arXiv subjects

Christian Beck

Publications and source records attributed to Christian Beck.

At least 37 records · Page 2Linked to original sources

arXiv2022

An overview on deep learning-based approximation methods for partial differential equations

It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem have been proposed and tested numerically on a number of examples of high-dimensional PDEs. This has given rise to a lively field of research in which deep learning-based methods and related Monte Carlo methods are applied to the approximation of high-dimensional PDEs. In this article we offer an introduction to this field of research by revisiting selected mathematical results related to deep learning approximation methods for PDEs and reviewing the main ideas of their proofs. We also provide a short overview of the recent literature in this area of research.

math.NA↗

arXiv2022

Information Shift Dynamics Described by Tsallis $q=3$ Entropy on a Compact Phase Space

Recent mathematical investigations have shown that under very general conditions exponential mixing implies the Bernoulli property. As a concrete example of a statistical mechanics which is exponentially mixing we consider a Bernoulli shift dynamics by Chebyshev maps of arbitrary order $N\geq 2$, which maximizes Tsallis $q=3$ entropy rather than the ordinary $q=1$ Boltzmann-Gibbs entropy. Such an information shift dynamics may be relevant in a pre-universe before ordinary space-time is created. We discuss symmetry properties of the coupled Chebyshev systems, which are different for even and odd $N$. We show that the value of the fine structure constant $α_{el}=1/137$ is distinguished as a coupling constant in this context, leading to uncorrelated behaviour in the spatial direction of the corresponding coupled map lattice for $N=3$.

cond-mat.stat-mech↗

arXiv2022

Lockdown effects on air quality in megacities during the first and second waves of COVID-19 pandemic

Air pollution is among the highest contributors to mortality worldwide, especially in urban areas. During spring 2020, many countries enacted social distancing measures in order to slow down the ongoing COVID-19 pandemic. A particularly drastic measure, the 'lockdown', urged people to stay at home and thereby prevent new COVID-19 infections during the first (2020) and second wave (2021) of the pandemic. In turn, it also reduced traffic and industrial activities. But how much did these lockdown measures improve air quality in large cities, and are there differences in how air quality was affected? Here, we analyse data from two megacities: London as an example for Europe and Delhi as an example for Asia. We consider data during first and second wave lockdowns and compare them to 2019 values. Overall, we find a reduction in almost all air pollutants with intriguing differences between the two cities except Delhi in 2021. In London, despite smaller average concentrations, we still observe high-pollutant states and an increased tendency towards extreme events (a higher kurtosis of the probability density during lockdown) during 2020 and low pollution levels during 2021. For Delhi, we observe a much stronger decrease of pollution concentrations, including high pollution states during 2020 and higher pollution levels in 2021. These results could help to design policies to improve long-term air quality in megacities.

physics.soc-ph↗

arXiv2022

The onset of molecule-spanning dynamics in a multi-domain protein

Protein dynamics has been investigated on a wide range of time scales. Nano- and picosecond dynamics have been assigned to local fluctuations, while slower dynamics have been attributed to larger conformational changes. However, it is largely unknown how local fluctuations can lead to global allosteric changes. Here we show that molecule-spanning dynamics on the 100 ns time scale precede larger allosteric changes. We assign global real-space movements to dynamic modes on the 100 ns time scales, which became possible by a combination of single-molecule fluorescence, quasi-elastic neutron scattering and all-atom MD simulations. Additionally, we demonstrate the effect of Sba1, a co-chaperone of Hsp90, on these molecule-spanning dynamics, which implies functional importance of such dynamics. Our integrative approach provides comprehensive insights into molecule-spanning dynamics on the nanosecond time scale for a multi-domain protein and indicates that such dynamics are the molecular basis for allostery and large conformational changes in proteins.

q-bio.BM↗

arXiv2022

Spatial heterogeneity of air pollution statistics

Air pollution is one of the leading causes of death globally, and continues to have a detrimental effect on our health. In light of these impacts, an extensive range of statistical modelling approaches has been devised in order to better understand air pollution statistics. However, the time-varying statistics of different types of air pollutants are far from being fully understood. The observed probability density functions (PDFs) of concentrations depend very much on the spatial location and on the pollutant substance. In this paper, we analyse a large variety of data from 3544 different European monitoring sites and show that the PDFs of nitric oxide ($NO$), nitrogen dioxide ($NO2$) and particulate matter ($PM10$ and $PM2.5$) concentrations generically exhibit heavy tails and are asymptotically well approximated by $q$-exponential distributions with a given width parameter $λ$. We observe that the power-law parameter $q$ and the width parameter $λ$ vary widely for the different spatial locations. For each substance, we find different patterns of parameter clouds in the $(q, λ)$ plane. These depend on the type of pollutants and on the environmental characteristics (urban/suburban/rural/traffic/industrial/background). This means the effective statistical physics description of air pollution exhibits a strong degree of spatial heterogeneity.

