Low-rank tensor methods for stochastic forward and inverse problems

Low-rank tensor methods for PDEs with
uncertain coefficients and
Bayesian Update surrogate
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 tensor methods for PDEs with uncertain coefficien

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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
4. Tensor calculus to compute QoI
Part II (Bayesian update):
1. Bayesian update surrogate
2. Examples
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KAUST
I received very rich collaboration experience as a co-organizator of:
3 UQ workshops,
2 Scalable Hierarchical Algorithms for eXtreme Computing
(SHAXC) workshops
1 HPC Conference (www.hpcsaudi.org, 2017)

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My interests and collaborations

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Motivation to do Uncertainty Quantification (UQ)
Motivation: there is an urgent need to quantify and reduce the
uncertainty in output quantities of computer simulations within
complex (multiscale-multiphysics) applications.
Typical challenges: classical sampling methods are often very
inefficient, whereas straightforward functional representations
are subject to the well-known Curse of Dimensionality.
My goal is systematic, mathematically founded, development of
UQ methods and low-rank algorithms relevant for applications.
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UQ and its relevance
Nowadays computational predictions are used in critical
engineering decisions and thanks to modern computers we are
able to simulate very complex phenomena. But, how reliable
are these predictions? Can they be trusted?
Example: Saudi Aramco currently has a simulator,
GigaPOWERS, which runs with 9 billion cells. How sensitive
are the simulation results with respect to the unknown reservoir
properties?
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Part I: Stochastic forward problem
Part I: Stochastic Galerkin method to solve
elliptic PDE with uncertain coefﬁcients

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PDE with uncertain coefficient and RHS
Consider
− div(κ(x, ω) u(x, ω)) = f(x, ω) in G × Ω, G ⊂ R2,
u = 0 on ∂G,
(1)
where κ(x, ω) - uncertain diffusion coefficient. Since κ positive,
usually κ(x, ω) = eγ(x,ω).
For well-posedness see [Sarkis 09, Gittelson 10, H.J.Starkloff
11, Ullmann 10].
Further we will assume that covκ(x, y) is given.
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My previous work
After applying the stochastic Galerkin method, obtain:
Ku = f, where all ingredients are represented in a tensor format
Compute max{u}, var(u), level sets of u, sign(u)
[1] Efficient Analysis of High Dimensional Data in Tensor Formats,
Espig, Hackbusch, A.L., Matthies and Zander, 2012.
Research which ingredients influence on the tensor rank of K
[2] Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats,
Wähnert, Espig, Hackbusch, A.L., Matthies, 2013.
Approximate κ(x, ω), stochastic Galerkin operator K in Tensor
Train (TT) format, solve for u, postprocessing
[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.
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Typical quantities of interest
Keeping all input and intermediate data in a tensor
representation one wants to perform different tasks:
evaluation for specific parameters (ω1, . . . , ωM),
finding maxima and minima,
finding ‘level sets’ (needed for histogram and probability
density).
Example of level set: all elements of a high dimensional tensor
from the interval [0.7, 0.8].
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Canonical and Tucker tensor formats
Deﬁnition and Examples of tensors
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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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Definition of tensor of order d
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.
A is an element of the linear space
Vn =
d
=1
V , V = RI
equipped with the Euclidean scalar product ·, · : Vn × Vn → R,
defined as
A, B :=
(i1...id )∈I
ai1...id
bi1...id
, for A, B ∈ Vn.
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Examples of rank-1 and rank-2 tensors
Rank-1:
f(x1, ..., xd ) = exp(f1(x1) + ... + fd (xd )) = d
j=1 exp(fj(xj))
Rank-2: f(x1, ..., xd ) = sin( d
j=1 xj), since
2i · sin( d
j=1 xj) = ei d
j=1 xj
− e−i d
j=1 xj
Rank-d function f(x1, ..., xd ) = x1 + x2 + ... + xd can be
approximated by rank-2: with any prescribed accuracy:
f ≈
d
j=1(1 + εxj)
ε
−
d
j=1 1
ε
+ O(ε), as ε → 0
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Tensor and Matrices
Rank-1 tensor
A = u1 ⊗ u2 ⊗ ... ⊗ ud =:
d
µ=1
uµ
Ai1,...,id
= (u1)i1
· ... · (ud )id
Rank-1 tensor A = u ⊗ v, matrix A = uvT , A = vuT , u ∈ Rn,
v ∈ Rm,
Rank-k tensor A = k
i=1 ui ⊗ vi, matrix A = k
i=1 uivT
i .
Kronecker product of n × n and m × m matrices is a new block
matrix A ⊗ B ∈ Rnm×nm, whose ij-th block is [AijB].
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Discretization of elliptic PDE
Now let us discretize our diffusion equation with
uncertain coefﬁcients
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Karhunen Loéve and Polynomial Chaos Expansions
Apply both
Karhunen Loéve Expansion (KLE):
κ(x, ω) = κ0(x) + ∞
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.
Polynomial Chaos Expansion (PCE)
κ(x, ω) = α κ(α)(x)Hα(θ), compute ξj(θ) = α∈J ξ
(α)
j Hα(θ),
where ξ
(α)
j = 1
κj G κ(α)(x)gj(x)dx.
Further compute ξ
(α)
j ≈ s
=1(ξ )j
∞
k=1(ξ , k )αk
.
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Final discretized stochastic PDE
Ku = f, where
K:= s
=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)
Examples of stochastic Galerkin matrices:
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Computing QoI in low-rank tensor format
Now, we consider how to
ﬁnd maxima in a high-dimensional tensor

