Convergence Rates of Gaussian ODE Filters

A recently-introduced class of probabilistic (uncertainty-aware) solvers for ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to initial value problems. These methods model the true solution $x$ and its first $q$ derivatives \emph{a priori} as a Gauss--Markov process $\boldsymbol{X}$, which is then iteratively conditioned on information about $\dot{x}$. This article establishes worst-case local convergence rates of order $q+1$ for a wide range of versions of this Gaussian ODE filter, as well as global convergence rates of order $q$ in the case of $q=1$ and an integrated Brownian motion prior, and analyses how inaccurate information on $\dot{x}$ coming from approximate evaluations of $f$ affects these rates. Moreover, we show that, in the globally convergent case, the posterior credible intervals are well calibrated in the sense that they globally contract at the same rate as the truncation error. We illustrate these theoretical results by numerical experiments which might indicate their generalizability to $q \in \{2,3,\dots\}$.Comment: 26 pages, 5 figure

The Cosmic Large-Scale Structure in X-rays (CLASSIX) Cluster Survey I: Probing galaxy cluster magnetic fields with line of sight rotation measures

To search for a signature of an intracluster magnetic field, we compare measurements of Faraday rotation of polarised extragalactic radio sources in the line of sight of galaxy clusters with those outside. We correlated a catalogue of 1383 rotation measures (RM) of extragalactic polarised radio sources with X-ray luminous galaxy clusters from the CLASSIX survey (combining REFLEX II and NORAS II). We compared the RM in the line of sight of clusters within their projected radii of r_500 with those outside and found a significant excess of the dispersion of the RM in the cluster regions. Since the observed RM is the result of Faraday rotation in several presumably uncorrelated magnetised cells of the intracluster medium, the observations correspond to quantities averaged over several magnetic field directions and strengths. Therefore the interesting quantity is the standard deviation of the RM for an ensemble of clusters. We found a standard deviation of the RM inside r_500 of about 120 +- 21 rad m^-2. This compares to about 56 +- 8 rad m^-2 outside. We show that the most X-ray luminous and thus most massive clusters contribute most to the observed excess RM. Modelling the electron density distribution in the intracluster medium with a self-similar model, we found that the dispersion of the RM increases with the column density, and we deduce a magnetic field value of about 2 - 6 (l/10kpc)^-1/2 microG assuming a constant magnetic field strength, where l is the size of the coherently magnetised intracluster medium cells. This magnetic field energy density amounts to a few percent of the average thermal energy density in clusters. When we assumed the magnetic energy density to be a constant fraction of the thermal energy density, we deduced a slightly lower value for this fraction of 3 - 10 (l/10kpc)^-1/2 per mille.Comment: 7 pages, 6 figures, in press, Astronomy and Astrophysics, 201

Financial integration, specialization and systemic risk

This paper studies the implications of cross-border financial integration for financial stability when banks' loan portfolios adjust endogenously. Banks can be subject to sectoral and aggregate domestic shocks. After integration they can share these risks in a complete interbank market. When banks have a comparative advantage in providing credit to certain industries, financial integration may induce banks to specialize in lending. An enhanced concentration in lending does not necessarily increase risk, because a well-functioning interbank market allows to achieve the necessary diversification. This greater need for risk sharing, though, increases the risk of cross-border contagion and the likelihood of widespread banking crises. However, even though integration increases the risk of contagion it improves welfare if it permits banks to realize specialization benefits. JEL Classification: D61, E44, G21.Financial integration, specialization, interbank market, financial contagion.

