Browsing by Author "Caruso, Noe"

Now showing 1 - 4 of 4
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    Convergence of the conjugate gradient method with unbounded operators
    (2019-08-27) 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.
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    Krilov solvability of unbounded inverse linear problems
    (SISSA, 2020-01-23) 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
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    On Krylov solutions to infinite-dimensional inverse linear problems
    (SISSA, 2018-11) Caruso, Noe; Michelangeli, Alessandro; Novati, Paolo; Mathematics
    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 the considered inverse problem. The presentation is based on theoretical results together with a series of model examples, and it is corroborated by specific numerical experiments.
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    Truncation and convergence issues for bounded linear inverse problems in Hilbert space
    (SISSA, 2018) Caruso, Noe; Michelangeli, Alessandro; Novati, Paolo; Mathematics
    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.