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Z. Qiu^1,2, F. Magoules^2,3 and D. Pelaez¹

¹Institut des Sciences Moleculaires d’Orsay, Universit´e Paris-Saclay, Orsay, Ile-de-France, France
²MICS, CentraleSupelec, Paris-Saclay University, Gif-sur-Yvette, France
²Faculty of Engineering and Information Technology, University of Pecs, Pecs, Hungary

doi:10.4203/ccc.6.13.4

Z. Qiu, F. Magoules, D. Pelaez, "Analytical Hierarchical Tucker Representation using Binary Trees", in P. Ivanyi, J. Kruis, B.H.V. Topping, (Editors), "Proceedings of the Seventeenth International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Edinburgh, UK, Online volume: CCC 6, Paper 13.4, 2023, doi:10.4203/ccc.6.13.4

Keywords: tensor decomposition, low-rank approximations, finite basis representation, singular value decomposition, analytical data-structure, binary tree.

Abstract

In this contribution we show that it is possible to achieve an analytical binary tree representation for a tensor stemming from an underlying scalar field. As initial datastructure we use a binary tree. This is obtained by a hierarchical Tucker (HT) decomposition of a reference tensor. To achieve this, tensor matricizations are followed by their truncated singular value decompositions. Then we fit the left singular vectors at each node using a set of auxiliary basis functions. These are system-dependent orthogonal polynomials. We call this finite basis representation (FBR). The resulting HT-FBR expression can be reconstructed to grids of any density, within the same domain of definition, while keeping the error reasonably/physically small. This paves the way to the direct optimisation of these compact analytical binary tree structures.

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Front Page Browse CCC CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Conferences ISSN 2753-3239 CCC: 6 PROCEEDINGS OF THE SEVENTEENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING Edited by: P. Ivanyi, J. Kruis and B.H.V. Topping Paper 13.4 Analytical Hierarchical Tucker Representation using Binary Trees Z. Qiu^1,2, F. Magoules^2,3 and D. Pelaez¹ ¹Institut des Sciences Moleculaires d’Orsay, Universit´e Paris-Saclay, Orsay, Ile-de-France, France ²MICS, CentraleSupelec, Paris-Saclay University, Gif-sur-Yvette, France ²Faculty of Engineering and Information Technology, University of Pecs, Pecs, Hungary doi:10.4203/ccc.6.13.4 Full Bibliographic Reference for this paper Z. Qiu, F. Magoules, D. Pelaez, "Analytical Hierarchical Tucker Representation using Binary Trees", in P. Ivanyi, J. Kruis, B.H.V. Topping, (Editors), "Proceedings of the Seventeenth International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Edinburgh, UK, Online volume: CCC 6, Paper 13.4, 2023, doi:10.4203/ccc.6.13.4 Keywords: tensor decomposition, low-rank approximations, finite basis representation, singular value decomposition, analytical data-structure, binary tree. Abstract In this contribution we show that it is possible to achieve an analytical binary tree representation for a tensor stemming from an underlying scalar field. As initial datastructure we use a binary tree. This is obtained by a hierarchical Tucker (HT) decomposition of a reference tensor. To achieve this, tensor matricizations are followed by their truncated singular value decompositions. Then we fit the left singular vectors at each node using a set of auxiliary basis functions. These are system-dependent orthogonal polynomials. We call this finite basis representation (FBR). The resulting HT-FBR expression can be reconstructed to grids of any density, within the same domain of definition, while keeping the error reasonably/physically small. This paves the way to the direct optimisation of these compact analytical binary tree structures. download the full-text of this paper (PDF, 11 pages, 380 Kb) go to the previous paper go to the next paper return to the table of contents return to the volume description
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