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In the present work, we use symbolic regression for automated modeling of dynamical systems. Symbolic regression is a powerful and general method suitable for data-driven identification of mathematical expressions. In particular, the structure and parameters of those expressions are identified simultaneously.
We consider two main variants of symbolic regression: sparse regression-based and genetic programming-based symbolic regression. Both are applied to identification, prediction and control of dynamical systems.
We introduce a new methodology for the data-driven identification of nonlinear dynamics for systems undergoing abrupt changes. Building on a sparse regression algorithm derived earlier, the model after the change is defined as a minimum update with respect to a reference model of the system identified prior to the change. The technique is successfully exemplified on the chaotic Lorenz system and the van der Pol oscillator. Issues such as computational complexity, robustness against noise and requirements with respect to data volume are investigated.
We show how symbolic regression can be used for time series prediction. Again, issues such as robustness against noise and convergence rate are investigated us- ing the harmonic oscillator as a toy problem. In combination with embedding, we demonstrate the prediction of a propagating front in coupled FitzHugh-Nagumo oscillators. Additionally, we show how we can enhance numerical weather predictions to commercially forecast power production of green energy power plants.
We employ symbolic regression for synchronization control in coupled van der Pol oscillators. Different coupling topologies are investigated. We address issues such as plausibility and stability of the control laws found. The toolkit has been made open source and is used in turbulence control applications.
Genetic programming based symbolic regression is very versatile and can be adapted to many optimization problems. The heuristic-based algorithm allows for cost efficient optimization of complex tasks.
We emphasize the ability of symbolic regression to yield white-box models. In contrast to black-box models, such models are accessible and interpretable which allows the usage of established tool chains.
The Sun is the nearest star to the Earth. It consists of an interior and an atmosphere. The convection zone is the outermost layer of the solar interior. A flux rope may emerge as a coherent structure from the convection zone into the solar atmosphere or be formed by magnetic reconnection in the atmosphere. A flux rope is a bundle of magnetic field lines twisting around an axis field line, creating a helical shape by which dense filament material can be supported against gravity. The flux rope is also considered as the key structure of the most energetic phenomena in the solar system, such as coronal mass ejections (CMEs) and flares. These magnetic flux ropes can produce severe geomagnetic storms. In particular, to improve the ability to forecast space weather, it is important to enrich our knowledge about the dynamic formation of flux ropes and the underlying physical mechanisms that initiate their eruption, such as a CME.
A confined eruption consists of a filament eruption and usually an associated are, but does not evolve into a CME; rather, the moving plasma is halted in the solar corona and usually seen to fall back. The first detailed observations of a confined filament eruption were obtained on 2002 May 27by the TRACE satellite in the 195 A band. So, in the Chapter 3, we focus on a flux rope instability model. A twisted flux rope can become unstable by entering the kink instability regime. We show that the kink instability, which occurs if the twist of a flux rope exceeds a critical value, is capable of initiating of an eruption. This model is tested against the well observed confined eruption on 2002 May 27 in a parametric magnetohydrodynamic (MHD) simulation study that comprises all phases of the event. Very good agreement with the essential observed properties is obtained, only except for a relatively poor matching of the initial filament height.
Therefore, in Chapter 4, we submerge the center point of the flux rope deeper below the photosphere to obtain a flatter coronal rope section and a better matching with the initial height profile of the erupting filament. This implies a more realistic inclusion of the photospheric line tying. All basic assumptions and the other parameter settings are kept the same as in Chapter 3. This complement of the parametric study shows that the flux rope instability model can yield an even better match with the observational data. We also focus in Chapters 3 and 4 on the magnetic reconnection during the confined eruption, demonstrating that it occurs in two distinct locations and phases that correspond to the observed brightenings and changes of topology, and consider the fate of the erupting flux, which can reform a (less twisted) flux rope.
