Repository: Freie Universit├Ąt Berlin, Math Department

Approximating turbulent and non-turbulent events with the Tensor Train decomposition method

von Larcher, T. and Klein, R. (2017) Approximating turbulent and non-turbulent events with the Tensor Train decomposition method. In: Turbulence in the Complex Conditions. Springer. (Submitted)

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Abstract

In recent research on multiscale problems low-rank multilevel approximation methods are found to attack high-dimensional problems successfully and they offer opportunities for compact representation of large data sets [11, 3]. Specifically, hierarchical tensor product decomposition methods such as the Tree-Tucker format, [4], and the Tensor Train format, [5, 13], are promising approaches for application to data that are concerned with cascade-of-scales problems, for instance in turbulent fluid dynamics. Beyond multilinear mathematics, those tensor formats are also successfully applied in e.g., physics or chemistry, where they are used in many body problems and quantum states. Tensors are multidimensional arrays or mathematically more precisely polylinear formats. For example, vectors are tensors of order d = 1, and tensors of order 3 or higher are generally denoted as higher-order tensors. Clearly, the storage requirement of a tensor depends on its order and on the mode sizes, that is, on the number of entries, n, per dimension. A d-dimensional tensor with mode sizes n results in a storage requirement of nd. Thus, in high dimensional problems or in so-called big data applications one has to deal with a massive storage requirement. Tensor product decomposition methods, first mentioned by [6], were developed to overcome that curse of dimensionality. Here, we test the capabilities of the Tensor Train decomposition to both, numerically computed and experimentally measured flow profile data. We aim at capturing coherent structures and self-similar patterns that might be hidden in the data, cf. [10]. Our study is concerned with the question of whether Tensor decomposition methods can support the development of improved understanding and quantitative characterisation of multiscale behavior of turbulent flows, cf. e.g. [14]. Results of tests using synthetic data to evaluate the suitability of the method to generally detect self-similar patterns are published in [17].

Item Type:Book Section
Additional Information:SFB 1114 Preprint: 11/2017
Subjects:Mathematical and Computer Sciences > Mathematics > Applied Mathematics
Divisions:Department of Mathematics and Computer Science > Institute of Mathematics > Geophysical Fluid Dynamics Group
ID Code:2199
Deposited By: Silvia Hoemke
Deposited On:09 Feb 2018 14:45
Last Modified:09 Feb 2018 15:01

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