Encapsulated PGD Algebraic Toolbox Operating with High-Dimensional Data
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ORIGINAL PAPER
Encapsulated PGD Algebraic Toolbox Operating with High‑Dimensional Data P. Díez1,2 · S. Zlotnik1,2 · A. García‑González1,2 · A. Huerta1,2 Received: 8 November 2019 / Accepted: 18 November 2019 © CIMNE, Barcelona, Spain 2019
Abstract 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.
1 Introduction Algebraic separation of multidimensional tensors has extensive applications in methods dealing with multidimensional data, and is therefore object of intensive research efforts [12, 13, 16]. Tensor separation is a very efficient strategy for data compression and opens the door to operate in higher dimensions following incremental-iterative approaches [1, 2]. These operations with high-dimensional tensors are a key ingredient in the solution of parametric problems in computational mechanics, in particular for stochastic models [7, 8]. Proper generalized decomposition (PGD) [4, 5] produces separated representations of multidimensional functions. In the discrete format, this is equivalent to obtain separated * P. Díez [email protected] 1
Laboratori de Càlcul Numèric, E.T.S. de Ingeniería de Caminos, Universitat Politècnica de Catalunya, Barcelona, Spain
International Centre for Numerical Methods in Engineering, CIMNE, Barcelona, Spain
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format of multidimensional tensor data. The initial idea is to obtain a separated representation of the (unknown) solution of some boundary-value problem. Here, the same idea is exploited t
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