΦβ(z(ξ))
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
2
4
6
κ
PDF
κ
f
κ
a
19

22. 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
20

23. 4*
Example: Updating of the parameter
Figure: Prior and posterior (updated) parameter κ.
Collaboration with Y. Marzouk, MIT, and TU Braunschweig.
Together with H. Najm, Sandia Lab, we try to compare our
technique with his advanced MCMC technique for chemical
combustion eqn.
21

24. 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]
22

25. 4*
Minimization of Uncert. in Num. Aerodynamics
Involved: 7 German Universities,
DLR, Airbus
Duration: 2007-2012.
1. Quantification of airfoil geometry-induced aerodynamic
uncertainties-comparison of approaches, Liu, Litvinenko,
Schillings, Schulz, JUQ 2017
2. Numerical Methods for Uncertainty Quantification and
Bayesian Update in Aerodynamics Litvinenko, Matthies,
chapter in Springer book, Vol 122, pp 262-285, 2013.
23

26. 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
24

27. 4*
Example: 3sigma intervals
Figure: 3σ interval, σ standard deviation, in each point of RAE2822
airfoil for the pressure (cp) and friction (cf) coefficients.
25

28. 4*
Mean and variance of density, tke, xv, zv, pressure
mean density, variance of density, mean turbul. kinetic energy,
x-velocity, z-velocity, a pressure (failed to compute)
26

