SISSA PreprintsThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.http://preprints.sissa.it:80/xmlui2020-01-23T07:48:30Z2020-01-23T07:48:30ZKrilov solvability of unbounded inverse linear problemsCaruso, NoeMichelangeli, Alessandrohttp://preprints.sissa.it:8180/xmlui/handle/1963/353412020-01-23T07:37:33Z2020-01-23T00:00:00ZKrilov solvability of unbounded inverse linear problems
Caruso, Noe; Michelangeli, Alessandro
The abstract issue of ‘Krylov solvability’ is extensively discussed for the inverse problem Af = g where A is a (possibly unbounded) linear operator on an inﬁnite-dimensional Hilbert space, and g is a datum in the range of A. The question consists of whether the solution f can be approximated in the Hilbert norm by ﬁnite linear combinations of g, Ag, A2g, . . . , and whether solutions of this sort exist and are unique. After revisiting the known picture when A is bounded, we study the general case of a densely deﬁned and closed A. Intrinsic operator-theoretic mechanisms are identiﬁed that guarantee or prevent Krylov solvability, with new features arising due to the unboundedness. Such mechanisms are checked in the self-adjoint case, where Krylov solvability is also proved by conjugate-gradient-based techniques
21 p.
2020-01-23T00:00:00ZTwo-dimensional Schrödinger operators with point interactions: Threshold expansions, zero modes and Lp-boundedness of wave operatorsCornean, Horia D.Michelangeli, AlessandroYajima, Kenjihttp://preprints.sissa.it:8180/xmlui/handle/1963/353402019-10-10T23:20:28Z2019-10-10T00:00:00ZTwo-dimensional Schrödinger operators with point interactions: Threshold expansions, zero modes and Lp-boundedness of wave operators
Cornean, Horia D.; Michelangeli, Alessandro; Yajima, Kenji
Mathematics Subject Classiffcation 2010: 35P15, 35J10, 47A40, 81Q10
2019-10-10T00:00:00ZPeriodic solutions of nearly integrable Hamiltonian systems bifurcating from infinite-dimensional toriFonda, AlessandroKlun, GiulianoSfecci, Andreahttp://preprints.sissa.it:8180/xmlui/handle/1963/353392019-09-12T23:20:25Z2019-09-12T00:00:00ZPeriodic solutions of nearly integrable Hamiltonian systems bifurcating from infinite-dimensional tori
Fonda, Alessandro; Klun, Giuliano; Sfecci, Andrea
We prove the existence of periodic solutions of some infinite-dimensional nearly integrable
Hamiltonian systems, bifurcating from infinite-dimensional tori, by the use of a generalization
of the Poincaré–Birkhoff Theorem.
Dedicated to Shair Ahmad, on the occasion of his 85th birthday
2019-09-12T00:00:00ZConvergence of the conjugate gradient method with unbounded operatorsCaruso, NoeMichelangeli, Alessandrohttp://preprints.sissa.it:8180/xmlui/handle/1963/353382019-08-27T23:20:21Z2019-08-27T00:00:00ZConvergence of the conjugate gradient method with unbounded operators
Caruso, Noe; Michelangeli, Alessandro
In the framework of inverse linear problems on infinite-dimensional Hilbert space, we prove the convergence of the conjugate gradient iterates to an exact solution to the inverse problem in the most general case where the self-adjoint, non-negative operator is unbounded and with minimal, technically unavoidable assumptions on the initial guess of the iterative algorithm. The convergence is proved to always hold in the Hilbert space norm (error convergence), as well as at other levels of regularity (energy norm, residual, etc.) depending on the regularity of the iterates. We also discuss, both analytically and through a selection of numerical tests, the main features and differences of our Convergence result as compared to the case, already available in the literature, where the operator is bounded.
2019-08-27T00:00:00Z