使用Tucker分解重建截面的新方法
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"一种使用Tucker分解重建截面的新方法"
在当前的科研领域中，数据的高维性和复杂性往往导致存储和计算的需求急剧增加。针对这一问题，一篇发表在《计算物理》(Journal of Computational Physics)2017年第345期的文章提出了一种新的方法，即利用Tucker分解来重建截面。这种方法特别适用于处理多变量函数，能够在保持数据精度的同时降低存储需求和计算成本。
Tucker分解是多维数据处理中的一个关键技术，它是一种低秩张量分解方法，与矩阵的奇异值分解(SVD)类似，但扩展到了多维数组。与传统的主成分分析(PCA)或Karhunen-Loève分解(KLD)不同，Tucker分解能够保留原始数据的更多结构信息，因此在数据压缩、降维以及重构方面具有显著优势。
文章中提到，对于多维函数的完整表示，随着维度d的增加，所需存储空间呈指数增长，这给计算带来了巨大挑战。在核工程的中子学研究中，截面数据的精确重建至关重要，因为这些数据涉及到反应堆性能的建模和分析。新方法通过Tucker分解将高维函数分解成多个低维组件，每个组件都可以独立处理，从而极大地降低了计算复杂性。
作者Thi Hieu Lua、Yvon Maday、Matthieu Guillot和Pierre Guérin分别来自法国电力公司(EDF Lab Paris-Saclay)的SINETICS部门、巴黎第六大学(UPMC)的Jacques-Louis Lions实验室以及布朗大学的应用数学系。他们在文章中详细阐述了如何运用Tucker分解对截面数据进行重构，以及这种方法在实际应用中的效果。
文章经过多次修订后于2017年5月10日被接受，并于5月15日在线发布。关键词包括：函数重构、低秩张量近似、Tucker分解、Karhunen-Loève分解以及截面。这项工作不仅对压缩感知理论有所贡献，还为解决高维数据处理的实际问题提供了新的工具，对于计算机科学、工程学以及需要处理大量多维数据的其他领域都具有重要价值。
192 T.H. Luu et al. / Journal of Computational Physics 345 (2017) 189–206
The tensor directional basis functions are not a priori chosen but tuned to each cross-section we consider. In particular,
the number of retained tensor directional basis functions (for a given accuracy) depends on the cross-section and also
on the phase direction. Thanks to the Karhunen–Loève decomposition technique, the most important information of each
cross-section is extracted, axis by axis, and represented into few tensor directional basis functions.
The
coefficients in the combination of the tensor products are determined by a system of linear equations traducing
interpolation equalities at some points. Note that the points are the same, whatever the cross-section, this allows us to limit
the number of APOLLO2 calculations performed on these points at the stage of determining the coefficients. In order to
determine the points on which this system depends, we rely on the idea presented in the empirical interpolation method
(EIM) [9].
With
these techniques, we can reconstruct the cross-sections with a high accuracy while reducing significantly the
calculation points and the storage in the neutron library.
In
the light to what was presented at the beginning of the introduction about the four concepts for approximation, our
approach allows us to minimize the number of data acquisition that each involves a use of the code APOLLO2. Since this
data is an important part of the storage, the storage size is therefore significantly reduced. The other part of our approach
is the definition of the tensor directional basis functions. The interpolation process through a tensor shape approach of
each cross-section allows us to minimize the complexity of the function reconstruction with further evaluation in various
different points.
The
paper is organized in the following manner:
Section
2 describes the theoretical background of the Karhunen–Loève decomposition on which our approach is based in
order to construct the tensor directional basis functions.
In
section 3, we present our proposed methodology for the Tucker decomposition in a general case.
Section
4 is reserved for practical applications to our problem: the reconstruction of cross-sections in neutronics. The
implementation procedure as well as the cost of Tucker model will be detailed.
In
section 5, we show the numerical results of a test case in order to compare the Tucker model with the multilinear
model using the following criteria: the number of calculation points, the storage in neutron libraries and the accuracy.
Section
6 is reserved for conclusion and discussion.
2. Theoretical background
2.1. Problem statement
From a mathematical point of view, our problem: the reconstruction of cross-sections, stands as the approximation of
about one thousand multivariate functions { f
k
(x)}
k
, defined on the domain . Here, x = (x
1
, ..., x
d
) ∈ ⊂ R
d
, d is the
number of parameters and =
1
× ...×
d
(
i,1≤i≤d
⊂ R).
The
objective is to acquire information, store the data, and propose a reconstruction of each f
k
with a high [accu-
racy]/[complexity]
ratio. Remember that we expect an absolute accuracy to be of the order of 10
−5
(or pcm).
Our
approach is based on a low-rank tensor technique in order to represent each function f
k
. Low-rank tensor rep-
resentations
(or formats) are widely used to treat the large-scale problems and there are many ways to express them,
depending on the specific domain of application. We refer to [8,10,11] for a general description of different formats. The
Karhunen–Loève decomposition will be at the basis of our approach so we recall in the following subsection some elements
of context.
2.1.1.
Karhunen–Loève decomposition
The
Karhunen–Loève decomposition that was introduced in statistics for continuous random processes [12,13] is also
known in various communities, as e.g. Proper Orthogonal Decomposition (POD) [14], Principal Component Analysis (PCA)
[15,16], Empirical Orthogonal Functions (EOFs) [17]. This decomposition is available for two-dimensional function spaces but
has no direct extension to high-dimensional ones. The interest of this decomposition is that it provides an optimal r-rank
approximation for two-variate functions, with respect to the L
2
-norm.
Let
f = f (x, y) be a two-variate function in L
2
(
x
×
y
). Through the Karhunen–Loève decomposition, f can be ex-
pressed
as follows:
f (x, y) =
+∞
n=1
λ
n
ϕ
n
(x)ψ
n
(y), ∀(x, y) ∈
x
×
y
(2)
where {ϕ
n
(x)}
+∞
n=1
(resp. {ψ
n
(y)}
+∞
n=1
) is a L
2
-orthonormal basis of L
2
(
x
) (resp. L
2
-orthonormal basis of L
2
(
y
)). Further-
more,
each (λ
n
, ϕ
n
(x)) (resp. (λ
n
, ψ
n
(y))) is a couple of eigenvalue–eigenfunction of a Hilbert–Schmidt operator K
(x)
f
(resp.
Hilbert–Schmidt operator K
(y)
f
) with the following definitions:
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