29. Part IV. INLA and advance numerics for
spatio-temporal statistics
27

30. 4*
Sparse matrices in INLA
Integrated Nested Laplace Approximations (INLA) ap-
proximates Bayesian inference for latent Gaussian mod-
els (LGMs).
Laplace method approximates high-dimensional integrals with a
second order Taylor expansion and computes the integral
analytically. A nested version of this idea, combined with
sparse data techniques is INLA.
My goal: improve sparse matrix techniques in R-INLA project
21 5
5 24
4 2
2 9
4 2
2 9
19 7
7 23
3
3
4
5
1 1
5 4
3
5
3 4 1 5
1 4
18 5
5 35
6
1
2 1
1
6
6
2
1 1
6
1
37 7
7 38
1 4
4 4
2 3 5
2 4
3 6
1 1
1
1
1
3
4
5 1
1 5 9
1
4 2
4 3
4 5
1
1
1
1
1 3
2 3
4 6
5 1
1 5
4 9
26 7
7 33
5 1
4 13
5 4
1
13
22 7
7 39
4 2
4 1
4
1
1
2
6 2
3 4
4
2
1
1
4 4
1
2
6 3
2 4
32 5
5 30
6 1
4 7
6 4
1 7
21 7
7 19
3 5
2 3
3 2
5 3
15 5
5 21
9 10
4
1
3 2
1
1 3
6
9
12 8
1 2
1 3
1 6 9
4 6
1 2
7 6 10
6 6
4 1
1 1
3
3 1
2
2 3
6
5
9
4 3 1
2
1
1 6
3
10 9
4
1 7
2
6
6 10
12
1
1 1
3
2 6
8 9
6
4
1
3 2
1
1 3
1
2 6
3
6 5
35 7
7 28
6 1
2 4
6 2
1
4
30 5
5
12 3
3 24
1
8
1
1
1
1 7
1 8 1 1
1
1
7
29 5
5 23
7 3
1 3
7 1
3 3
27 5
5 25
1 1
4 1
2
1
14
1
2
1
1
1 3
1 2
1 2
3 1
2 1
1 4 1
1
4 2
1
1 14
1 1
1
1 1
2
3 2
2
1
1
3
2 1
1 4
1 1
15 3
3 18 1 3
1
3 35
1
18 1
3 5
1 18
3
1 5
20 5
5 19 5
5 32
1
6 1
2
2 5
1 6 2 2
1 5
31 5
5 30
5 2
2 6
5 2
2 6
21 5
5 16 5
5 29
24 19
2 3
4 2
3 5 6
3
1
2 2
1 1
2 2
1 5
2
6 5
2 6
7
18
27 18
6 10
2
3 4
1 7 15 17
10 10
7
2 4
1
1 1
1 1 2
2 3 14
18 7
5 1
1 5 1
2 5
1 2
2 1 6
2
2 2
1
3 5
5 1 2
1 4 3
12
24
2
4 3
2 5
3 6
1
2
2 1
1 5
3
2
1
2
6 2
5
6
2 7
19 18
10
7 2
2
1
1 1
1 1
4 2 3
10 14
27
6
2
3 1
4
7
10 15
18 17
18
5
1
1
5
1
2
2
2
1
2
1 2
2 1
5 6
3
5
1 1
2 4
5 3
7 12
31 6
6 30
6 2
1 3
6 1
2 3
22 5
5 36
1 2
5 1
1 1
1 1
1
1
1
7 1
1 5
2 1
1
1
1 1
1 1
1 7
1
27 6
6
24 7
7 25
3
2 8 3
3 2
8
3
31 6
6
24 6
6 22
1 3
4 2
2 7 2 1 1
1 2
2
1 1
2
3 1
1 6 3
1
4 2
2 7
3 2
1 2 1
2 1
1
1
3 1
1 6
2 3
25 4
4 36
5 1
2 5
5 2
1 5
23 6
6 23
4
1
2
1
1
5 1
4 1
2
1
5
1 1
23 3
3 25
7 2
2 6
7 2
2 6
26 5
5 41
5 9
1
5 2
2 3 2 2
1
3 1 6
13
3 4
4 2
1 1
2 2 5
4 12
1
1 11
5 4
1
5 1
1
1
1 1
6
3
5
5 2
2 3
1 2
3
1 1
2 6
9 13
4
1
1
12 11
3
4
1 2
1
2 2
4 5
5
5
1
1
1
1 1 6
1
4 3
38
2 1
1 4
2 1
1 4
13 4
4 22
6 2
1 5
6 1
2 5
28 4
4 27
1
4 1
1 3
4
4 4
1 2
2
2
5 2
4 1
1 3
1
1 2
2
4
4 4
5
2 2
35 4
4 35
6 4
1
1
12 2
6 1
4
12
1 2
31 6
6
12 3
3 26
2 2
6 1
2 6 1
2 1
1 1
1 4
1 2
2 2 5
2 2
5 2
1
2 1
1
7 2
4
2
6 2
1 6
2 1
1
1 2
2 2
4 5
2
1
1 1
2
5
1 2
1
2
1 7
2
2 4
38 5
5 37
5 1
1
6
5
1
1 6
38 7
7
15 3
3 18
1
6 2
1
1
1
7 1
6
1 2
1
1 1 7
1
36 6
6 38
5 2
1 4
5 1
2 4
23 4
4 24
28

31. 4*
Two ways to find a low-rank tensor approx. of Mat´ern cov.
Low-rank approximation of Mat´ern covariance
29

32. Part V-a. Parallel algorithms
in environmental statistics
30

33. 4*
Improving of the moisture model
Task: to improve statistical model, which predicts moisture
Given: 2D-Grid with n ≈ 2.5Mi locations
−120 −110 −100 −90 −80 −70
253035404550
Soil moisture
longitude
latitude
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
High-resolution daily soil moisture data at the top layer of the
Mississippi basin, U.S.A., 01.01.2014
Important for agriculture, defense, ...
1. A. Litvinenko, HLIBCov: Parallel Hierarchical Matrix Approximation of Large Covariance Matrices and Likelihoods
with Applications in Parameter Identification, preprint arXiv:1709.08625, 2017
2. A. Litvinenko, Y. Sun, M.G. Genton, D. Keyes, Likelihood Approximation With Hierarchical Matrices For Large
Spatial Datasets, preprint arXiv:1709.04419, 2017
31