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Maximum norm and corresponding index
Let u = r
j=1
d
µ=1 ujµ ∈ Rr , compute
u ∞ := max
i:=(i1,...,id )∈I
|ui| = max
i:=(i1,...,id )∈I
r
j=1
d
µ=1
ujµ iµ
.
Computing u ∞ is equivalent to the following e.v. problem.
Let i∗
:= (i∗
1 , . . . , i∗
d ) ∈ I, #I = d
µ=1 nµ.
u ∞ = |ui∗ | =
r
j=1
d
µ=1
ujµ i∗
µ
and e(i∗
)
:=
d
µ=1
ei∗
µ
,
where ei∗
µ
∈ Rnµ the i∗
µ-th canonical vector in Rnµ (µ ∈ N≤d ).
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Then
u e(i∗
)
=


r
j=1
d
µ=1
ujµ




d
µ=1
ei∗
µ

 =
r
j=1
d
µ=1
ujµ ei∗
µ
=
r
j=1
d
µ=1
(ujµ)i∗
µ
ei∗
µ
=


r
j=1
d
µ=1
(ujµ)i∗
µ


ui∗ =
d
µ=1
e(i∗
µ) = ui∗ e(i∗
)
.
Thus, we obtained an “eigenvalue problem”:
u e(i∗
)
= ui∗ e(i∗
)
.
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Computing u ∞, u ∈ Rr by vector iteration
By deﬁning the following diagonal matrix
D(u) :=
r
j=1
d
µ=1
diag (ujµ) µ µ∈N≤nµ
(2)
with representation rank r, obtain D(u)v = u v.
Now apply the well-known vector iteration method (with rank
truncation) to
D(u)e(i∗
)
= ui∗ e(i∗
)
,
obtain u ∞.
[Approximate iteration, Khoromskij, Hackbusch, Tyrtyshnikov 05],
and [Espig, Hackbusch 2010]
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How to compute the mean value in CP format
Let u = r
j=1
d
µ=1 ujµ ∈ Rr , then the mean value u can be
computed as a scalar product
u =


r
j=1
d
µ=1
ujµ

 ,


d
µ=1
1
nµ
˜1µ

 =
r
j=1
d
µ=1
ujµ, ˜1µ
nµ
=
(3)
=
r
j=1
d
µ=1
1
nµ
nµ
k=1
(ujµ)k , (4)
where ˜1µ := (1, . . . , 1)T ∈ Rnµ .
Numerical cost is O r · d
µ=1 nµ .
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Numerical Experiments
2D L-shape domain, N = 557 dofs.
Total stochastic dimension is Mu = Mk + Mf = 20, there are
|J | = 231 PCE coefﬁcients
u =
231
j=1
uj,0 ⊗
20
µ=1
ujµ ∈ R557
⊗
20
µ=1
R3
.
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Level sets
Now we compute level sets
sign(b u ∞1 − u)
for b ∈ {0.2, 0.4, 0.6, 0.8}.
Tensor u has 320 ∗ 557 ≈ 2 · 1012 entries ≈ 16 TB of
memory.
The computing time of one level set was 10 minutes.
Intermediate ranks of sign(b u ∞1 − u) and of rank(uk )
were less than 24.
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Part II
Part II: Bayesian update
We will speak about Gauss-Markov-Kalman ﬁlter for the
Bayesian updating of parameters in comput. model.

