Overview of our latest works in applying low-rank tensor techniques to a) solving PDEs with uncertain coefficients (or multi-parametric PDEs) b) postprocessing high-dimensional data c) compute the largest element, level sets, TOP5% elelments

1. Low-rank tensor methods for analysis of high
dimensional data
Alexander Litvinenko and Mike Espig
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http://sri-uq.kaust.edu.sa/
Extreme Computing Research Center, KAUST
Alexander Litvinenko and Mike Espig Low-rank tensor methods for analysis of high dimensional da

2. 4*
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)

3. 4*
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,
W¨ahnert, 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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4. 4*
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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5. 4*
Canonical and Tucker tensor formats
Definition and Examples of tensors
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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*
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.
Let T := d
µ=1 Rnµ ,
RR(T ) := R
i=1
d
µ=1 viµ ∈ T : viµ ∈ Rnµ ,
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8. 4*
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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9. 4*
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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10. 4*
Computing QoI in low-rank tensor format
Now, we consider how to
find maxima in a high-dimensional tensor

11. 4*
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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12. 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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13. 4*
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
(1)
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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14. 4*
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
=
(2)
=
r
j=1
d
µ=1
1
nµ
nµ
k=1
(ujµ)k , (3)
where ˜1µ := (1, . . . , 1)T ∈ Rnµ .
Numerical cost is O r · d
µ=1 nµ .
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15. 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, (4)
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 .

16. 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].

17. 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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18. 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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19. 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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20. 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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21. 4*
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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22. 4*
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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23. 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
(5)
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
(6)
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.

24. 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)