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- 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.
- ItemDisentangling nonlinear geomagnetic variability during magnetic storms and quiescence by timescale dependent recurrence properties(Les Ulis : EDP Sciences, 2020) Alberti, Tommaso; Lekscha, Jaqueline; Consolini, Giuseppe; De Michelis, Paola; Donner, Reik V.Understanding the complex behavior of the near-Earth electromagnetic environment is one of the main challenges of Space Weather studies. This includes both the correct characterization of the different physical mechanisms responsible for its configuration and dynamics as well as the efforts which are needed for a correct forecasting of several phenomena. By using a nonlinear multi-scale dynamical systems approach, we provide here new insights into the scale-to-scale dynamical behavior of both quiet and disturbed periods of geomagnetic activity. The results show that a scale-dependent dynamical transition occurs when moving from short to long timescales, i.e., from fast to slow dynamical processes, the latter being characterized by a more regular behavior, while more dynamical anomalies are found in the behavior of the fast component. This suggests that different physical processes are typical for both dynamical regimes: the fast component, being characterized by a more chaotic and less predictable behavior, can be related to the internal dynamical state of the near-Earth electromagnetic environment, while the slow component seems to be less chaotic and associated with the directly driven processes related to the interplanetary medium variability. Moreover, a clear difference has been found between quiet and disturbed periods, the former being more complex than the latter. These findings support the view that, for a correct forecasting in the framework of Space Weather studies, more attention needs to be devoted to the identification of proxies describing the internal dynamical state of the near-Earth electromagnetic environment. © T. Alberti et al., Published by EDP Sciences 2020.
- ItemLensing from small-scale travelling ionospheric disturbances observed using LOFAR(Les Ulis : EDP Sciences, 2022) Boyde, Ben; Wood, Alan; Dorrian, Gareth; Fallows, Richard A.; Themens, David; Mielich, Jens; Elvidge, Sean; Mevius, Maaijke; Zucca, Pietro; Dabrowski, Bartosz; Krankowski, Andrzej; Vocks, Christian; Bisi, MarioObservations made using the LOw-Frequency ARray (LOFAR) between 10:15 and 11:48 UT on the 15th of September 2018 over a bandwidth of approximately 25-65 MHz contain discrete pseudo-periodic features of ionospheric origin. These features occur within a period of approximately 10 min and collectively last roughly an hour. They are strongly frequency dependent, broadening significantly in time towards the lower frequencies, and show an overlaid pattern of diffraction fringes. By modelling the ionosphere as a thin phase screen containing a wave-like disturbance, we are able to replicate the observations, suggesting that they are associated with small-scale travelling ionospheric disturbances (TIDs). This modelling indicates that the features observed here require a compact radio source at a low elevation and that the TID or TIDs in question have a wavelength <~30 km. Several features suggest the presence of deviations from an idealised sinusoidal wave form. These results demonstrate LOFAR-s capability to identify and characterise small-scale ionospheric structures.
- 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.
- ItemA LOFAR observation of ionospheric scintillation from two simultaneous travelling ionospheric disturbances(Les Ulis : EDP Sciences, 2020) Fallows, Richard A.; Forte, Biagio; Astin, Ivan; Allbrook, Tom; Arnold, Alex; Wood, Alan; Dorrian, Gareth; Mevius, Maaijke; Rothkaeh, Hanna; Matyjasiak, Barbara; Krankowski, Andrzej; Anderson, James M.; Asgekar, Ashish; Avruch, I. Max; Bentum, Mark; Bisi, Mario M.; Butcher, Harvey R; Ciardi, Benedetta; Dabrowski, Bartosz; Damstra, Sieds; de Gasperin, Francesco; Duscha, Sven; Eislöffel, Jochen; Franzen, Thomas M.O.; Garrett, Michael A.; Griessmeier, Jean-Matthias; Gunst, Andre W.; Hoeft, Matthias; Horandel, Jorg R.; Iacobelli, Marco; Intema, Huib T.; Koopmans, Leon V.E.; Maat, Peter; Mann, Gottfried; Nelles, Anna; Paas, Harm; Pandey, Vishambhar N.; Reich, Wolfgang; Rowlinson, Antonia; Ruiter, Mark; Schwarz, Dominik J.; Serylak, Maciej; Shulevski, Aleksander; Smirnov, Oleg M.; Soida, Marian; Steinmetz, Matthias; Thoudam, Satyendra; Toribio, M. Carmen; van Ardenne, Arnold; van Bemmel, Ilse M.; van der Wiel, Matthijs H.D.; van Haarlem, Michiel P.; Vermeulen, Rene C.; Vocks, Christian; Wijers, Ralph A.M.J.; Wucknitz, Olaf; Zarka, Philippe; Zucca, PietroThis paper presents the results from one of the first observations of ionospheric scintillation taken using the Low-Frequency Array (LOFAR). The observation was of the strong natural radio source Cassiopeia A, taken overnight on 18–19 August 2013, and exhibited moderately strong scattering effects in dynamic spectra of intensity received across an observing bandwidth of 10–80 MHz. Delay-Doppler spectra (the 2-D FFT of the dynamic spectrum) from the first hour of observation showed two discrete parabolic arcs, one with a steep curvature and the other shallow, which can be used to provide estimates of the distance to, and velocity of, the scattering plasma. A cross-correlation analysis of data received by the dense array of stations in the LOFAR “core” reveals two different velocities in the scintillation pattern: a primary velocity of ~20–40 ms−1 with a north-west to south-east direction, associated with the steep parabolic arc and a scattering altitude in the F-region or higher, and a secondary velocity of ~110 ms−1 with a north-east to south-west direction, associated with the shallow arc and a scattering altitude in the D-region. Geomagnetic activity was low in the mid-latitudes at the time, but a weak sub-storm at high latitudes reached its peak at the start of the observation. An analysis of Global Navigation Satellite Systems (GNSS) and ionosonde data from the time reveals a larger-scale travelling ionospheric disturbance (TID), possibly the result of the high-latitude activity, travelling in the north-west to south-east direction, and, simultaneously, a smaller-scale TID travelling in a north-east to south-west direction, which could be associated with atmospheric gravity wave activity. The LOFAR observation shows scattering from both TIDs, at different altitudes and propagating in different directions. To the best of our knowledge this is the first time that such a phenomenon has been reported.
- 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.
- ItemTrends in recurrence analysis of dynamical systems(Les Ulis : EDP Sciences, 2023) Marwan, Norbert; Kraemer, K. HaukeThe last decade has witnessed a number of important and exciting developments that had been achieved for improving recurrence plot-based data analysis and to widen its application potential. We will give a brief overview about important and innovative developments, such as computational improvements, alternative recurrence definitions (event-like, multiscale, heterogeneous, and spatio-temporal recurrences) and ideas for parameter selection, theoretical considerations of recurrence quantification measures, new recurrence quantifiers (e.g. for transition detection and causality detection), and correction schemes. New perspectives have recently been opened by combining recurrence plots with machine learning. We finally show open questions and perspectives for futures directions of methodical research.
- 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.
- 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.
- ItemIdentification of potential precursors for the occurrence of Large-Scale Traveling Ionospheric Disturbances in a case study during September 2017(Les Ulis : EDP Sciences, 2020) Ferreira, Arthur Amaral; Borries, Claudia; Xiong, Chao; Borges, Renato Alves; Mielich, Jens; Kouba, DanielTraveling Ionospheric Disturbances (TIDs) reflect changes in the ionospheric electron density which are caused by atmospheric gravity waves. These changes in the electron density impact the functionality of different applications such as precise navigation and high-frequency geolocation. The Horizon 2020 project TechTIDE establishes a warning system for the occurrence of TIDs with the motivation to mitigate their impact on communication and navigation applications. This requires the identification of appropriate indicators for the generation of TIDs and for this purpose we investigate potential precursors for the TID occurrence. This paper presents a case study of the double main phase geomagnetic storm, starting from the night of 7th September and lasting until the end of 8th September 2017. Detrended Total Electron Content (TEC) derived from Global Navigation Satellite System (GNSS) measurements from more than 880 ground stations in Europe was used to identify the occurrence of different types of large scale traveling ionospheric disturbances (LSTIDs) propagating over the European sector. In this case study, LSTIDs were observed more frequently and with higher amplitude during periods of enhanced auroral activity, as indicated by increased electrojet index (IE) from the International Monitor for Auroral Geomagnetic Effects (IMAGE). Our investigation suggests that Joule heating due to the dissipation of Pedersen currents is the main contributor to the excitation of the observed LSTIDs. We observe that the LSTIDs are excited predominantly after strong ionospheric perturbations at high-latitudes. Ionospheric parameters including TEC gradients, the Along Arc TEC Rate (AATR) index and the Rate Of change of TEC index (ROTI) have been analysed for their suitability to serve as a precursor for LSTID occurrence in mid-latitude Europe, aiming for near real-time indication and warning of LSTID activity. The results of the presented case study suggest that the AATR index and TEC gradients are promising candidates for near real-time indication and warning of the LSTIDs occurrence in mid-latitude Europe since they have a close relation to the source mechanisms of LSTIDs during periods of increased auroral activity.