The Sun also produces series of homologous eruptions, i.e. eruptions which occur repetitively in the same active region and are of similar morphology. Therefore, in Chapter 5, we employ the reformed flux rope as a new initial condition, to investigate the possibility of subsequent homologous eruptions. Free magnetic energy is built up by imposing motions in the bottom boundary, such as converging motions, leading to flux cancellation. We apply converging motions in the sunspot area, such that a small part of the flux from the sunspots with different polarities is transported toward the polarity inversion line (PIL) and cancels with each other. The reconnection associated with the cancellation process forms more helical magnetic flux around the reformed flux rope, which leads to a second and a third eruption. In this study, we obtain the first MHD simulation results of a homologous sequence of eruptions that show a transition from a confined to two ejective eruptions, based on the reformation of a flux rope after each eruption.
This publication based thesis, which consists of seven published articles, summarizes my contributions to the research field of laser excited ultrafast structural dynamics. The coherent and incoherent lattice dynamics on microscopic length scales are detected by ultrashort optical and X-ray pulses. The understanding of the complex physical processes is essential for future improvements of technological applications. For this purpose, tabletop soruces and large scale facilities, e.g. synchrotrons, are employed to study structural dynamics of longitudinal acoustic strain waves and heat transport. The investigated effects cover timescales from hundreds of femtoseconds up to several microseconds.
The main part of this thesis is dedicated to the investigation of tailored phonon wave packets propagating in perovskite nanostructures. Tailoring is achieved either by laser excitation of nanostructured bilayer samples or by a temporal series of laser pulses. Due to the propagation of longitudinal acoustic phonons, the out-of-plane lattice spacing of a thin film insulator-metal bilayer sample is modulated on an ultrafast timescale. This leads to an ultrafast modulation of the X-ray diffraction efficiency which is employed as a phonon Bragg switch to shorten hard X-ray pulses emitted from a 3rd generation synchrotron.
In addition, we have observed nonlinear mixing of high amplitude phonon wave packets which originates from an anharmonic interatomic potential. A chirped optical pulse sequence excites a narrow band phonon wave packet with specific momentum and energy. The second harmonic generation of these phonon wave packets is followed by ultrafast X-ray diffraction. Phonon upconversion takes place because the high amplitude phonon wave packet modulates the acoustic properties of the crystal which leads to self steepening and to the successive generation of higher harmonics of the phonon wave packet.
Furthermore, we have demonstrated ultrafast strain in direction parallel to the sample surface. Two consecutive so-called transient grating excitations displaced in space and time are used to coherently control thermal gradients and surface acoustic modes. The amplitude of the coherent and incoherent surface excursion is disentangled by time resolved X-ray reflectivity measurements. We calibrate the absolute amplitude of thermal and acoustic surface excursion with measurements of longitudinal phonon propagation. In addition, we develop a diffraction model which allows for measuring the surface excursion on an absolute length scale with sub-Äangström precision. Finally, I demonstrate full coherent control of an excited surface deformation by amplifying and suppressing thermal and coherent excitations at the surface of a laser-excited Yttrium-manganite sample.
Modern gamma-ray telescopes, provide the main stream of data for astrophysicists in quest of detecting the sources of gamma rays such as active galactic nuclei (AGN). Many blazars have been detected with gamma-ray telescopes such as HESS, VERITAS, MAGIC and Fermi satellite as sources of gamma-rays with the energy E ≥ 100 GeV. These very-high-energy photons interact with extragalactic background light (EBL) producing ultra-relativistic electron-positron pairs. Observations with Fermi-LAT indicate that the GeV gamma-ray flux from some blazars is lower than that predicted from the full electromagnetic cascade. The pairs can induce electrostatic and electromagnetic instabilities. In this case, wave-particle interactions can reduce the energy of the pairs. Therefore, the collective plasma effects can also substantially suppress the GeV-band gamma-ray emission affecting as well the IGMF constraints. Using Particle in cell (PIC) simulations, we have revisited the issue of plasma instabilities induced by electron-positron beams in the fully ionized intergalactic medium. This problem is related to pair beams produced by TeV radiation of blazars. The main objective of our study is to clarify the feedback of the beam-driven instabilities on the pairs. The present dissertation provides new results regarding the plasma instabilities from blazar induced pair beams interacting with intergalactic medium. This clarifies the relevance of plasma instabilities and improves our understanding of blazars.
