Encapsulated PGD Algebraic Toolbox Operating with High-Dimensional Data
In its original conception, proper generalized decomposition (PGD) provides explicit parametric solutions, denoted as computational vademecums or digital abacuses, to parametric boundary value problems. The PGD approach is extended here to devise a set of algebraic tools enabling to operate with multidimensional tensor data. These tools are designed to store, compress and perform basic operations (in particular divisions) with tensors in separable format. These tools are directly producing the computational vademecums for the resulting high-dimensional tensor data. Thus, the general methodology enables performing nontrivial operations (storage, compression, division, solving linear systems of equations...) for multidimensional tensor data. The idea is based on the principle of the PGD separation, that produces a separable least squares approximation of any multidimensional function. The PGD compression is a particular case, extensively used in practice to compact the separable solution without loss of accuracy. Here, this concept is applied to algebraic tensor structures that are also seen as functions in multidimensional Cartesian domains. Moreover, a straightforward extension of this concept is devised to operate with multidimensional objects stored in the separable format. That allows creating a toolbox of PGD arithmetic operators that is publicly released at https://git.lacan.upc.edu/zlotnik/algebraicPGDtools. Numerical tests demonstrate the performance and efficiency of the toolbox, both for tensor data handling and operation and also in applications pertaining to the discretized version of boundary value problems.
This work is partially funded by Generalitat de Catalunya (Grant No. 1278 SGR 2017-2019) and Ministerio de Economía y Empresa and Ministerio de Ciencia, Innovación y Universidades (Grant No. DPI2017-85139-C2-2-R).
Compliance with Ethical Standards
Conflict of interest
The authors declare that they have no conflict of interest.
- 8.Espig M, Hackbusch W, Litvinenko A, Matthies HG, Zander E (2012) Efficient analysis of high dimensional data in tensor formats. In: Garcke J, Griebel M (eds) Sparse grids and applications, vol 88. Lecture notes in computational science and engineering. Springer, Berlin, Heidelberg, pp 31–56. https://doi.org/10.1007/978-3-642-31703-3_2 CrossRefGoogle Scholar
- 14.Modesto D, Zlotnik S, Huerta A (2015) Proper generalized decomposition for parameterized Helmholtz problems in heterogeneous and unbounded domains: application to harbor agitation. Comput Methods Appl Mech Eng 295:127–149. https://doi.org/10.1016/j.cma.2015.03.026 MathSciNetCrossRefzbMATHGoogle Scholar
- 17.Rozza G (2014) Fundamentals of reduced basis method for problems governed by parametrized PDEs and applications. In: Chinesta F, Ladevèze P (eds) Separated representations and pgd-based model reduction. CISM international centre for mechanical sciences, vol 554. Springer, Vienna, pp 153–227. https://doi.org/10.1007/978-3-7091-1794-1_4 CrossRefGoogle Scholar
- 18.Sibileau A, García-González A, Auricchio F, Morganti S, Díez P (2018) Explicit parametric solutions of lattice structures with proper generalized decomposition (PGD): applications to the design of 3D-printed architectured materials. Comput Mech 62(4):871–891. https://doi.org/10.1007/s00466-017-1534-9 MathSciNetCrossRefzbMATHGoogle Scholar