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Mathematical setup
Consider
K(u; q) = f ⇒ u = S(f; q),
where S is solution operator.
Operator depends on parameters q ∈ Q,
hence state u ∈ U is also function of q:
Measurement operator Y with values in Y:
y = Y(q; u) = Y(q, S(f; q)).
Examples of measurements:
y(ω) = D0
u(ω, x)dx, or u in few points
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Random QoI
With state u a RV, the quantity to be measured
y(ω) = Y(q(ω), u(ω)))
is also uncertain, a random variable.
Noisy data: ˆy + (ω),
where ˆy is the “true” value and a random error .
Forecast of the measurement: z(ω) = y(ω) + (ω).
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Conditional probability and expectation
Classically, Bayes’s theorem gives conditional probability
P(Iq|Mz) =
P(Mz|Iq)
P(Mz)
P(Iq) (or πq(q|z) =
p(z|q)
Zs
pq(q));
Expectation with this posterior measure is conditional
expectation.
Kolmogorov starts from conditional expectation E (·|Mz),
from this conditional probability via P(Iq|Mz) = E χIq
|Mz .
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Conditional expectation
The conditional expectation is deﬁned as
orthogonal projection onto the closed subspace L2(Ω, P, σ(z)):
E(q|σ(z)) := PQ∞ q = argmin˜q∈L2(Ω,P,σ(z)) q − ˜q 2
L2
The subspace Q∞ := L2(Ω, P, σ(z)) represents the available
information.
The update, also called the assimilated value
qa(ω) := PQ∞ q = E(q|σ(z)), is a Q-valued RV
and represents new state of knowledge after the measurement.
Doob-Dynkin: Q∞ = {ϕ ∈ Q : ϕ = φ ◦ z, φ measurable}.
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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, obtain:
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∂
∂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 .
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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 |.
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We can rewrite the system above in the compact form:
[Φ] [diag(...wi...)] [Φ]T




...
ϕβ
...



 = [Φ]


w0q(ξ0)
...
wNq(ξN)


[Φ] ∈ RJα×N, [diag(...wi...)] ∈ RN×N, [Φ] ∈ RJα×N.
Solving this system, obtain vector of coefﬁcients (...ϕβ...)T for
all β.
Finally, the assimilated parameter qa will be
qa = qf + ˜ϕ(ˆy) − ˜ϕ(z), (5)
z(ξ) = y(ξ) + ε(ω), ˜ϕ = β∈Jp
ϕβΦβ(z(ξ))
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Example: Lorenz 1963 problem (chaotic system of ODEs)
˙x = σ(ω)(y − x)
˙y = x(ρ(ω) − z) − y
˙z = xy − β(ω)z
Initial state q0(ω) = (x0(ω), y0(ω), z0(ω)) are uncertain.
Solving in t0, t1, ..., t10, Noisy Measur. → UPDATE, solving in
t11, t12, ..., t20, Noisy Measur. → UPDATE,...
IDEA of the Bayesian Update (BU):
Take qf (ω) = q0(ω).
Linear BU: qa = qf + K · (z − y)
Non-Linear BU: qa = qf + H1 · (z − y) + (z − y)T · H2 · (z − y).
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Trajectories of x,y and z in time. After each update (new
information coming) the uncertainty drops. [O. Pajonk, B. V. Rosic, A.
Litvinenko, and H. G. Matthies, 2012]
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Example: Lorenz problem
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
x
f
xa
y
f
ya
z
f
za
Figure: quadratic BU surrogate, measure the state (x(t), y(t), z(t)).
Prior and posterior after one update.
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Example: Lorenz Problem
10 5 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
x1
x2
15 10 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
y
y1
y2
5 10 15
0
0.1
0.2
0.3
0.4
0.5
0.6
z
z1
z2
Figure: Comparison of the posterior functions computed by linear and
quadratic BU after second update.
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Example: Lorenz Problem
20 0 20
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
x
50 0 50
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
y
0 10 20
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
z
x
f
xa
y
f
ya
z
f
za
Figure: Quadratic measurement (x(t)2
, y(t)2
, z(t)2
): Comparison of a
priori and a posterior for NLBU
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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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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
Figure: Original and updated solutions, mean value plus/minus 1,2,3
standard deviations
[graphics are built in the stochastic Galerkin library sglib, written by E. Zander in TU Braunschweig]
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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.
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4*
Future plans and possible collaboration
Future plans and possible collaboration ideas

4*
Future plans, Idea N1
Possible collaboration work with Troy Butler: To develop a
low-rank adaptive goal-oriented Bayesian update technique. The
solution of the forward and inverse problems will be considered as a
whole adaptive process, controlled by error/uncertainty estimators.
z
(y - z) q
f ε
forward update
low-rank and adaptive
y
f z
(y - z)
ε
forward
y q.....
low-rank and adaptive
... q
update
Stochastic forward spatial discret.
stochastic discret.
low-rank approx.
Inverse problem
Errors
inverse operator approx.