Light-driven diffusioosmosis
(2018)
The emergence of microfluidics created the need for precise and remote control of micron-sized objects. I demonstrate how light-sensitive motion can be induced at the micrometer scale by a simple addition of a photosensitive surfactant, which makes it possible to trigger hydrophobicity with light. With point-like laser irradiation, radial inward and outward hydrodynamic surface flows are remotely switched on and off. In this way, ensembles of microparticles can be moved toward or away from the irradiation center. Particle motion is analyzed according to varying parameters, such as surfactant and salt concentration, illumination condition, surface hydrophobicity, and surface structure.
The physical origin of this process is the so-called light-driven diffusioosmosis (LDDO), a phenomenon that was discovered in the framework of this thesis and is described experimentally and theoretically in this work. To give a brief explanation, a focused light irradiation induces a local photoisomerization that creates a concentration gradient at the solid-liquid interface. To compensate for the change in osmotic pressure near the surface, a hydrodynamic flow along the surface is generated. Surface-surfactant interaction largely governs LDDO. It is shown that surfactant adsorption depends on the isomerization state of the surfactant. Photoisomerization, therefore, triggers a surfactant attachment or detachment from the surface. This change is considered to be one of the reasons for the formation of LDDO flow.
These flows are introduced not only by a focused laser source but also by global irradiation. Porous particles show reversible repulsive and attractive interactions when dispersed in the solution of photosensitive surfactant. Repulsion and attraction is controlled by the irradiation wavelength. Illumination with red light leads to formation of aggregates, while illumination with blue light leads to the formation of a well-separated grid with equal interparticle distances, between 2µm and 80µm, depending on the particle surface density. These long-range interactions are considered to be a result of an increase or decrease of surfactant concentration around each particle, depending on the irradiation wavelength. Surfactant molecules adsorb inside the pores of the particles. A light-induced photoisomerization changes adsorption to the pores and drives surfactant molecules to the outside. The concentration gradients generate symmetric flows around each single particle resulting in local LDDO. With a break of the symmetry (i.e., by closing one side of the particle with a metal cap), one can achieve active self-propelled particle motion.
In this thesis, we treat the extreme Newman-Penrose components of both the Maxwell field (s=±1) and the linearized gravitational perturbations (or "linearized gravity" for short) (s=±2) in the exterior of a slowly rotating Kerr black hole. Upon different rescalings, we can obtain spin s components which satisfy the separable Teukolsky master equation (TME). For each of these spin s components defined in Kinnersley tetrad, the resulting equations by performing some first-order differential operator on it once and twice (twice only for s=±2), together with the TME, are in the form of an "inhomogeneous spin-weighted wave equation" (ISWWE) with different potentials and constitute a linear spin-weighted wave system. We then prove energy and integrated local energy decay (Morawetz) estimates for this type of ISWWE, and utilize them to achieve both a uniform bound of a positive definite energy and a Morawetz estimate for the regular extreme Newman-Penrose components defined in the regular Hawking-Hartle tetrad.
We also present some brief discussions on mode stability for TME for the case of real frequencies. This says that in a fixed subextremal Kerr spacetime, there is no nontrivial separated mode solutions to TME which are purely ingoing at horizon and purely outgoing at infinity. This yields a representation formula for solutions to inhomogeneous Teukolsky equations, and will play a crucial role in generalizing the above energy and Morawetz estimates results to the full subextremal Kerr case.
In the here presented work we discuss a series of results that are all in one way or another connected to the phenomenon of trapping in black hole spacetimes.
First we present a comprehensive review of the Kerr-Newman-Taub-NUT-de-Sitter family of black hole spacetimes and their most important properties. From there we go into a detailed analysis of the bahaviour of null geodesics in the exterior region of a sub-extremal Kerr spacetime. We show that most well known fundamental properties of null geodesics can be represented in one plot. In particular, one can see immediately that the ergoregion and trapping are separated in phase space.
