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Spectral Tensor-Train Decomposition for low-rank surrogate models
Bigoni, Daniele; Engsig-Karup, Allan Peter; Marzouk, Youssef M.
Publication date:2014
Document VersionPublisher's PDF, also known as Version of record
Link back to DTU Orbit
Citation (APA):Bigoni, D., Engsig-Karup, A. P., & Marzouk, Y. M. (2014). Spectral Tensor-Train Decomposition for low-ranksurrogate models. Poster session presented at Spatial Statistics and Uncertainty Quantification onSupercomputers, Bath, United Kingdom.
Spectral tensor-train decompositionfor low-rank surrogate modelsDaniele Bigoni⇤1, Allan P. Engsig-Karup1, Youssef M. Marzouk21 Department of Applied Mathematics and Computer Science, Technical University of Denmark2 Department of Aeronautics and Astronautics, Massachusetts Institute of Technology⇤ Corresponding author: [email protected]
IntroductionThe construction of surrogate models is very important as a mean of acceleration in computational methods
for uncertainty quantification (UQ). When the forward model is particularly expensive compared to the
accuracy loss due to the use of a surrogate – as for example in computational fluid dynamics (CFD) – the
latter can be used for the forward propagation of uncertainty [7] and the solution of inference problems [4].
Figure 1: TT-cross
Software: http://www.compute.dtu.dk/⇠dabi/Python PyPi: TensorToolbox
Problem settingWe consider f 2 L
2
([a, b]
d
), where d � 1 andassume f is a computationally expensive func-tion. Let ⇠ 2 [a, b]
d be random variables enteringthe formulation of a parametric problem. In thecontext of UQ, we might want to:•Compute relevant statistics• Inquire the sensitivity of f to ⇠
• Infer the distribution of ⇠In most real problems, these goals require anhigh number of evaluations of f . Often the con-struction of the surrogate and its evaluation inplace of the original f provides a good payoff.
Tensor-train decompositionLet f be evaluated at all points on a tensor gridX =
Nd
j=1
x
j
, where x
j
= (x
i
j
)
n
j
i
j
=1
for j 2 [1, d].Let A = f (X ).
Discrete tensor-train approximation [5]
For r = (1, r
1
, . . . , r
d�1
, 1), let ATT
be s.t.
A(i
1
, . . . , i
d
) = ATT
(i
1
, . . . , i
d
) + ETT
(i
1
, . . . , i
d
)
ATT
=
rX
↵0,...,↵d
=1
G
1
(↵
0
, i
1
,↵
1
) . . . G
d
(↵
d�1
, i
d
,↵
d
)
The construction can be built through the evalu-ation of f on the most important fibers (Fig. 1),detected using the TT-cross algorithm [6].For example, let f (x, y) = 1
x+y+1
sin(4⇡(x + y))
Figure 2: TT-cross: selection of fibers.
•Existence of low-rank best approximation•Memory complexity: linear in d
•Computational complexity: linear in d
It tackles the curse of dimensionality.
References[1] BIGONI, D., MARZOUK, Y. M., AND ENGSIG-KARUP, A. P. Spectral
tensor-train decomposition.[2] ENGSIG-KARUP, A., MADSEN, M. G., AND GLIMBERG, S. L. A mas-
sively parallel GPU-accelerated model for analysis of fully nonlinearfree surface waves. International Journal for Numerical Methods in
Fluids 70, 1 (2011), 20–36.[3] ENGSIG-KARUP, A. P. Analysis of efficient preconditioned defect
correction methods for nonlinear water waves. International Journal
for Numerical Methods in Fluids, January (2014), 749–773.[4] MARZOUK, Y. M., AND NAJM, H. N. Dimensionality reduction and
polynomial chaos acceleration of Bayesian inference in inverse prob-lems. Journal of Computational Physics 228, 6 (Apr. 2009), 1862–1902.
[5] OSELEDETS, I. Tensor-train decomposition. SIAM Journal on Scien-
tific Computing 33, 5 (2011), 2295–2317.[6] OSELEDETS, I., AND TYRTYSHNIKOV, E. TT-cross approximation for
multidimensional arrays. Linear Algebra and its Applications 432, 1(Jan. 2010), 70–88.
