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search.filters.applied.f.access: openAccess × DDC: 510 × Type: article × Publisher: Les Ulis : EDP Sciences ×

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Now showing 1 - 5 of 5
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    Convergence bounds for empirical nonlinear least-squares
    (Les Ulis : EDP Sciences, 2022) Eigel, Martin; Schneider, Reinhold; Trunschke, Philipp
    We consider best approximation problems in a nonlinear subset ℳ of a Banach space of functions (𝒱,∥•∥). The norm is assumed to be a generalization of the L 2-norm for which only a weighted Monte Carlo estimate ∥•∥n can be computed. The objective is to obtain an approximation v ∈ ℳ of an unknown function u ∈ 𝒱 by minimizing the empirical norm ∥u − v∥n. We consider this problem for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and is independent of the specified nonlinear least squares setting. Several model classes are examined and the analytical statements about the RIP are compared to existing sample complexity bounds from the literature. We find that for well-studied model classes our general bound is weaker but exhibits many of the same properties as these specialized bounds. Notably, we demonstrate the advantage of an optimal sampling density (as known for linear spaces) for sets of functions with sparse representations.
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    Consistency and convergence for a family of finite volume discretizations of the Fokker–Planck operator
    (Les Ulis : EDP Sciences, 2021) Heida, Martin; Kantner, Markus; Stephan, Artur
    We introduce a family of various finite volume discretization schemes for the Fokker–Planck operator, which are characterized by different Stolarsky weight functions on the edges. This family particularly includes the well-established Scharfetter–Gummel discretization as well as the recently developed square-root approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the choice of the Stolarsky weights. We show that the Scharfetter–Gummel scheme has the analytically best convergence properties but also that there exists a whole branch of Stolarsky means with the same convergence quality. We show by numerical experiments that for small convection the choice of the optimal representative of the discretization family is highly non-trivial, while for large gradients the Scharfetter–Gummel scheme stands out compared to the others.
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    On forward and inverse uncertainty quantification for models involving hysteresis operators
    (Les Ulis : EDP Sciences, 2020) Klein, Olaf; Davino, Daniele; Visone, Ciro
    Parameters within hysteresis operators modeling real world objects have to be identified from measurements and are therefore subject to corresponding errors. To investigate the influence of these errors, the methods of Uncertainty Quantification (UQ) are applied. Results of forward UQ for a play operator with a stochastic yield limit are presented. Moreover, inverse UQ is performed to identify the parameters in the weight function in a Prandtl-Ishlinskiĭ operator and the uncertainties of these parameters.
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    Temporal cavity solitons in a delayed model of a dispersive cavity ring laser
    (Les Ulis : EDP Sciences, 2020) Pimenov, Alexander; Amiranashvili, Shalva; Vladimirov, Andrei G.; Eleuteri, Michela; Krejčí, Pavel; Rachinskii, Dmitrii
    Nonlinear localised structures appear as solitary states in systems with multistability and hysteresis. In particular, localised structures of light known as temporal cavity solitons were observed recently experimentally in driven Kerr-cavities operating in the anomalous dispersion regime when one of the two bistable spatially homogeneous steady states exhibits a modulational instability. We use a distributed delay system to study theoretically the formation of temporal cavity solitons in an optically injected ring semiconductor-based fiber laser, and propose an approach to derive reduced delay-differential equation models taking into account the dispersion of the intracavity fiber delay line. Using these equations we perform the stability and bifurcation analysis of injection-locked continuous wave states and temporal cavity solitons.
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    Newton and Bouligand derivatives of the scalar play and stop operator
    (Les Ulis : EDP Sciences, 2020) Brokate, Martin
    We prove that the play and the stop operator possess Newton and Bouligand derivatives, and exhibit formulas for those derivatives. The remainder estimate is given in a strengthened form, and a corresponding chain rule is developed. The construction of the Newton derivative ensures that the mappings involved are measurable. © The authors. Published by EDP Sciences, 2020.