We then consider the sets of future/past trapped null geodesics in the exterior region of a sub-extremal Kerr-Newman-Taub-NUT spacetime. We show that from the point of view of any timelike observer outside of such a black hole, trapping can be understood as two smooth sets of spacelike directions on the celestial sphere of the observer. Therefore the topological structure of the trapped set on the celestial sphere of any observer is identical to that in Schwarzschild.
We discuss how this is relevant to the black hole stability problem.
In a further development of these observations we introduce the notion of what it means for the shadow of two observers to be degenerate. We show that, away from the axis of symmetry, no continuous degeneration exists between the shadows of observers at any point in the exterior region of any Kerr-Newman black hole spacetime of unit mass. Therefore, except possibly for discrete changes, an observer can, by measuring the black holes shadow, determine the angular momentum and the charge of the black hole under observation, as well as the observer's radial position and angle of elevation above the equatorial plane. Furthermore, his/her relative velocity compared to a standard observer can also be measured. On the other hand, the black hole shadow does not allow for a full parameter resolution in the case of a Kerr-Newman-Taub-NUT black hole, as a continuous degeneration relating specific angular momentum, electric charge, NUT charge and elevation angle exists in this case.
We then use the celestial sphere to show that trapping is a generic feature of any black hole spacetime.
In the last chapter we then prove a generalization of the mode stability result of Whiting (1989) for the Teukolsky equation for the case of real frequencies. The main result of the last chapter states that a separated solution of the Teukolsky equation governing massless test fields on the Kerr spacetime, which is purely outgoing at infinity, and purely ingoing at the horizon, must vanish. This has the consequence, that for real frequencies, there are linearly independent fundamental solutions of the radial Teukolsky equation which are purely ingoing at the horizon, and purely outgoing at infinity, respectively. This fact yields a representation formula for solutions of the inhomogenous Teukolsky equation, and was recently used by Shlapentokh-Rothman (2015) for the scalar wave equation.
The purpose of Probabilistic Seismic Hazard Assessment (PSHA) at a construction site is to provide the engineers with a probabilistic estimate of ground-motion level that could be equaled or exceeded at least once in the structure’s design lifetime. A certainty on the predicted ground-motion allows the engineers to confidently optimize structural design and mitigate the risk of extensive damage, or in worst case, a collapse. It is therefore in interest of engineering, insurance, disaster mitigation, and security of society at large, to reduce uncertainties in prediction of design ground-motion levels.
In this study, I am concerned with quantifying and reducing the prediction uncertainty of regression-based Ground-Motion Prediction Equations (GMPEs). Essentially, GMPEs are regressed best-fit formulae relating event, path, and site parameters (predictor variables) to observed ground-motion values at the site (prediction variable). GMPEs are characterized by a parametric median (μ) and a non-parametric variance (σ) of prediction. μ captures the known ground-motion physics i.e., scaling with earthquake rupture properties (event), attenuation with distance from source (region/path), and amplification due to local soil conditions (site); while σ quantifies the natural variability of data that eludes μ. In a broad sense, the GMPE prediction uncertainty is cumulative of 1) uncertainty on estimated regression coefficients (uncertainty on μ,σ_μ), and 2) the inherent natural randomness of data (σ). The extent of μ parametrization, the quantity, and quality of ground-motion data used in a regression, govern the size of its prediction uncertainty: σ_μ and σ.
In the first step, I present the impact of μ parametrization on the size of σ_μ and σ. Over-parametrization appears to increase the σ_μ, because of the large number of regression coefficients (in μ) to be estimated with insufficient data. Under-parametrization mitigates σ_μ, but the reduced explanatory strength of μ is reflected in inflated σ. For an optimally parametrized GMPE, a ~10% reduction in σ is attained by discarding the low-quality data from pan-European events with incorrect parametric values (of predictor variables).
