Automatic Deformation of Riemann-Hilbert Problems with Applications to the Painlev\'e II Transcendents

The stability and convergence rate of Olver's collocation method for the numerical solution of Riemann-Hilbert problems (RHPs) is known to depend very sensitively on the particular choice of contours used as data of the RHP. By manually performing contour deformations that proved to be successful in the asymptotic analysis of RHPs, such as the method of nonlinear steepest descent, the numerical method can basically be preconditioned, making it asymptotically stable. In this paper, however, we will show that most of these preconditioning deformations, including lensing, can be addressed in an automatic, completely algorithmic fashion that would turn the numerical method into a black-box solver. To this end, the preconditioning of RHPs is recast as a discrete, graph-based optimization problem: the deformed contours are obtained as a system of shortest paths within a planar graph weighted by the relative strength of the jump matrices. The algorithm is illustrated for the RHP representing the Painlev\'e II transcendents.Comment: 20 pages, 16 figure

A posteriori error estimates for elliptic problems in two and three space dimensions

Let $u \in H$ be the exact solution of a given selfadjoint elliptic boundary value problem, which is approximated by some $\tilde u \in \mathcal{S}$, $\mathcal{S}$ being a suitable finite-element space. Efficient and reliable a posteriors estimates of the error $\| {u - \tilde u} \|$, measuring the (local) quality of $\tilde u$, play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh. A well-known class of error estimates can be derived systematically by localizing the discretized defect problem by using domain decomposition techniques. In this paper, we provide a guideline for the theoretical analysis of such error estimates. We further clarify the relation to other concepts. Our analysis leads to new error estimates, which are specially suited to three space dimensions. The theoretical results are illustrated by numerical computations

Energy Level Crossings in Molecular Dynamics

Energy level crossings are the landmarks that separate classical from quantum mechanical modeling of molecular systems. They induce non-adiabatic transitions between the otherwise adiabatically decoupled electronic level spaces. This review covers results on the analysis of propagation through level crossings of codimension two, a mathematical justification of surface hopping algorithms, and a spectral study of a linear isotropic system

Complex magnetism of B20-MnGe: from spin-spirals, hedgehogs to monopoles

B20 compounds are the playground for various non-trivial magnetic textures such as skyrmions, which are topologically protected states. Recent measurements on B20-MnGe indicate no clear consensus on its magnetic behavior, which is characterized by the presence of either spin-spirals or 3-dimensional objects interpreted to be a cubic lattice of hedgehogs and anti-hedgehogs. Utilizing a massively parallel linear scaling all-electron density functional algorithm, we find from full first-principles simulations on cells containing thousands of atoms that upon increase of the compound volume, the state with lowest energy switches across different magnetic phases: ferromagnetic, spin-spiral, hedgehog and monopole

Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles

The density function for the joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles is found in terms of a Painlev\'e II transcendent and its associated isomonodromic system. As a corollary, the density function for the spacing between these two eigenvalues is similarly characterized.The particular solution of Painlev\'e II that arises is a double shifted B\"acklund transformation of the Hasting-McLeod solution, which applies in the case of the distribution of the largest eigenvalue at the soft edge. Our deductions are made by employing the hard-to-soft edge transitions to existing results for the joint distribution of the first and second eigenvalue at the hard edge \cite{FW_2007}. In addition recursions under $a \mapsto a+1$ of quantities specifying the latter are obtained. A Fredholm determinant type characterisation is used to provide accurate numerics for the distribution of the spacing between the two largest eigenvalues.Comment: 26 pages, 1 Figure, 2 Table

Single-block rockfall dynamics inferred from seismic signal analysis

International audienceSeismic monitoring of mass movements can significantly help to mitigate the associated hazards; however, the link between event dynamics and the seismic signals generated is not completely understood. To better understand these relationships, we conducted controlled releases of single blocks within a soft-rock (black marls) gully of the Rioux-Bourdoux torrent (French Alps). A total of 28 blocks, with masses ranging from 76 to 472 kg, were used for the experiment. An instrumentation combining video cameras and seismometers was deployed along the travelled path. The video cameras allow reconstructing the trajectories of the blocks and estimating their velocities at the time of the different impacts with the slope. These data are compared to the recorded seismic signals. As the distance between the falling block and the seismic sensors at the time of each impact is known, we were able to determine the associated seismic signal amplitude corrected for propagation and attenuation effects. We compared the velocity, the potential energy lost, the kinetic energy and the momentum of the block at each impact to the true amplitude and the radiated seismic energy. Our results suggest that the amplitude of the seismic signal is correlated to the momentum of the block at the impact. We also found relationships between the potential energy lost, the kinetic energy and the seismic energy radiated by the impacts. Thanks to these relationships, we were able to retrieve the mass and the velocity before impact of each block directly from the seismic signal. Despite high uncertainties, the values found are close to the true values of the masses and the velocities of the blocks. These relationships allow for gaining a better understanding of the physical processes that control the source of high-frequency seismic signals generated by rockfalls

Data and Services at the Integrated Climate Data Center (ICDC) at the University of Hamburg

KlimawandelEarth observation data obtained from remote sensing sensors and in-situ data archives are fundamental for our current understanding of the Earth’s climate system. Such data are an important pre-requisite for Earth System research and should be easy to access and easy to use. In addition such data should be quality assessed and attached with information about uncertainties and long-term stability. If these data sets are stored in a self-explanatory, easy-to-use format, their usefulness and scientific value increase. This is the guideline for the Integrated Climate Data Center (ICDC) at the Center for Earth System Research and Sustainability (CEN), University of Hamburg. ICDC offers a reliable, quick and easy data access along with expert support for users and data providers. The ICDC provides several types of worldwide accessible in situ and satellite Earth observation data of the atmosphere, ocean, land surface, and cryosphere via the web portal http://icdc.zmaw.de. Recently, data from socio-economic sciences have been integrated into ICDC’s data base to enhance interdisciplinary collaboration. On ICDC’s web portal, each data set has its own page. It contains the data access points, a short data description, information about spatiotemporal coverage and resolution, data quality, important reference documents and contacts, and about how to cite the data set. The data are converted into netCDF or ASCII format. Consistency and quality checks are carried out – often in the framework of international collaborations. Literature studies are conducted to learn about potential limitations or preferred application areas of the data offered. The data sets can be accessed through the web page via FTP, HTTP or OPeNDAP. Using the Live Access Server, users can visualize data as maps, along transects and profiles, zoom into key regions, and create time series. In both fields, visualization and data access, ICDC tries to provide fast response times and high reliability

Effective-Range Dependence of Resonantly Interacting Fermions

We extract the leading effective range corrections to the equation of state of the unitary Fermi gas from ab initio fixed-node quantum Monte Carlo (FNQMC) calculations in a periodic box using a density functional theory (DFT), and show them to be universal by considering several two-body interactions. Furthermore, we find that the DFT is consistent with the best available unbiased QMC calculations, analytic results, and experimental measurements of the equation of state. We also discuss the asymptotic effective-range corrections for trapped systems and present the first QMC results with the correct asymptotic scaling.Comment: 11 pages, 5 figures: Updated to match published versio