arXiv subjects

Alex Pereira da Silva

Publications and source records attributed to Alex Pereira da Silva.

2 recordsLinked to original sources

arXiv2017

On the generation of groups of bounded linear operators on Fréchet spaces

In this paper we present a general method for generation of uniformly continuous groups on abstract Fréchet spaces (without appealing to spectral theory) and apply it to a such space of distributions, namely ${\mathscr F}L^{2}_{loc}(\mathbb{R}^{n})$, so that the linear evolution problem \begin{equation*} \left\{\begin{array}{l} u_{t} = a(D)u, t \in \mathbb{R} \\ u(0) = u_0 \end{array} \right. \end{equation*}always has a unique solution in such a space, for every pseudodifferential operator $a(D)$ with constant coefficients. We also provide necessary and sufficient conditions so that the spaces $L^{2}$ and ${\mathscr E}'$ are left invariant by this group; and we conclude that the solution of the heat equation on ${\mathscr F}L^{2}_{loc}(\mathbb{R}^{n})$ for all $t \in \mathbb{R}$ extends the standard solution on Hilbert spaces for $t \geqslant 0$.

math.AP↗

arXiv2015

Rank-1 Tensor Approximation Methods and Application to Deflation

Because of the attractiveness of the canonical polyadic (CP) tensor decomposition in various applications, several algorithms have been designed to compute it, but efficient ones are still lacking. Iterative deflation algorithms based on successive rank-1 approximations can be used to perform this task, since the latter are rather easy to compute. We first present an algebraic rank-1 approximation method that performs better than the standard higher-order singular value decomposition (HOSVD) for three-way tensors. Second, we propose a new iterative rank-1 approximation algorithm that improves any other rank-1 approximation method. Third, we describe a probabilistic framework allowing to study the convergence of deflation CP decomposition (DCPD) algorithms based on successive rank-1 approximations. A set of computer experiments then validates theoretical results and demonstrates the efficiency of DCPD algorithms compared to other ones.

math.NA↗