physics.ao-ph↗

arXiv2021

Weak correlation between fluctuations in protein diffusion inside bacteria

A weak correlation between the diffusion-exponent fluctuations and the temperature fluctuations is discussed based on recent experimental observations for protein diffusion inside bacteria. Its existence is shown to be essential for describing the statistical properties of the fluctuations. It is also quantified how largely the fluctuations are modulated by the weak correlation.

cond-mat.stat-mech↗

arXiv2021

Complexity and Persistence of Price Time Series of the European Electricity Spot Market

The large variability of renewable power sources is a central challenge in the transition to a sustainable energy system. Electricity markets are central for the coordination of electric power generation. These markets rely evermore on short-term trading to facilitate the balancing of power generation and demand and to enable systems integration of small producers. Electricity prices in these spot markets show pronounced fluctuations, featuring extreme peaks as well as occasional negative prices. In this article, we analyse electricity price time series from the European EPEX market, in particular the hourly day-ahead, hourly intraday, and 15-min intraday market prices. We quantify the fluctuations, correlations, and extreme events and reveal different time scales in the dynamics of the market. The short-term fluctuations show remarkably different characteristics for time scales below and above 12 hours. Fluctuations are strongly correlated and persistent below 12 hours, which contributes to extreme price events and a strong multifractal behaviour. On longer time scales, they get anti-correlated and price time series revert to their mean, witnessed by a stark decrease of the Hurst coefficient after 12 hours. The long-term behaviour is strongly influenced by the evolution of a large-scale weather patterns with a typical time scale of four days. We elucidate this dependence in detail using a classification into circulation weather types. The separation in time scales enables a superstatistical treatment, which confirms the characteristic time scale of four days, and motivates the use of $q$-Gaussian distributions as the best fit to the empiric distribution of electricity prices.

q-fin.ST↗

arXiv2021

Velocities of an Erupting Filament

Solar filaments exist as stable structures for extended periods of time before many of them form the core of a CME. We examine the properties of an erupting filament on 2017 May 29--30 with high-resolution He I 10830 A and Halpha spectra from the Dunn Solar Telescope, full-disk Dopplergrams of He I 10830 A from the Chromospheric Telescope, and EUV and coronograph data from SDO and STEREO. Pre-eruption line-of-sight velocities from an inversion of He I with the HAZEL code exhibit coherent patches of 5 Mm extent that indicate counter-streaming and/or buoyant behavior. During the eruption, individual, aligned threads appear in the He I velocity maps. The distribution of velocities evolves from Gaussian to strongly asymmetric. The maximal optical depth of He I 10830 A decreased from tau = 1.75 to 0.25, the temperature increased by 13 kK, and the average speed and width of the filament increased from 0 to 25 km s-1 and 10 to 20 Mm, respectively. All data sources agree that the filament rose with an exponential acceleration reaching 7.4 m s-2 that increased to a final velocity of 430 km s-1 at 22:24 UT; a CME was associated with this filament eruption. The properties during the eruption favor a kink/torus instability, which requires the existence of a flux rope. We conclude that full-disk chromospheric Dopplergrams can be used to trace the initial phase of on-disk filament eruptions in real-time, which might potentially be useful for modelling the source of any subsequent CMEs.

