Possible applications of low-rank tensors in statistics and UQ (my talk in Bonn, Germany)

Possible applications of low-rank tensors in statisticsand UQ

Alexander Litvinenko,Extreme Computing Research Center and Uncertainty

Quantification Center, KAUST(joint work with H.G. Matthies, MIT and KAUST)

Center for UncertaintyQuantification


Center for Uncertainty Quantification Logo Lock-up

http://sri-uq.kaust.edu.sa/

http://sri-uq.kaust.edu.sa/

4*

Problem 1. Predict temperature, velocity, salinity

Grid: 50Mi locations on 50 levels, 4*(X*Y*Z) = 4*500*500*50=50Mi.

High-resolution time-dependent data about Red Sea: zonal velocity and

temperature




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Problem 1. Apply low-rank tensor for

1. Kriging estimates := CsyC−1yy y

2. Estimation of variance σ, is the diagonal of conditional cov.matrix

Css|y = diag(Css − CsyC−1yy Cys

)

,

3. Gestatistical optimal design

ϕA := n−1trace{Css|y}

ϕC := cT(Css − CsyC−1yy Cys

)c

,




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Problem 2. Stochastic Galerkin Operator

Problem 2. Stochastic Galerkin Operator




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Discretization of stoch. PDE − div(κ(p, x)∇u(p, x)) = f (x ,p)

Pictures 1, 2 (poor and rich discretization of p):

(∑

i=1

∆i ⊗ Ki) · (x ⊗ e) = (f ⊗ e) (1)

Picture 3:

(∑

i=1

Ki ⊗∆i) · (x ⊗ e) = (f ⊗ e) (2)Center for UncertaintyQuantification



1 / 1




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Problem 3. Predict moisture, estimate covariance parameters

Grid: 1830× 1329 = 2, 432, 070 locations with 2,153,888observations and 278,182 missing values.

−120 −110 −100 −90 −80 −70

25

30

35

40

45

50

Soil moisture

longitude

latit

ude

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 Mississippibasin, U.S.A., 01.01.2014 (Chaney et al., in review).

Important for agriculture, defense. Moisture is very heterogeneous.




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Problem 4: Identifying uncertain parameters

Given: a vector of measurements z = (z1, ..., zn)T with acovariance matrix C (θ∗) = C (σ2, ν, `).To identify: uncertain parameters (σ2, ν, `).Plan: Maximize the log-likelihood function

L(θ) = −1

2

(N log2π + log det{C (θ)}+ zTC (θ)−1z

),

On each iteration i we have a new matrix C (θi ).




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Solution: Estimation of uncertain parameters

H-matrix rank

3 7 9

cov. le

ngth

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

Box-plots for ` = 0.0334 (domain [0, 1]2) vs different H-matrixranks k = {3, 7, 9}.Which H-matrix rank is sufficient for identification of parametersof a particular type of cov. matrix?




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0 10 20 30 40−4000

−3000

−2000

−1000

0

1000

2000

parameter θ, truth θ*=12

Log−

likelih

ood(θ

)

Shape of Log−likelihood(θ)

log(det(C))

zTC

−1z

Log−likelihood

Figure : Minimum of negative log-likelihood (black) is atθ = (·, ·, `) ≈ 12 (σ2 and ν are fixed)




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Problem 5: Multivariate characteristic function

Multivariate characteristic function




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Problem 5: Multivariate characteristic function

The multivariate characteristic function ϕX(t) of a d-dimensionalrandom vector X = (X1, ...,Xd) with X1,...,Xd independent, is

ϕX(t) =

∫

Rd

pX(y)exp(i〈y, t〉)dy, t = (t1, ..., td) ∈ Rd , (1)

The probability density is

pX(y) =1

(2π)d

∫

Rd

exp(−i〈y, t〉)ϕX(t)dt, y ∈ Rd (2)




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Elliptically contoured multivariate stable distribution

The characteristic function ϕX(t) of the elliptically contouredmultivariate stable distribution is defined as follow:

ϕX(t) = exp

(i(t1, t2) · (µ1, µ2)T −

((t1, t2)

(σ21 00 σ22

)(t1, t2)T

)α/2),

(3)Now the question is to find a separation of

((t1, t2)

(σ21 00 σ22

)(t1, t2)T

)α/2≈

R∑

ν=1

φν,1(t1) · φν,2(t2), (4)




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Multivariate distribution

Let ϕX(t) of some multivariate d-dimensional distribution isapproximated as follow:

ϕX(t) ≈R∑

`=1

d⊗

µ=1

ϕX`,µ(tµ). (5)

pX(y) ≈∫

Rd

exp(−i〈y, t〉)ϕX(t)dt (6)

≈∫

Rd

exp(−id∑

j=1

yj tj)R∑

`=1

d⊗

µ=1

ϕX`,µ(tµ)dt1...dtd (7)

≈R∑

`=1

d⊗

µ=1

∫

Rexp(−iyµtµ)ϕX`,µ(tµ)dtµ ≈

R∑

`=1

d⊗

µ=1

pX`,µ(yµ)

(8)




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Literature

1. PCE of random coefficients and the solution of stochastic partialdifferential equations in the Tensor Train format, S. Dolgov, B. N.Khoromskij, A. Litvinenko, H. G. Matthies, 2015/3/11, arXiv:1503.032102. Efficient analysis of high dimensional data in tensor formats, M. Espig,W. Hackbusch, A. Litvinenko, H.G. Matthies, E. Zander Sparse Grids andApplications, 31-56, 40, 20133. Application of hierarchical matrices for computing the Karhunen-Loeveexpansion, B.N. Khoromskij, A. Litvinenko, H.G. Matthies, Computing84 (1-2), 49-67, 31, 20094. Efficient low-rank approximation of the stochastic Galerkin matrix intensor formats, M. Espig, W. Hackbusch, A. Litvinenko, H.G. Matthies,P. Waehnert, Comp. & Math. with Appl. 67 (4), 818-829, 2012




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Possible applications of low-rank tensors in statistics and UQ (my talk in Bonn, Germany)

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