Algorithmic Verification of Continuous and Hybrid Systems

We provide a tutorial introduction to reachability computation, a class of computational techniques that exports verification technology toward continuous and hybrid systems. For open under-determined systems, this technique can sometimes replace an infinite number of simulations.Comment: In Proceedings INFINITY 2013, arXiv:1402.661

Suitability of post-Newtonian/numerical-relativity hybrid waveforms for gravitational wave detectors

This article presents a study of the sufficient accuracy of post-Newtonian and numerical relativity waveforms for the most demanding usage case: parameter estimation of strong sources in advanced gravitational wave detectors. For black hole binaries, these detectors require accurate waveform models which can be constructed by fusing an analytical post-Newtonian inspiral waveform with a numerical relativity merger-ringdown waveform. We perform a comprehensive analysis of errors that enter such "hybrid waveforms". We find that the post-Newtonian waveform must be aligned with the numerical relativity waveform to exquisite accuracy, about 1/100 of a gravitational wave cycle. Phase errors in the inspiral phase of the numerical relativity simulation must be controlled to less than about 0.1rad. (These numbers apply to moderately optimistic estimates about the number of GW sources; exceptionally strong signals require even smaller errors.) The dominant source of error arises from the inaccuracy of the investigated post-Newtonian Taylor-approximants. Using our error criterium, even at 3.5-th post-Newtonian order, hybridization has to be performed significantly before the start of the longest currently available numerical waveforms which cover 30 gravitational wave cycles. The current investigation is limited to the equal-mass, zero-spin case and does not take into account calibration errors of the gravitational wave detectors.Comment: 32 pages, 12 figures, submitted to CQG for the NRDA2010 conference proceedings, added new figure (fig. 5) since last versio

Analysis of parametric biological models with non-linear dynamics

In this paper we present recent results on parametric analysis of biological models. The underlying method is based on the algorithms for computing trajectory sets of hybrid systems with polynomial dynamics. The method is then applied to two case studies of biological systems: one is a cardiac cell model for studying the conditions for cardiac abnormalities, and the second is a model of insect nest-site choice.Comment: In Proceedings HSB 2012, arXiv:1208.315

Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).Peer ReviewedPostprint (author's final draft

A nested hybridizable discontinuous Galerkin method for computing second-harmonic generation in three-dimensional metallic nanostructures

In this paper, we develop a nested hybridizable discontinuous Galerkin (HDG) method to numerically solve the Maxwell's equations coupled with the hydrodynamic model for the conduction-band electrons in metals. By means of a static condensation to eliminate the degrees of freedom of the approximate solution defined in the elements, the HDG method yields a linear system in terms of the degrees of freedom of the approximate trace defined on the element boundaries. Furthermore, we propose to reorder these degrees of freedom so that the linear system accommodates a second static condensation to eliminate a large portion of the degrees of freedom of the approximate trace, thereby yielding a much smaller linear system. For the particular metallic structures considered in this paper, the resulting linear system obtained by means of nested static condensations is a block tridiagonal system, which can be solved efficiently. We apply the nested HDG method to compute the second harmonic generation (SHG) on a triangular coaxial periodic nanogap structure. This nonlinear optics phenomenon features rapid field variations and extreme boundary-layer structures that span multiple length scales. Numerical results show that the ability to identify structures which exhibit resonances at $\omega$ and $2\omega$ is paramount to excite the second harmonic response.Comment: 31 pages, 7 figure

Analyzing Binary Black hole Spacetimes

With the first ever detection of gravitational waves from merging black-hole binaries by LIGO (Laser Interferometer GravitationalWave Observatory), a new era of gravitational wave astronomy was started. With its increased sensitivity, LIGO will see many more black-hole binaries in the future. To detect the gravitational waves and elucidate the properties of their sources, one needs theoretical waveform templates. These, in turn, require solving Einstein field equations, at least approximately. Approximate techniques like post-Newtonian theory and black-hole perturbation theory can produce waveforms that are accurate for certain phases of binaries evolution. Numerical relativity, on the other hand, can in principle produce accurate waveforms models for the full binary evolution. However, such simulations are computationally very expensive for the slow inspiral phase. To overcome this issue, we hybridized numerical relativity obtained by solving the Einstein field equations during the late-inspiral, plunge, and ringdown phase and post-Newtonian waveforms for the early-inspiral phase. Here we focus on hybridizing waveforms for precessing black-hole binaries. In this work we also developed a new tool to test the accuracy limits of approximate a binary black-hole spacetimes constructed using analytical approximate techniques. Our method is based on direct comparison to a numerically generated solution to the Einstein field equations

Status of NINJA: the Numerical INJection Analysis project

The 2008 NRDA conference introduced the Numerical INJection Analysis project (NINJA), a new collaborative effort between the numerical relativity community and the data analysis community. NINJA focuses on modeling and searching for gravitational wave signatures from the coalescence of binary system of compact objects. We review the scope of this collaboration and the components of the first NINJA project, where numerical relativity groups shared waveforms and data analysis teams applied various techniques to detect them when embedded in colored Gaussian noise