astro-ph.SR↗

arXiv2021

Heating of the solar chromosphere in a sunspot light bridge by electric currents

Context: Resistive Ohmic dissipation has been suggested as a mechanism for heating the solar chromosphere, but few studies have established this association. Aim: We aim to determine how Ohmic dissipation by electric currents can heat the solar chromosphere. Methods: We combine high-resolution spectroscopic Ca II data from the Dunn Solar Telescope and vector magnetic field observations from the Helioseismic and Magnetic Imager (HMI) to investigate thermal enhancements in a sunspot light bridge. The photospheric magnetic field from HMI was extrapolated to the corona using a non-force-free field technique that provided the three-dimensional distribution of electric currents, while an inversion of the chromospheric Ca II line with a local thermodynamic equilibrium and a nonlocal thermodynamic equilibrium spectral archive delivered the temperature stratifications from the photosphere to the chromosphere. Results: We find that the light bridge is a site of strong electric currents, of about 0.3 A/m^2 at the bottom boundary, which extend to about 0.7 Mm while decreasing monotonically with height. These currents produce a chromospheric temperature excess of about 600-800 K relative to the umbra. Only the light bridge, where relatively weak and highly inclined magnetic fields emerge over a duration of 13 hr, shows a spatial coincidence of thermal enhancements and electric currents. The temperature enhancements and the Cowling heating are primarily confined to a height range of 0.4-0.7 Mm above the light bridge. The corresponding increase in internal energy of 200 J/m^3 can be supplied by the heating in about 10 min. Conclusions: Our results provide direct evidence for currents heating the lower solar chromosphere through Ohmic dissipation.

astro-ph.SR↗

arXiv2021

Solving the Kolmogorov PDE by means of deep learning

Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and the natural sciences. In particular, SDEs and Kolmogorov PDEs, respectively, are highly employed in models for the approximative pricing of financial derivatives. Kolmogorov PDEs and SDEs, respectively, can typically not be solved explicitly and it has been and still is an active topic of research to design and analyze numerical methods which are able to approximately solve Kolmogorov PDEs and SDEs, respectively. Nearly all approximation methods for Kolmogorov PDEs in the literature suffer under the curse of dimensionality or only provide approximations of the solution of the PDE at a single fixed space-time point. In this paper we derive and propose a numerical approximation method which aims to overcome both of the above mentioned drawbacks and intends to deliver a numerical approximation of the Kolmogorov PDE on an entire region $[a,b]^d$ without suffering from the curse of dimensionality. Numerical results on examples including the heat equation, the Black-Scholes model, the stochastic Lorenz equation, and the Heston model suggest that the proposed approximation algorithm is quite effective in high dimensions in terms of both accuracy and speed.

math.NA↗

arXiv2021

Deep splitting method for parabolic PDEs

In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handle extremely high-dimensional PDEs. We test the method on different examples from physics, stochastic control and mathematical finance. In all cases, it yields very good results in up to 10,000 dimensions with short run times.

math.NA↗

arXiv2021

Fluctuations of water quality time series in rivers follow superstatistics

Superstatistics is a general method from nonequilibrium statistical physics which has been applied to a variety of complex systems, ranging from hydrodynamic turbulence to traffic delays and air pollution dynamics. Here, we investigate water quality time series (such as dissolved oxygen concentrations and electrical conductivity) as measured in rivers, and provide evidence that they exhibit superstatistical behaviour. Our main example are time series as recorded in the river Chess in South East England. Specifically, we use seasonal detrending and empirical mode decomposition (EMD) to separate trends from fluctuations for the measured data. With either detrending method, we observe heavy-tailed fluctuation distributions, which are well described by a log-normal superstatistics for dissolved oxygen. Contrarily, we find a double peaked non-standard superstatistics for the electrical conductivity data, which we model using two combined $χ^2$-distributions.

physics.ao-ph↗

arXiv2021

Phase and amplitude synchronisation in power-grid frequency fluctuations in the Nordic Grid

Monitoring and modelling the power grid frequency is key to ensuring stability in the electrical power system. Many tools exist to investigate the detailed deterministic dynamics and especially the bulk behaviour of the frequency. However, far less attention has been paid to its stochastic properties, and there is a need for a cohesive framework that couples both short-time scale fluctuations and bulk behaviour. Moreover, commonly assumed uncorrelated stochastic noise is predominantly employed in modelling in energy systems. In this publication, we examine the stochastic properties of six synchronous power-grid frequency recording with high-temporal resolution of the Nordic Grid from September 2013, focusing on the increments of the frequency recordings. We show that these increments follow non-Gaussian statistics and display spatial and temporal correlations. Furthermore, we report two different physical synchronisation phenomena: a very short timescale phase synchronisation ($<2\,$s) followed by a slightly larger timescale amplitude synchronisation ($2\,$s-$5\,$s). Overall, these results provide guidance on how to model fluctuations in power systems.

physics.soc-ph↗

arXiv2021

Spatio-temporal complexity of power-grid frequency fluctuations

Power-grid systems constitute one of the most complex man-made spatially extended structures. These operate with strict operational bounds to ensure synchrony across the grid. This is particularly relevant for power-grid frequency, which operates strictly at $50\,$Hz ($60\,$Hz). Nevertheless, small fluctuations around the mean frequency are present at very short time scales $<2$ seconds and can exhibit highly complex spatio-temporal behaviour. Here we apply superstatistical data analysis techniques to measured frequency fluctuations in the Nordic Grid. We study the increment statistics and extract the relevant time scales and superstatistical distribution functions from the data. We show that different synchronous recordings of power-grid frequency have very distinct stochastic fluctuations with different types of superstatistics at different spatial locations, and with transitions from one superstatistics to another when the time lag of the increment statistics is changed.

