Retarded boundary integral equations on the sphere: exact and numerical solution

In this paper we consider the three-dimensional wave equation in unbounded domains with Dirichlet boundary conditions. We start from a retarded single-layer potential ansatz for the solution of these equations which leads to the retarded potential integral equation on the bounded surface of the scatterer. We formulate an algorithm for the space-time Galerkin discretization with smooth and compactly supported temporal basis functions, which were introduced in Sauter & Veit (2013, Numer. Math., 145-176). For the debugging of an implementation and for systematic parameter tests it is essential to have at hand some explicit representations and some analytic properties of the exact solutions for some special cases. We will derive such explicit representations for the case where the scatterer is the unit ball. The obtained formulas are easy to implement and we will present some numerical experiments for these cases to illustrate the convergence behaviour of the proposed metho

A Galerkin method for retarded boundary integral equations with smooth and compactly supported temporal basis functions

We consider retarded boundary integral formulations of the three-dimensional wave equation in unbounded domains. Our goal is to apply a Galerkin method in space and time in order to solve these problems numerically. In this approach the computation of the system matrix entries is the major bottleneck. We will propose new types of finite-dimensional spaces for the time discretization. They allow variable time-stepping, variable order of approximation and simplify the quadrature problem arising in the generation of the system matrix substantially. The reason is that the basis functions of these spaces are globally smooth and compactly supported. In order to perform numerical tests concerning our new basis functions we consider the special case that the boundary of the scattering problem is the unit sphere. In this case explicit solutions of the problem are available which will serve as reference solutions for the numerical experiment

Fast quadrature techniques for retarded potentials based on TT/QTT tensor approximation

We consider the Galerkin approach for the numerical solution of retarded boundary integral formulations of the three dimensional wave equation in unbounded domains. Recently smooth and compactly supported basis functions in time were introduced which allow the use of standard quadrature rules in order to compute the entries of the boundary element matrix. In this paper we use TT and QTT tensor approximations to increase the effciency of these quadrature rules. Various numerical experiments show the substantial reduction of the computational cost that is needed to obtain accurate approximations for the arising integrals

Real world experience reflecting current guidelines

Background: In case of suspected acute coronary syndrome (ACS), international guidelines recommend to obtain a 12-lead ECG as soon as possible after first medical contact, to administrate platelet aggregation inhibitors and antithrombins, and to transfer the patient as quickly as possible to an emergency department. Methods: A German emergency care service database was retrospectively analysed from 2014 to 2016. Data were tested for normal distribution and the Mann-Whitney test was used for statistical analysis. Results are presented as medians (IQR). Results: A total of 1424 patients with suspected ACS were included in the present analysis. A 12-lead ECG was documented in 96% of patients (n = 1369). The prehospital incidence of ST-segment elevation myocardial infarction (STEMI) was 18% (n = 250). In 981 patients (69%), acetylsalicylic acid (ASA), unfractionated heparin (UFH), or ASA and UFH was given. Time in prehospital care differed significantly between non-STEMI (NSTEMI) ACS (37 [IQR 30, 44] min) and STEMI patients (33 [IQR 26, 40] min, n = 1395, p < 0.0001). Most of NSTEMI ACS and STEMI patients were brought to the emergency care unit, while 30% of STEMI patients were directly handed over to a cardiac catheterization laboratory. Conclusions: Prehospital ECG helps to identify patients with STEMI, which occurs in 18% of suspected ACS. Patients without ST-elevations suffered from longer prehospital care times. Thus, it is tempting to speculate that ST-elevations in patients prompt prehospital medical teams to act more efficiently while the absence of ST-elevations even in patients with suspected ACS might cause unintended delays. Moreover, this analysis suggests the need for further efforts to make the cardiac catheterization laboratory the standard hand-over location for all STEMI patients.Einführung: Internationale Leitlinien empfehlen die umgehende Ableitung eines 12-Kanal-Elektrokardiogramms bei Patienten mit akuter Angina-pectoris-Symptomatik. Ebenfalls wird die präklinische Gabe von Azetylsalizylsäure, Heparin sowie der sofortige Transport in ein Krankenhaus mit der Möglichkeit einer koronarinvasiven Diagnostik empfohlen. Methoden: Es wurden die Einsatzprotokolle einer Rettungsstellendatenbank retrospektiv für den Zeitraum von 2014–2016 auf diese Merkmale untersucht. Die Daten wurden auf Normalverteilung getestet und für die statistische Analyse wurde ein Mann-Whitney-Test verwendet. Die Ergebnisse sind als Mediane (Interquartilsabstand, IQR) dargestellt. Ergebnisse: Es wurden 1424 Patienten mit Angina-pectoris-Symptomatik eingeschlossen. In 96 % (n = 1369) der Fälle wurde ein 12-Kanal-Elektrokardiogramm aufgezeichnet. Präklinisch wurden 250 Patienten (18 %) mit ST-Strecken-Hebungs-Infarkt (STEMI) diagnostiziert. Insgesamt 981 Patienten (69 %) erhielten Azetylsalizylsäure bzw. Heparin oder Azetylsalizylsäure und Heparin. Die präklinischen Zeiten unterschieden sich signifikant zwischen Patienten mit akutem Koronarsyndrom ohne Ischämiezeichen im Elektrokardiogramm (NSTEMI ACS; 37 min [IQR 30,44]) und Patienten mit ST-Hebungen (STEMI; 33 min [IQR 26,40]). Der Großteil der Patienten wurde in der Notaufnahme übergeben, während 30 % der STEMI-Patienten direkt in das Herzkatheterlabor transferiert wurden. Schlussfolgerung: Die Inzidenz von ST-Hebungen liegt im präklinischen Bereich bei Patienten mit Angina-pectoris bei 18 %. Patienten ohne Ischämiezeichen wurden langsamer einem Krankenhaus zugeführt als Patienten mit STEMI. Dieses Ergebnis kann auf die geltenden Leitlinien zurückzuführen sein, da bei STEMI-Patienten die Zeit von Diagnosestellung bis zur koronarinvasiven Diagnostik <120 min betragen soll

