Sparse convolution quadrature for time domain boundary integral formulations of the wave equation

Many important physical applications are governed by the wave equation. The formulation as time domain boundary integral equations involves retarded potentials. For the numerical solution of this problem, we employ the convolution quadrature method for the discretization in time and the Galerkin boundary element method for the space discretization. We introduce a simple a priori cut-off strategy where small entries of the system matrices are replaced by zero. The threshold for the cut-off is determined by an a priori analysis which will be developed in this paper. This analysis will also allow to estimate the effect of additional perturbations such as panel clustering and numerical integration on the overall discretization error. This method reduces the storage complexity for time domain integral equations from O(M2N) to O(M2N½ logM), where N denotes the number of time steps and M is the dimension of the boundary element spac

Probing the gluon density of the proton in the exclusive photoproduction of vector mesons at the LHC: A phenomenological analysis

The current uncertainty on the gluon density extracted from the global parton analysis is large in the kinematical range of small values of the Bjorken - $x$ variable and low values of the hard scale $Q^2$. An alternative to reduces this uncertainty is the analysis of the exclusive vector meson photoproduction in photon - hadron and hadron - hadron collisions. This process offers a unique opportunity to constrain the gluon density of the proton, since its cross section is proportional to the gluon density squared. In this paper we consider current parametrizations for the gluon distribution and estimate the exclusive vector meson photoproduction cross section at HERA and LHC using the leading logarithmic formalism. We perform a fit of the normalization of the $\gamma h$ cross section and the value of the hard scale for the process and demonstrate that the current LHCb experimental data are better described by models that assume a slow increasing of the gluon distribution at small - $x$ and low $Q^2$.Comment: 8 pages, 6 figures, 1 table. Version published in European Physical Journal

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

Damping of electromagnetic waves due to electron-positron pair production

The problem of the backreaction during the process of electron-positron pair production by a circularly polarized electromagnetic wave propagating in a plasma is investigated. A model based on the relativistic Boltzmann-Vlasov equation with a source term corresponding to the Schwinger formula for the pair creation rate is used. The damping of the wave, the nonlinear up-shift of its frequency due to the plasma density increase and the effect of the damping on the wave polarization and on the background plasma acceleration are investigated as a function of the wave amplitude.Comment: 11 pages, 5 figures; revtex

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

Two-scale composite finite element method for Dirichlet problems on complicated domains

In this paper, we define a new class of finite elements for the discretization of problems with Dirichlet boundary conditions. In contrast to standard finite elements, the minimal dimension of the approximation space is independent of the domain geometry and this is especially advantageous for problems on domains with complicated micro-structures. For the proposed finite element method we prove the optimal-order approximation (up to logarithmic terms) and convergence estimates valid also in the cases when the exact solution has a reduced regularity due to re-entering corners of the domain boundary. Numerical experiments confirm the theoretical results and show the potential of our proposed metho

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

Diagnostic error increases mortality and length of hospital stay in patients presenting through the emergency room

Background: Diagnostic errors occur frequently, especially in the emergency room. Estimates about the consequences of diagnostic error vary widely and little is known about the factors predicting error. Our objectives thus was to determine the rate of discrepancy between diagnoses at hospital admission and discharge in patients presenting through the emergency room, the discrepancies’ consequences, and factors predicting them. Methods: Prospective observational clinical study combined with a survey in a University-affiliated tertiary care hospital. Patients’ hospital discharge diagnosis was compared with the diagnosis at hospital admittance through the emergency room and classified as similar or discrepant according to a predefined scheme by two independent expert raters. Generalized linear mixed-effects models were used to estimate the effect of diagnostic discrepancy on mortality and length of hospital stay and to determine whether characteristics of patients, diagnosing physicians, and context predicted diagnostic discrepancy. Results: 755 consecutive patients (322 [42.7%] female; mean age 65.14 years) were included. The discharge diagnosis differed substantially from the admittance diagnosis in 12.3% of cases. Diagnostic discrepancy was associated with a longer hospital stay (mean 10.29 vs. 6.90 days; Cohen’s d 0.47; 95% confidence interval 0.26 to 0.70; P = 0.002) and increased patient mortality (8 (8.60%) vs. 25(3.78%); OR 2.40; 95% CI 1.05 to 5.5 P = 0.038). A factor available at admittance that predicted diagnostic discrepancy was the diagnosing physician’s assessment that the patient presented atypically for the diagnosis assigned (OR 3.04; 95% CI 1.33–6.96; P = 0.009). Conclusions: Diagnostic discrepancies are a relevant healthcare problem in patients admitted through the emergency room because they occur in every ninth patient and are associated with increased in-hospital mortality. Discrepancies are not readily predictable by fixed patient or physician characteristics; attention should focus on context

Finite elements on degenerate meshes: inverse-type inequalities and applications

In this paper we obtain a range of inverse-type inequalities which are applicable to finite-element functions on general classes of meshes, including degenerate meshes obtained by anisotropic refinement. These are obtained for Sobolev norms of positive, zero and negative order. In contrast to classical inverse estimates, negative powers of the minimum mesh diameter are avoided. We give two applications of these estimates in the context of boundary elements: (i) to the analysis of quadrature error in discrete Galerkin methods and (ii) to the analysis of the panel clustering algorithm. Our results show that degeneracy in the meshes yields no degradation in the approximation properties of these method