[7] XIU, D., AND KARNIADAKIS, G. E. The Wiener-Askey polynomialchaos for stochastic differential equations. Tech. rep., DTIC Docu-ment, 2003.
Functional TT-decompositionUsing the spectral theory on (non-symmetric)Hilbert-Schmidt kernels, we can constructa functional counterpart of the discrete TT-approximation.
Functional tensor-train approximation [1]
For r = (1, r
1
, . . . , r
d�1
, 1), let fTT
be s.t.
f (x) = f
TT
(x) +R
TT
(x)
f
TT
(x) =
rX
↵0,...,↵d
=1
�
1
(↵
0
, x
1
,↵
1
) · · · �d
(↵
d�1
, x
d
,↵
d
)
where �
i
(↵
i�1
, ·,↵i
) are orthogonal (see [1]).
f
TT
is constructed through the eigenvalue de-composition of Hermitian integral operators de-fined in terms of f . It can be proved that [1]:• for fixed r, f
TT
is optimal• if @
�
f
@x
�11 ···@x�d
d
exists and is continuous, then�
k
(↵
k�1
, ·,↵k
) 2 C�
k
(I
k
) for all k, ↵k�1
and ↵
k
.The latter statement can be relaxed:
FTT-decomposition and Sobolev spaces [1]
Let I ⇢ Rd be closed and bounded, and f 2L
2
!
(I) be a Holder continuous function with ex-ponent > 1/2 such that f 2 Hk
!
(I). Then f
TT
issuch that �
j
(↵
j�1
, ·,↵j
) 2 Hk
!
j
(I
j
) for all j, ↵j�1
and ↵
j
.
Spectral TT-decompositionLet P
N
: L
2
!
(I) ! span
⇣{�
i
}Ni=0
⌘where {�
i
}Ni=0
areorthogonal polynomials:
STT-Projection
P
N
f
TT
=
NX
i=0
c
i
�
i
c
i
=
rX
↵0,...,↵d
=1
�
1
(↵
0
, i
1
,↵
1
) . . . �
d
(↵
d�1
, i
d
,↵
d
)
�
n
(↵
n�1
, i
n
,↵
n
) =
Z
I
n
�
n
(↵
n�1
, x
n
,↵
n
)�
i
n
(x
n
)dx
n
Let ⇧N
: L
2
!
(I) ! span
⇣{l
i
}Ni=0
⌘, {l
i
}Ni=0
being theLagrange polynomials:
STT-Interpolation
⇧
N
f
TT
=
rX
↵0,...,↵d
=1
�
1
(↵
0
, x
1
,↵
1
) · · · �d
(↵
d�1
, x
d
,↵
d
)
�
n
(↵
n�1
, x
n
,↵
n
) = L
(n)
�
n
(↵
n�1
, x
n
,↵
n
)
where L
(n) is the Lagrange interpolation matrix.
Conclusions•Tackles the curse of dimensionality.•Spectral convergence on smooth functions.
Ongoing works•Anisotropic heterogeneous adaptivity.•Ordering problem.•Application in the fields of coastal engineering
[2, 3] and geoscience.
Numerical Examples
102 103 104 105
# func. eval
10-1410-1310-1210-1110-1010-910-810-710-610-510-410-310-210-1
L2er
r
Oscillatory
d=10d=50d=100d=200
��������
�����
�����
103 104 105 106
# func. eval
10-6
10-5
10-4
10-3
10-2
10-1
100
101
L2er
r
Corner Peak
d=10d=15d=20
�������� ���� Genz functions:
f
1
(x) = cos
0
@2⇡w
1
+
dX
i=1
c
i
x
i
1
A
f
2
(x) =
0
@1 +
dX
i=1
c
i
x
i
1
A�(d+1)
The method shows spectralconvergence on both the tests,even on f
2
, when there is noanalytical low-rank represen-tation.For d = 5, we compare thenon-adaptive STT-Projectionwith the anisotropically adap-tive Smolyak Sparse Grid.
Ordering problemTT and STT are negativelyaffected by the wrong or-dering of the dimensions,leading to an increasedcomputational cost and se-vere loss of accuracy.We propose a strategy tofind a good ordering.
(a) Vicinity matrix (b) Undirected graph (c) Hierarchical clustering
We construct a vicinity matrix based on the 2nd order ranks of thetensor. We then need to solve the Traveling Salesman Problem.