In case of regions with scarce ground-motion recordings, without under-parametrization, the only way to mitigate σ_μ is to substitute long-term earthquake data at a location with short-term samples of data across several locations – the Ergodic Assumption. However, the price of ergodic assumption is an increased σ, due to the region-to-region and site-to-site differences in ground-motion physics. σ of an ergodic GMPE developed from generic ergodic dataset is much larger than that of non-ergodic GMPEs developed from region- and site-specific non-ergodic subsets - which were too sparse to produce their specific GMPEs. Fortunately, with the dramatic increase in recorded ground-motion data at several sites across Europe and Middle-East, I could quantify the region- and site-specific differences in ground-motion scaling and upgrade the GMPEs with 1) substantially more accurate region- and site-specific μ for sites in Italy and Turkey, and 2) significantly smaller prediction variance σ. The benefit of such enhancements to GMPEs is quite evident in my comparison of PSHA estimates from ergodic versus region- and site-specific GMPEs; where the differences in predicted design ground-motion levels, at several sites in Europe and Middle-Eastern regions, are as large as ~50%.
Resolving the ergodic assumption with mixed-effects regressions is feasible when the quantified region- and site-specific effects are physically meaningful, and the non-ergodic subsets (regions and sites) are defined a priori through expert knowledge. In absence of expert definitions, I demonstrate the potential of machine learning techniques in identifying efficient clusters of site-specific non-ergodic subsets, based on latent similarities in their ground-motion data. Clustered site-specific GMPEs bridge the gap between site-specific and fully ergodic GMPEs, with their partially non-ergodic μ and, σ ~15% smaller than the ergodic variance.
The methodological refinements to GMPE development produced in this study are applicable to new ground-motion datasets, to further enhance certainty of ground-motion prediction and thereby, seismic hazard assessment. Advanced statistical tools show great potential in improving the predictive capabilities of GMPEs, but the fundamental requirement remains: large quantity of high-quality ground-motion data from several sites for an extended time-period.
Earth's climate varies continuously across space and time, but humankind has witnessed only a small snapshot of its entire history, and instrumentally documented it for a mere 200 years. Our knowledge of past climate changes is therefore almost exclusively based on indirect proxy data, i.e. on indicators which are sensitive to changes in climatic variables and stored in environmental archives. Extracting the data from these archives allows retrieval of the information from earlier times. Obtaining accurate proxy information is a key means to test model predictions of the past climate, and only after such validation can the models be used to reliably forecast future changes in our warming world. The polar ice sheets of Greenland and Antarctica are one major climate archive, which record information about local air temperatures by means of the isotopic composition of the water molecules embedded in the ice. However, this temperature proxy is, as any indirect climate data, not a perfect recorder of past climatic variations. Apart from local air temperatures, a multitude of other processes affect the mean and variability of the isotopic data, which hinders their direct interpretation in terms of climate variations. This applies especially to regions with little annual accumulation of snow, such as the Antarctic Plateau. While these areas in principle allow for the extraction of isotope records reaching far back in time, a strong corruption of the temperature signal originally encoded in the isotopic data of the snow is expected. This dissertation uses observational isotope data from Antarctica, focussing especially on the East Antarctic low-accumulation area around the Kohnen Station ice-core drilling site, together with statistical and physical methods, to improve our understanding of the spatial and temporal isotope variability across different scales, and thus to enhance the applicability of the proxy for estimating past temperature variability. The presented results lead to a quantitative explanation of the local-scale (1–500 m) spatial variability in the form of a statistical noise model, and reveal the main source of the temporal variability to be the mixture of a climatic seasonal cycle in temperature and the effect of diffusional smoothing acting on temporally uncorrelated noise. These findings put significant limits on the representativity of single isotope records in terms of local air temperature, and impact the interpretation of apparent cyclicalities in the records. Furthermore, to extend the analyses to larger scales, the timescale-dependency of observed Holocene isotope variability is studied. This offers a deeper understanding of the nature of the variations, and is crucial for unravelling the embedded true temperature variability over a wide range of timescales.
The solar activity and its consequences affect space weather and Earth’s climate. The solar activity exhibits a cyclic behaviour with a period of about 11 years. The solar cycle properties are governed by the dynamo taking place in the interior of the Sun, and they are distinctive. Extending the knowledge about solar cycle properties into the past is essential for understanding the solar dynamo and forecasting space weather. It can be acquired through the analysis of historical sunspot drawings. Sunspots are the dark areas, which are associated with strong magnetic fields, on the solar surface. Sunspots are the oldest and longest available observed features of solar activity.
