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Tensor methods have emerged as an indispensable tool for the numerical solution of high-dimensional problems in computational science, and in particular problems arising in stochastic and parametric analyses. In many practical situations, the approximation of functions of multiple parameters (or random variables) is made computationally tractable by using low-rank tensor formats. Here, we present some results on rank-structured approximations and we discuss the connection between best approximation problems in tree-based low-rank formats and the problem of finding optimal low-dimensional subspaces for the projection of a tensor. Then, we present constructive algorithms that adopt a subspace point of view for the computation of sub-optimal low-rank approximations with respect to a given norm. These algorithms are based on the construction of sequences of suboptimal but nested subspaces.

Keywords: high dimensional problems - tensor numerical methods - projection-based model order reduction - low-rank tensor formats - greedy algorithms - proper generalized decomposition - uncertainty quantification - parametric equations
Tensor methods have emerged as an indispensable tool for the numerical solution of high-dimensional problems in computational science, and in particular problems arising in stochastic and parametric analyses. In many practical situations, the approximation of functions of multiple parameters (or random variables) is made computationally tractable by using low-rank tensor formats. Here, we present some results on rank-structured approximations ...

65D15 ; 35J50 ; 41A63 ; 65N12 ; 15A69 ; 46B28 ; 46A32 ; 41A46 ; 41A15

Smooth parametrization consists in a subdivision of mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the talk is to provide a short overview of some results and open problems on smooth parametrization and its applications in several apparently separated domains: Smooth Dynamics, Diophantine Geometry, and Approximation Theory. The structure of the results, open problems, and conjectures in each of these domains shows in many cases a remarkable similarity, which I'll try to stress. Sometimes this similarity can be easily explained, sometimes the reasons remain somewhat obscure, and it motivates some natural questions discussed in the talk. I plan to present also some new results, connecting smooth parametrization with "Remez-type" (or "Norming") inequalities for polynomials restricted to analytic varieties. Smooth parametrization consists in a subdivision of mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the talk is to provide a short overview of some results and open problems on smooth parametrization and its applications in several apparently separated domains: Smooth Dynamics, Diophantine Geometry, and Approximation ...

37C05 ; 11Gxx ; 41A46

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