A method for dense packing discovery

The problem of packing a system of particles as densely as possible is foundational in the field of discrete geometry and is a powerful model in the material and biological sciences. As packing problems retreat from the reach of solution by analytic constructions, the importance of an efficient numerical method for conducting \textit{de novo} (from-scratch) searches for dense packings becomes crucial. In this paper, we use the \textit{divide and concur} framework to develop a general search method for the solution of periodic constraint problems, and we apply it to the discovery of dense periodic packings. An important feature of the method is the integration of the unit cell parameters with the other packing variables in the definition of the configuration space. The method we present led to improvements in the densest-known tetrahedron packing which are reported in [arXiv:0910.5226]. Here, we use the method to reproduce the densest known lattice sphere packings and the best known lattice kissing arrangements in up to 14 and 11 dimensions respectively (the first such numerical evidence for their optimality in some of these dimensions). For non-spherical particles, we report a new dense packing of regular four-dimensional simplices with density $\phi=128/219\approx0.5845$ and with a similar structure to the densest known tetrahedron packing.Comment: 15 pages, 5 figure

Numerical solution of exterior Maxwell problems by Galerkin BEM and Runge-Kutta convolution quadrature

In this paper we consider time-dependent electromagnetic scattering problems from conducting objects. We discretize the time-domain electric field integral equation using Runge-Kutta convolution quadrature in time and a Galerkin method in space. We analyze the involved operators in the Laplace domain and obtain convergence results for the fully discrete scheme. Numerical experiments indicate the sharpness of the theoretical estimate

Random projections and the optimization of an algorithm for phase retrieval

Iterative phase retrieval algorithms typically employ projections onto constraint subspaces to recover the unknown phases in the Fourier transform of an image, or, in the case of x-ray crystallography, the electron density of a molecule. For a general class of algorithms, where the basic iteration is specified by the difference map, solutions are associated with fixed points of the map, the attractive character of which determines the effectiveness of the algorithm. The behavior of the difference map near fixed points is controlled by the relative orientation of the tangent spaces of the two constraint subspaces employed by the map. Since the dimensionalities involved are always large in practical applications, it is appropriate to use random matrix theory ideas to analyze the average-case convergence at fixed points. Optimal values of the gamma parameters of the difference map are found which differ somewhat from the values previously obtained on the assumption of orthogonal tangent spaces.Comment: 15 page