Influence of polarizability on metal oxide properties studied by molecular dynamics simulations

We have studied the dependence of metal oxide properties in molecular dynamics (MD) simulations on the polarizability of oxygen ions. We present studies of both liquid and crystalline structures of silica (SiO2), magnesia (MgO) and alumina (Al2O3). For each of the three oxides, two separately optimized sets of force fields were used: (i) Long-range Coulomb interactions between oxide and metal ions combined with a short-range pair potential. (ii) Extension of force field (i) by adding polarizability to the oxygen ions. We show that while an effective potential of type (i) without polarizable oxygen ions can describe radial distributions and lattice constants reasonably well, potentials of type (ii) are required to obtain correct values for bond angles and the equation of state. The importance of polarizability for metal oxide properties decreases with increasing temperature.Comment: 8 pages, 7 figure

Financial integration, specialization and systemic risk

This paper studies the implications of cross-border financial integration for financial stability when banks' loan portfolios adjust endogenously. Banks can be subject to sectoral and aggregate domestic shocks. After integration they can share these risks in a complete interbank market. When banks have a comparative advantage in providing credit to certain industries, they will exploit the enhanced risk sharing opportunities through more specialization in lending. The enhanced concentration in lending does not increase risk, because a well-functioning interbank market allows to achieve the necessary diversification. The greater need for risk sharing through it increases, however, the risk of cross-border contagion. Better risk sharing and greater risk of contagion tend to offset each other and financial integration improves welfare since specialization benefits are realized. --Financial integration,specialization,interbank market,financial contagion

Quantum simulations with ultracold atoms: beyond standard optical lattices

Many outstanding problems in quantum physics, such as high-Tc superconductivity or quark confinement, are still - after decades of research - awaiting commonly accepted explanations. One reason is that such systems are often difficult to control, show an intermingling of several effects, or are not easily accessible to measurement. To arrive at a deeper understanding of the physics at work, researchers typically derive simplified models designed to capture the most striking phenomena of the system under consideration. However, due to the exponential complexity of Hilbert space, even some of the simplest of such models pose formidable challenges to analytical and numerical calculations. In 1982, Feynman proposed to solve such quantum models with experimental simulation on a physically distinct, specifically engineered quantum system [Int. J. Theor.Phys. 21, 467]. Designed to be governed by the same underlying equations as the original model, it is hoped that direct measurements on these so called quantum simulators (QSs) will allow to gather deep insights into outstanding problems of physics and beyond. In this thesis, we identify four requirements that a useful QS has to fulfill, relevance, control, reliability, and efficiency. Focusing on these, we review the state of the art of two popular approaches, digital QSs (i.e., special purpose quantum computers) and analog QSs (devices with always-on interactions). Further, focusing on possibilities to increase control over QSs, we discuss a scheme to engineer quantum correlations between mesoscopic numbers of spinful particles in optical lattices. This technique, based on quantum polarization spectroscopy, may be useful for state preparation and quantum information protocols. Additionally, employing several analytical and numerical methods for the calculation of many-body ground states, we demonstrate the variety of condensed-matter problems that can be attacked with QSs consisting of ultracold ions or neutral atoms in optical lattices. The chosen examples, some of which have already been realized in experiment, include such diverse settings as frustrated antiferromagnetism, quantum phase transitions in exotic lattice geometries, topological insulators, non-Abelian gauge-fields, orbital order of ultracold Fermions, and systems with long-range interactions. The experimental realization of all of these models requires techniques which go beyond standard optical lattices, e.g., time-periodic driving of lattices with exotic geometry, loading ultracold atoms into higher bands, or immersing trapped ions into an optical lattice. The chosen models, motivated by important open questions of quantum physics, pose difficult problems for classical computers, but they may be amenable in the near future to quantum simulation with ultracold atoms or ions. While the experimental control over relevant models has increased dramatically in the last years, the reliability and efficiency of QSs has received considerably less attention. As a second important part of this thesis, we emphasize the need to consider these aspects under realistic experimental conditions. We discuss specific situations where terms that have typically been neglected in the description of the QS introduce systematic errors and even lead to novel physics. Further, we characterize in a generic example the influence of quenched disorder on an analog QS. Its performance for simulating universal behavior near a quantum phase transition seems satisfactory for low disorder. Moreover, our results suggest a connection between the reliability and efficiency of a QS: it works less reliable exactly in those interesting regimes where classical calculations are less efficient. If QSs fulfill all of our four requirements, they may revolutionize our approach to quantum-mechanical problems, allowing to solve the behavior of complex Hamiltonians, and to design nano-scale materials and chemical compounds from the ground up