34. 4*
Improving of the moisture model
Goal: To improve estimation of un-
known statistical parameters in a spa-
tial soil moisture field.
Log-likelihood function with Mat´ern C(θ) and available data Z:
L(θ) = −
n
2
log(2π) −
1
2
log |C(θ)| −
1
2
Z C(θ)−1
Z.
To identify: unknown parameters θ := (σ2, ν, ).
Collaboration with statisticians: M. Genton, Y. Sun, R. Huser, from KAUST.
n = 512K, matrix setup 261 sec., compression rate 99.98% (0.4 GB against 2006 GB). H-LU is done in
843 sec., error 2 · 10−3
.
32

35. 4*
Parameter identification
64 32 16 8 4 2
n, samples in thousands
0.07
0.08
0.09
0.1
0.11
0.12
ℓ
64 32 16 8 4 2
n, samples in thousands
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
νSynthetic data with known parameters
( ∗, ν∗, σ2∗
) = (0.5, 1, 0.5). Boxplots for and ν for
n = 1, 000 × {64, 32, ..., 4, 2}; 100 replicates
33

36. Part V-b. High Performance Computing
and UQ
34

37. 4*
Parallel algorithms with distributed data
Consider density driven groundwater flow with uncertain
porosity and permeability
· (ρq) = 0, (3)
∂t (φc) + · (cq − D c) = 0. (4)
0
200
400
600
0
50
100
150
0.09
0.095
0.1
0.105
0.11
3
150
4
600
×10 -5
100
5
400
6
50 200
0 0
Mean and variance of porosity
The mean (left) and the variances (right) of the porosity,
[0, 600] × [0, 150], φ(x) ≈ 0.1,
var(φ(x)) ∈ (3.2 · 10−5, 5.2 · 10−5).
We compute ≈ 800 scenarios in parallel, each scenario on 32
cores in parallel. Total: 800 × 32 = 25600 cores.
35

38. 4*
Mean and variance of concentration
The mean and variance of the concentration, computed via QMC.
The number of time steps is 1500; E(c) ∈ (0, 1), var(c) ∈ (0, 0.068).
Two different realizations of concentration: with 5 fingers (left) and 4
fingers (right).
36

39. 4*
Conclusion
Introduced:
1. UQ and many examples,
2. Big data require a compact representation. We suggest
low-rank tensor data techniques.
3. Tensor techniques requires to redefine standard statistical
algorithms
4. Bayesian update surrogate ϕ (as a linear, quadratic,...
approximation)
5. Applications in aerodynamics, geostatistics, reservoirs,
environmental statistics
6. HPC applications: parallel algorithms for UQ on a
supercomputer
37

40. 4*
Literature
1. A. Litvinenko, Application of Hierarchical matrices for solving multiscale
problems, PhD Thesis, Leipzig University, Germany,
https://www.wire.tu-bs.de/mitarbeiter/litvinen/diss.pdf, 2006
2. B.N. Khoromskij, A. Litvinenko, Domain decomposition based H-matrix
preconditioners for the skin problem, Domain Decomposition Methods
in Science and Engineering XVII, pp 175-182, 2006
3. A. Litvinenko, Documentation for the Hierarchical Domain
Decomposition (HDD) method, Technical report, Vol 5, pp 1-33,
Max-Planck Institute for applied mathematics in the science in Leipzig,
http://www.mis.mpg.de/publications/other-series/tr/report-0506.html,
2006
4. A. Litvinenko, Partial inversion of elliptic operator to speed up
computation of likelihood in Bayesian inference, arXiv preprint
1708.02207, https://arxiv.org/abs/1708.02207, 2017
5. W. Nowak, A. Litvinenko, Kriging and spatial design accelerated by
orders of magnitude: Combining low-rank covariance approximations
with FFT-techniques, Mathematical Geosciences 45 (4), 411-435, 2013
38