Abstract
Low-rank multilevel approximation methods are often suited to attack high-dimensional problems successfully and they allow very compact representation of large data sets. Specifically, hierarchical tensor product decomposition methods, e.g., the Tree-Tucker format and the Tensor Train format emerge as a promising approach for application to data that are concerned with cascade-of-scales problems as, e.g., 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. Here, we focus on two particular objectives, that is, we aim at capturing self-similar structures that might be hidden in the data and we present the reconstruction capabilities of the Tensor Train decomposition method tested with 3D channel turbulence flow data.
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Acknowledgements
This research has been funded by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114 ‘Scaling Cascades in Complex Systems’, Project B04 ‘Multiscale Tensor decomposition methods for partial differential equations’. The authors thank Prof. Illia Horenko (CRC 1114 Mercator Fellow) as well as Prof. Reinhold Schneider and Prof. Harry Yserentant for rich discussions and for steady support. Data analysis was conducted using the Tensor library xerus developed by Huber and Wolf [7]. The channel turbulence data were generated and processed using resources of the North-German Supercomputing Alliance (HLRN), Germany, and of the Department of Mathematics and Computer Science, Freie Universität Berlin, Germany. The authors thank Raphael Badel and Christian Hege (both at Zuse Institute Berlin, Germany) for steady support in data processing and data visualisation.
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von Larcher, T., Klein, R. (2019). Approximating Turbulent and Non-turbulent Events with the Tensor Train Decomposition Method. In: Gorokhovski, M., Godeferd, F. (eds) Turbulent Cascades II. ERCOFTAC Series, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-030-12547-9_30
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DOI: https://doi.org/10.1007/978-3-030-12547-9_30
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