In this paper we analyze the behavior of the LSQR algorithm for the solution of compact operator equations in Hilbert spaces. We present results concerning existence of Krylov solutions and the rate of convergence in terms of an ℓp sequence where p depends on the summability of the singular values of the operator. Under stronger regularity requirements we also consider the decay of the error. Finally we study the approximation of the dominant singular values of the operator attainable with the bidiagonal matrices generated by the Lanczos bidiagonalization and the arising low rank approximations. Some numerical experiments on classical test problems are presented.

VL - 583 UR - https://www.sciencedirect.com/science/article/pii/S0024379519303714 ER - TY - JOUR T1 - On Krylov solutions to infinite-dimensional inverse linear problems JF - Calcolo Y1 - 2019 A1 - Noe Caruso A1 - Alessandro Michelangeli A1 - Paolo Novati AB -We discuss, in the context of inverse linear problems in Hilbert space, the notion of the associated infinite-dimensional Krylov subspace and we produce necessary and sufficient conditions for the Krylov-solvability of a given inverse problem, together with a series of model examples and numerical experiments.

VL - 56 ER - TY - JOUR T1 - Spontaneous morphing of equibiaxially pre-stretched elastic bilayers: The role of sample geometry JF - International Journal of Mechanical Sciences Y1 - 2018 A1 - Noe Caruso A1 - Aleksandar Cvetković A1 - Alessandro Lucantonio A1 - Giovanni Noselli A1 - Antonio DeSimone KW - Bifurcation KW - Elastic bilayer KW - Pre-stretch KW - Shape programming AB -An elastic bilayer, consisting of an equibiaxially pre-stretched sheet bonded to a stress-free one, spontaneously morphs into curved shapes in the absence of external loads or constraints. Using experiments and numerical simulations, we explore the role of geometry for square and rectangular samples in determining the equilibrium shape of the system, for a fixed pre-stretch. We classify the observed shapes over a wide range of aspect ratios according to their curvatures and compare measured and computed values, which show good agreement. In particular, as the bilayer becomes thinner, a bifurcation of the principal curvatures occurs, which separates two scaling regimes for the energy of the system. We characterize the transition between these two regimes and show the peculiar features that distinguish square from rectangular samples. The results for our model bilayer system may help explaining morphing in more complex systems made of active materials.

VL - 149 UR - https://www.sciencedirect.com/science/article/pii/S0020740317311761 ER - TY - RPRT T1 - Truncation and convergence issues for bounded linear inverse problems in Hilbert space Y1 - 2018 A1 - Noe Caruso A1 - Alessandro Michelangeli A1 - Paolo Novati AB - We present a general discussion of the main features and issues that (bounded) inverse linear problems in Hilbert space exhibit when the dimension of the space is infinite. This includes the set-up of a consistent notation for inverse problems that are genuinely infinite-dimensional, the analysis of the finite-dimensional truncations, a discussion of the mechanisms why the error or the residual generically fail to vanish in norm, and the identification of practically plausible sufficient conditions for such indicators to be small in some weaker sense. The presentation is based on theoretical results together with a series of model examples and numerical tests. PB - SISSA UR - http://preprints.sissa.it/handle/1963/35326 U1 - 35637 U2 - Mathematics U4 - 1 ER -