Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces

By Antonio Falcó, Anthony Nouy. 2011

99 citations recorded by the source corpus.

Tensor-based methods are receiving a growing interest in scientific computing for the numerical solution of problems defined in high dimensional tensor product spaces. A family of methods called proper generalized decompositions (PGD) methods have been recently introduced for the a priori construction of tensor approximations of the solution of such problems. In this paper, we give a mathematical analysis of a family of progressive and updated PGDs for a particular class of problems associated with the minimization of a convex functional over a reflexive tensor Banach space.

Subfields: Computational Mathematics, Computational Mechanics, Computational Theory and Mathematics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Convergence results in this direction can be found in [3, 40, 41, 85–87, 180].”
Non-intrusive low-rank separated approximation of high-dimensional stochastic models (2012, result): “Similar representations, although with different solution strategies, have been later investigated in [60, 41, 28].”
Geometric Structures in Tensor Representations (Final Release) (2013, unknown): “An important feature of these formats, in the framework of topological tensor spaces, is the existence of a best approximation in each ﬁxed set of tensors with bounded rank [11].”
Survey of multifidelity methods in uncertainty propagation, inference, and optimization (2018, methodology): “We refer to the review papers [46, 49], the literature [6, 45, 50, 77, 153, 154] and the references therein for details on PGD and its applications.”
Greedy algorithms for high-dimensional non-symmetric linear problems (2012, background): “· ,r (d) k scales like Ndand the total size of the discretization problems is nNd. This numerical strategy has been extensively studied for the resolution of (linear or nonlinear) elliptic problems [20, 13, 10, 5, 18]. More precisely, let ube deﬁned as the unique solution of a minimization problem ”
On Local Convergence of Alternating Schemes for Optimization of Convex Problems in the Tensor Train Format (2013, background): “Many desirable theoretical features of this approach have recently been verified: Results on the existence of best approximations and minimizers of convex functionals from [12, 13] also apply to the hierarchical Tucker format; extending results from [21], manifold properties can be verified [43].”