Multiple Tensor Train Approximation of Parametric Constitutive Equations in Elasto-Viscoplasticity

Abstract

This work presents a novel approach to construct surrogate models of parametric differential algebraic equations based on a tensor representation of the solutions. The procedure consists of building simultaneously an approximation given in tensor-train format, for every output of the reference model. A parsimonious exploration of the parameter space coupled with a compact data representation allows alleviating the curse of dimensionality. The approach is thus appropriate when many parameters with large domains of variation are involved. The numerical results obtained for a nonlinear elasto-viscoplastic constitutive law show that the constructed surrogate model is sufficiently accurate to enable parametric studies such as the calibration of material coefficients. Math. Comput. Appl. 2019, 24, 17 2 of 17 the conducting of parametric studies. In particular, the robustness of the calibration process can be dramatically improved using surrogate model approaches.The idea of representing the set of all possible parameter-dependent solutions of ODEs and PDEs as a multiway tensor was pioneered with the introduction of the Proper Generalized Decomposition (PGD) [6][7][8]. In this representation, each dimension corresponds to a spatial/temporal coordinate or a parameter coefficient. The resulting tensor is never assembled explicitly, but instead remains an abstract object for which a low-rank approximation based on a canonical polyadic decomposition [9] is computed. The PGD method further alleviates the curse of dimensionality by introducing a multidimensional weak formulation over the entire parameter space, and the solutions are sought in a particular form where all variables are separated. When differential operators admit a tensor decomposition, the PGD method is very efficient because the multiple integrals involved in the multidimensional weak form of the equations can be rewritten as a sum of products of simple integrals.Unfortunately, realistic constitutive equations or even less sophisticated elasto-viscoplastic models admit no tensor decomposition with respect to the material coefficients and the time variables. An extension of the PGD to highly nonlinear laws is therefore non-trivial. However, many other tensor decomposition approaches have been successfully proposed to approximate functions or solutions of differential equations defined over high-dimensional spaces. We refer the reader to [10-12] for detailed reviews on tensor decomposition techniques and their applications.Among the existing formats-CP decomposition [9,13,14], Tucker decomposition [11,15], hierarchical Tucker decomposition [11,16]-this work investigates the Tensor-Train (TT) decomposition [17,18]. The TT-cross algorithm, introduced in [17] and further developed in [19,20], is a sampling procedure to build an approximation of a given tensor under the tensor-train format. Sampling procedures in parameter space have proven their ability to reduce nonlinear and non-separable DAEs by using the Proper Orthogonal Decomposition (POD) [21], the gappy...