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    Inverse learning in Hilbert scales
    (Dordrecht [u.a.] : Springer Science + Business Media B.V, 2023) Rastogi, Abhishake; Mathé, Peter
    We study linear ill-posed inverse problems with noisy data in the framework of statistical learning. The corresponding linear operator equation is assumed to fit a given Hilbert scale, generated by some unbounded self-adjoint operator. Approximate reconstructions from random noisy data are obtained with general regularization schemes in such a way that these belong to the domain of the generator. The analysis has thus to distinguish two cases, the regular one, when the true solution also belongs to the domain of the generator, and the ‘oversmoothing’ one, when this is not the case. Rates of convergence for the regularized solutions will be expressed in terms of certain distance functions. For solutions with smoothness given in terms of source conditions with respect to the scale generating operator, then the error bounds can then be made explicit in terms of the sample size.
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    Well-being in amyotrophic lateral sclerosis: A pilot experience sampling study
    (Lausanne : Frontiers Research Foundation, 2014) Real, R.G.; Dickhaus, T.; Ludolph, A.; Hautzinger, M.; Kübler, A.
    Objective: The aim of this longitudinal study was to identify predictors of instantaneous well-being in patients with amyotrophic lateral sclerosis (ALS). Based on flow theory well-being was expected to be highest when perceived demands and perceived control were in balance, and that thinking about the past would be a risk factor for rumination which would in turn reduce well-being. Methods: Using the experience sampling method, data on current activities, associated aspects of perceived demands, control, and well-being were collected from 10 patients with ALS three times a day for two weeks. Results: Results show that perceived control was uniformly and positively associated with well-being, but that demands were only positively associated with well-being when they were perceived as controllable. Mediation analysis confirmed thinking about the past, but not thinking about the future, to be a risk factor for rumination and reduced well-being. Discussion: Findings extend our knowledge of factors contributing to well-being in ALS as not only perceived control but also perceived demands can contribute to well-being. They further show that a focus on present experiences might contribute to increased well-being.