physics.soc-ph↗

arXiv2021

Statistical Characterization of Airplane Delays

The aviation industry is of great importance for a globally connected economy. Customer satisfaction with airlines and airport performance is considerably influenced by how much flights are delayed. But how should the delay be quantified with thousands of flights for each airport and airline? Here, we present a statistical analysis of arrival delays at several UK airports between 2018 and 2020. We establish a procedure to compare both mean delay and extreme events among airlines and airports, identifying a power-law decay of large delays. Furthermore, we note drastic changes in plane delay statistics during the COVID-19 pandemic. Finally, we find that delays are described by a superposition of simple distributions, leading to a superstatistics.

physics.soc-ph↗

arXiv2021

Superstatistical modelling of protein diffusion dynamics in bacteria

A recent experiment [Sadoon AA, Wang Y. 2018 Phys. Rev. E 98, 042411] has revealed that nucleoid associated proteins (i.e., DNA-binding proteins) exhibit highly heterogeneous diffusion processes in bacteria where not only the diffusion constant but also the anomalous diffusion exponent fluctuates for the various proteins. The distribution of displacements of such proteins is observed to take a q-Gaussian form, which decays as a power law. Here, a statistical model is developed for the diffusive motion of the proteins within the bacterium, based on a superstatistics with two variables. This model hierarchically takes into account the joint fluctuations of both the anomalous diffusion exponents and the diffusion constants. A fractional Brownian motion is discussed as a possible local model. Good agreement with the experimental data is obtained.

cond-mat.stat-mech↗

arXiv2021

Full history recursive multilevel Picard approximations for ordinary differential equations with expectations

We consider ordinary differential equations (ODEs) which involve expectations of a random variable. These ODEs are special cases of McKean-Vlasov stochastic differential equations (SDEs). A plain vanilla Monte Carlo approximation method for such ODEs requires a computational cost of order $\varepsilon^{-3}$ to achieve a root-mean-square error of size $\varepsilon$. In this work we adapt recently introduced full history recursive multilevel Picard (MLP) algorithms to reduce this computational complexity. Our main result shows for every $δ>0$ that the proposed MLP approximation algorithm requires only a computational effort of order $\varepsilon^{-(2+δ)}$ to achieve a root-mean-square error of size $\varepsilon$.

math.NA↗

arXiv2020

The formation of an atypical sunspot light bridge as a result of large-scale flux emergence

We use a combination of full-disk data from the Solar Dynamics Observatory and high-resolution data from the Dunn Solar Telescope (DST) to study the formation, structure, and evolution of an atypical light bridge (LB) in a regular sunspot. The LB results from the emergence of magnetic flux with one footpoint rooted in a pore outside the parent sunspot that appears about 17 hrs before the LB. The pore has a polarity opposite to that of the sunspot and recedes away from it at a speed of about 0.4 km/s. This is accompanied by the development of an elongated magnetic channel in the outer penumbra which triggers the formation of the LB when it reaches the inner penumbral boundary. The LB is a nearly horizontal structure with a field strength of about 1.2 kG that exhibits long-lived photospheric blue-shifts of about 0.85 km/s along its entire length.The emergence of the LB leads to dynamic surges in the chromosphere and transition region about 13 min later. We derived the photospheric and chromospheric structure of the LB in the DST data from spectral line parameters and inversions of He i at 1083 nm, Si i at 1082.7 nm, Ca ii IR at 854 nm and Halpha at 656 nm, and speckle-reconstructed imaging at 700 nm and 430 nm. The LB shows an elongated filamentary shape in the photosphere without lateral extrusions. The thermal inversion of Ca ii IR reveals the LB to be about 600-800 K hotter than the umbra. Different sections of the LB are elevated to heights between 400 and 700 km. Our results indicate that the LB formation is part of a flux emergence event with the LB envelope reaching a height of about 29 Mm before dissolving after about 13 hr. We suggest that the existence of persistent, large-scale photospheric blue-shifts in LBs is the most likely criterion to distinguish between flux emergence events and overturning convection in field-free umbral intrusions.

astro-ph.SR↗