4*
Edge between Green functions in PDEs and covariance
matrices.
Possible collaboration with statistical group, Doug Nychka
(NCAR), Havard Rue
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4*
Data assimilation techniques, Bayesian update surrogare.
Develop non-linear, non-Gaussian Bayesian update
approximation for gPCE coefﬁcients.
Possible collaboration with Jan Mandel, Troy Butler, Kody Law,
Y. Marzouk, H. Najm, TU Braunschweig and KAUST

4*
Collaborators
1. Uncertainty quantiﬁcation and Bayesian Update: Prof. H.
Matthies, Bojana V. Rosic, Elmar Zander, Oliver Pajonk
from TU Braunschweig, Germany,
2. Low-rank tensor calculus: Mike Espig from RWTH Aachen,
Boris and Venera Khoromskij from MPI Leipzig
3. Spatial and environmental statistics: Marc Genton, Ying
Sun, Raphael Huser, Brian Reich, Ben Shaby and David
Bolin.
4. Some others: UQ, data assimilation, high-dimensional
problems/statistics

4*
Conclusion
Introduced low-rank tensor methods to solve elliptic PDEs
with uncertain coefﬁcients,
Explained how to compute the maximum, the mean, level
sets,... in low-rank tensor format,
Derived Bayesian update surrogate ϕ (as a linear,
quadratic, cubic etc approximation), i.e. compute
conditional expectation of q, given measurement y.
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4*
Example: Canonical rank d, whereas TT rank 2
d-Laplacian over uniform tensor grid. It is known to have the
Kronecker rank-d representation,
∆d = A⊗IN ⊗...⊗IN +IN ⊗A⊗...⊗IN +...+IN ⊗IN ⊗...⊗A ∈ RI⊗d ⊗I⊗d
(6)
with A = ∆1 = tridiag{−1, 2, −1} ∈ RN×N, and IN being the
N × N identity. Notice that for the canonical rank we have rank
kC(∆d ) = d, while TT-rank of ∆d is equal to 2 for any
dimension due to the explicit representation
∆d = (∆1 I) ×
I 0
∆1 I
× ... ×
I 0
∆1 I
×
I
∆1
(7)
where the rank product operation ”×” is deﬁned as a regular
matrix product of the two corresponding core matrices, their
blocks being multiplied by means of tensor product. The similar
bound is true for the Tucker rank rankTuck (∆d ) = 2.

4*
Advantages and disadvantages
Denote k - rank, d-dimension, n = # dofs in 1D:
1. CP: ill-posed approx. alg-m, O(dnk), hard to compute
approx.
2. Tucker: reliable arithmetic based on SVD, O(dnk + kd )
3. Hierarchical Tucker: based on SVD, storage O(dnk + dk3),
truncation O(dnk2 + dk4)
4. TT: based on SVD, O(dnk2) or O(dnk3), stable
5. Quantics-TT: O(nd ) → O(dlogq
n)

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, (8)
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 .

4*
Computing QoI in low-rank tensor format
Now, we consider how to
ﬁnd ‘level sets’,
for instance, all entries of tensor u from interval [a, b].

4*
Definitions of characteristic and sign functions
1. To compute level sets and frequencies we need
characteristic function.
2. To compute characteristic function we need sign function.
The characteristic χI(u) ∈ T of u ∈ T in I ⊂ R is for every multi-
index i ∈ I pointwise defined as
(χI(u))i :=
1, ui ∈ I,
0, ui /∈ I.
Furthermore, the sign(u) ∈ T is for all i ∈ I pointwise defined
by
(sign(u))i :=



1, ui > 0;
−1, ui < 0;
0, ui = 0.
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4*
sign(u) is needed for computing χI(u)
Lemma
Let u ∈ T , a, b ∈ R, and 1 = d
µ=1
˜1µ, where
˜1µ := (1, . . . , 1)t ∈ Rnµ .
(i) If I = R<b, then we have χI(u) = 1
2 (1 + sign(b1 − u)).
(ii) If I = R>a, then we have χI(u) = 1
2(1 − sign(a1 − u)).
(iii) If I = (a, b), then we have
χI(u) = 1
2 (sign(b1 − u) − sign(a1 − u)).
Computing sign(u), u ∈ Rr , via hybrid Newton-Schulz iteration
with rank truncation after each iteration.
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4*
Level Set, Frequency
Definition (Level Set, Frequency)
Let I ⊂ R and u ∈ T . The level set LI(u) ∈ T of u respect to I is
pointwise defined by
(LI(u))i :=
ui, ui ∈ I ;
0, ui /∈ I ,
for all i ∈ I.
The frequency FI(u) ∈ N of u respect to I is defined as
FI(u) := # supp χI(u).
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4*
Computation of level sets and frequency
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 .
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