A semi-Lagrangian Vlasov solver in tensor train format

In this article, we derive a semi-Lagrangian scheme for the solution of the Vlasov equation represented as a low-parametric tensor. Grid-based methods for the Vlasov equation have been shown to give accurate results but their use has mostly been limited to simulations in two dimensional phase space due to extensive memory requirements in higher dimensions. Compression of the solution via high-order singular value decomposition can help in reducing the storage requirements and the tensor train (TT) format provides efficient basic linear algebra routines for low-rank representations of tensors. In this paper, we develop interpolation formulas for a semi-Lagrangian solver in TT format. In order to efficiently implement the method, we propose a compression of the matrix representing the interpolation step and an efficient implementation of the Hadamard product. We show numerical simulations for standard test cases in two, four and six dimensional phase space. Depending on the test case, the memory requirements reduce by a factor $10^2-10^3$ in four and a factor $10^5-10^6$ in six dimensions compared to the full-grid method

Direct Mapping of Massive Compact Objects in Extragalactic Dark Halos

A significant fraction of non-baryonic or baryonic dark matter in galactic halos may consist of MASsive Compact Objects (MASCOs) with mass M=10^{1-4}M_{sun}. Possible candidates for such compact objects include primordial black holes or remnants of primordial (Population III) stars. We propose a method for directly detecting MASCOs in extragalactic halos, using the VLBI techniques with extremely high resolution. If a galactic halo comprising a large number of MASCOs produces multiple images of a background radio-loud QSO by gravitational lensing, then a high-resolution radio map of each macro-lensed image should reveal microlensing effects by MASCOs. To assess their observational feasibility, we simulate microlensing of the radio-loud, four-image lensed QSO, B1422+231, assuming angular resolution of ~0.01 mas. MASCOs are represented by point masses. For comparison, we also simulate microlensing of B1422+231 by singular isothermal spheres. We find that the surface brightness of the macro-lensed images shows distinct spatial patterns on the scale of the Einstein radius of the perturbers. In the case of point-mass perturbers, many tiny dark spots also appear in the macro-lensed images associated with a decrease in the surface brightness toward the fringe of the original QSO image, whereas no such spots are available in the SIS models. Based on the size, position and magnified or demagnified patterns of images, we shall be able to determine the mass and density profile of a MASCO as well as its spatial distribution and abundance in a galactic halo.Comment: 4 pages, 3 figure

Efficient Explicit Time Stepping of High Order Discontinuous Galerkin Schemes for Waves

This work presents algorithms for the efficient implementation of discontinuous Galerkin methods with explicit time stepping for acoustic wave propagation on unstructured meshes of quadrilaterals or hexahedra. A crucial step towards efficiency is to evaluate operators in a matrix-free way with sum-factorization kernels. The method allows for general curved geometries and variable coefficients. Temporal discretization is carried out by low-storage explicit Runge-Kutta schemes and the arbitrary derivative (ADER) method. For ADER, we propose a flexible basis change approach that combines cheap face integrals with cell evaluation using collocated nodes and quadrature points. Additionally, a degree reduction for the optimized cell evaluation is presented to decrease the computational cost when evaluating higher order spatial derivatives as required in ADER time stepping. We analyze and compare the performance of state-of-the-art Runge-Kutta schemes and ADER time stepping with the proposed optimizations. ADER involves fewer operations and additionally reaches higher throughput by higher arithmetic intensities and hence decreases the required computational time significantly. Comparison of Runge-Kutta and ADER at their respective CFL stability limit renders ADER especially beneficial for higher orders when the Butcher barrier implies an overproportional amount of stages. Moreover, vector updates in explicit Runge--Kutta schemes are shown to take a substantial amount of the computational time due to their memory intensity

An intelligent real time 3D vision system for robotic welding tasks

MARWIN is a top-level robot control system that has been designed for automatic robot welding tasks. It extracts welding parameters and calculates robot trajectories directly from CAD models which are then verified by real-time 3D scanning and registration. MARWIN's 3D computer vision provides a user-centred robot environment in which a task is specified by the user by simply confirming and/or adjusting suggested parameters and welding sequences. The focus of this paper is on describing a mathematical formulation for fast 3D reconstruction using structured light together with the mechanical design and testing of the 3D vision system and show how such technologies can be exploited in robot welding tasks

Robot trajectory planning using OLP and structured light 3D machine vision

This paper proposes a new methodology for robotic offline programming (OLP) addressing the issue of automatic program generation directly from 3D CAD models and verification through online 3D reconstruction. Limitations of current OLP include manufacturing tolerances between CAD and workpieces and inaccuracies in workpiece placement and modelled work cell. These issues are addressed and demonstrated through surface scanning, registration, and global and local error estimation. The method allows the robot to adjust the welding path designed from the CAD model to the actual workpiece. Alternatively, for non-repetitive tasks and where a CAD model is not available, it is possible to interactively define the path online over the scanned surface

GEMPIC: Geometric ElectroMagnetic Particle-In-Cell Methods

We present a novel framework for Finite Element Particle-in-Cell methods based on the discretization of the underlying Hamiltonian structure of the Vlasov-Maxwell system. We derive a semi-discrete Poisson bracket, which retains the defining properties of a bracket, anti-symmetry and the Jacobi identity, as well as conservation of its Casimir invariants, implying that the semi-discrete system is still a Hamiltonian system. In order to obtain a fully discrete Poisson integrator, the semi-discrete bracket is used in conjunction with Hamiltonian splitting methods for integration in time. Techniques from Finite Element Exterior Calculus ensure conservation of the divergence of the magnetic field and Gauss' law as well as stability of the field solver. The resulting methods are gauge invariant, feature exact charge conservation and show excellent long-time energy and momentum behaviour. Due to the generality of our framework, these conservation properties are guaranteed independently of a particular choice of the Finite Element basis, as long as the corresponding Finite Element spaces satisfy certain compatibility conditions.Comment: 57 Page