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- ItemNewton and Bouligand derivatives of the scalar play and stop operator(Les Ulis : EDP Sciences, 2020) Brokate, MartinWe 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.
- ItemTemporal 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, DmitriiNonlinear 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.
- ItemSecond-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis(Les Ulis : EDP Sciences, 2021) Colli, Pierluigi; Signori, Andrea; Sprekels, JürgenThis paper concerns a distributed optimal control problem for a tumor growth model of Cahn–Hilliard type including chemotaxis with possibly singular potentials, where the control and state variables are nonlinearly coupled. First, we discuss the weak well-posedness of the system under very general assumptions for the potentials, which may be singular and nonsmooth. Then, we establish the strong well-posedness of the system in a reduced setting, which however admits the logarithmic potential: this analysis will lay the foundation for the study of the corresponding optimal control problem. Concerning the optimization problem, we address the existence of minimizers and establish both first-order necessary and second-order sufficient conditions for optimality. The mathematically challenging second-order analysis is completely performed here, after showing that the solution mapping is twice continuously differentiable between suitable Banach spaces via the implicit function theorem. Then, we completely identify the second-order Fréchet derivative of the control-to-state operator and carry out a thorough and detailed investigation about the related properties.
- ItemConvergence bounds for empirical nonlinear least-squares(Les Ulis : EDP Sciences, 2022) Eigel, Martin; Schneider, Reinhold; Trunschke, PhilippWe 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.
- ItemOn forward and inverse uncertainty quantification for models involving hysteresis operators(Les Ulis : EDP Sciences, 2020) Klein, Olaf; Davino, Daniele; Visone, CiroParameters 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.
- ItemGrain boundary assisted photocurrent collection in thin film solar cells(Les Ulis : EDP Sciences, 2015) Harndt, Susanna; Kaufmann, Christian A.; Lux-Steiner, Martha C.; Klenk, Reiner; Nürnberg, ReinerThe influence of absorber grain boundaries on the photocurrent transport in chalcopyrite based thin film solar cells has been calculated using a two dimensional numerical model. Considering extreme cases, the variation in red response is more expressed than in one dimensional models. These findings may offer an explanation for the strong influence of buffer layer preparation on the spectral response of cells with small grained absorbers.
- ItemConsistency and convergence for a family of finite volume discretizations of the Fokker–Planck operator(Les Ulis : EDP Sciences, 2021) Heida, Martin; Kantner, Markus; Stephan, ArturWe 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.
- ItemOptimization with learning-informed differential equation constraints and its applications(Les Ulis : EDP Sciences, 2022) Dong, Guozhi; Hintermüller, Michael; Papafitsoros, KostasInspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through data-driven techniques are studied. A particular focus is on the analysis and on numerical methods for problems with machine-learned components. For a rather general context, an error analysis is provided, and particular properties resulting from artificial neural network based approximations are addressed. Moreover, for each of the two inspiring applications analytical details are presented and numerical results are provided.