One of the longest available records of sunspot drawings is the collection by Samuel Heinrich Schwabe during 1825–1867. The sunspot sizes measured from digitized Schwabe drawings are not to scale and need to be converted into physical sunspot areas. We employed a statistical approach assuming that the area distribution of sunspots was the same in the 19th century as it was in the 20th century. Umbral areas for about 130 000 sunspots observed by Schwabe were obtained. The annually averaged sunspot areas correlate reasonably well with the sunspot number. Tilt angles and polarity separations of sunspot groups were calculated assuming them to be bipolar. There is, of course, no polarity information in the observations. We derived an average tilt angle by attempting to exclude unipolar groups with a minimum separation of the two surmised polarities and an outlier rejection method, which follows the evolution of each group and detects the moment, when it turns unipolar as it decays. As a result, the tilt angles, although displaying considerable natural scatter, are on average 5.85° ± 0.25°, with the leading
polarity located closer to the equator, in good agreement with tilt angles obtained from 20th century data sets. Sources of uncertainties in the tilt angle determination are discussed and need to be addressed whenever different data sets are combined.
Digital images of observations printed in the books Rosa Ursina and Prodromus pro sole mobili by Christoph Scheiner, as well as the drawings from Scheiner’s letters to Marcus Welser, are analyzed to obtain information on the positions and sizes of sunspots that appeared before the Maunder minimum. In most cases, the given orientation of the ecliptic is used to set up the heliographic coordinate system for the drawings. Positions and sizes are measured manually displaying the drawings on a computer screen. Very early drawings have no indication of the solar orientation. A rotational matching using common spots of adjacent days is used in some cases, while in other cases, the assumption that images were aligned with a zenith–horizon coordinate system appeared to be the most likely. In total, 8167 sunspots were measured. A distribution of sunspot latitudes versus time (butterfly diagram) is obtained for Scheiner’s observations. The observations of 1611 are very inaccurate, but the drawings of 1612 have at least an indication of the solar orientation, while the remaining part of the spot positions from 1618–1631 have good to very good accuracy. We also computed 697 tilt angles of apparent bipolar sunspot groups, which were observed in the period 1618–1631. We find that the average tilt angle of nearly 4° does not significantly differ from the 20th century values.
The solar cycle properties seem to be related to the tilt angles of sunspot groups, and it is an important parameter in the surface flux transport models. The tilt angles of bipolar sunspot groups from various historical sets of solar drawings including from Schwabe and Scheiner are analyzed. Data by Scheiner, Hevelius, Staudacher, Zucconi, Schwabe, and Spörer deliver a series of average tilt angles spanning a period of 270 years, in addition to previously found values for 20th-century data obtained by other authors. We find that the average tilt angles before the Maunder minimum were not significantly different from modern values. However, the average tilt angles of a period 50 years after the Maunder minimum, namely for cycles 0 and 1, were much lower and near zero. The typical tilt angles before the Maunder minimum suggest that abnormally low tilt angles were not responsible for driving the solar cycle into a grand minimum.
With the Schwabe (1826–1867) and Spörer (1866–1880) sunspot data, the butterfly diagram of sunspot groups extends back till 1826. A recently developed method, which separates the wings of the butterfly diagram based on the long gaps present in sunspot group occurrences at different latitudinal bands, is used to separate the wings of the butterfly diagram. The cycle-to-cycle variation in the start (F), end (L), and highest (H) latitudes of the wings with respect to the strength of the wings are analyzed. On the whole, the wings of the stronger cycles tend to start at higher latitudes and have a greater extent. The time spans of the wings and the time difference between the wings in the northern hemisphere display a quasi-periodicity of 5–6 cycles. The average wing overlap is zero in the southern hemisphere, whereas it is 2–3 months in the north. A marginally significant oscillation of about 10 solar cycles is found in the asymmetry of the L latitudes. This latest, extended database of butterfly wings provides new observational constraints, regarding the spatio-temporal distribution of sunspot occurrences over the solar cycle, to solar dynamo models.