Optimal computation of brightness integrals parametrized on the unit sphere

We compare various approaches to find the most efficient method for the practical computation of the lightcurves (integrated brightnesses) of irregularly shaped bodies such as asteroids at arbitrary viewing and illumination geometries. For convex models, this reduces to the problem of the numerical computation of an integral over a simply defined part of the unit sphere. We introduce a fast method, based on Lebedev quadratures, which is optimal for both lightcurve simulation and inversion in the sense that it is the simplest and fastest widely applicable procedure for accuracy levels corresponding to typical data noise. The method requires no tessellation of the surface into a polyhedral approximation. At the accuracy level of 0.01 mag, it is up to an order of magnitude faster than polyhedral sums that are usually applied to this problem, and even faster at higher accuracies. This approach can also be used in other similar cases that can be modelled on the unit sphere. The method is easily implemented in lightcurve inversion by a simple alteration of the standard algorithm/software.Comment: Astronomy and Astrophysics, in pres

Reach Set Approximation through Decomposition with Low-dimensional Sets and High-dimensional Matrices

Approximating the set of reachable states of a dynamical system is an algorithmic yet mathematically rigorous way to reason about its safety. Although progress has been made in the development of efficient algorithms for affine dynamical systems, available algorithms still lack scalability to ensure their wide adoption in the industrial setting. While modern linear algebra packages are efficient for matrices with tens of thousands of dimensions, set-based image computations are limited to a few hundred. We propose to decompose reach set computations such that set operations are performed in low dimensions, while matrix operations like exponentiation are carried out in the full dimension. Our method is applicable both in dense- and discrete-time settings. For a set of standard benchmarks, it shows a speed-up of up to two orders of magnitude compared to the respective state-of-the art tools, with only modest losses in accuracy. For the dense-time case, we show an experiment with more than 10.000 variables, roughly two orders of magnitude higher than possible with previous approaches

Geometric Rounding and Feature Separation in Meshes

Geometric rounding of a mesh is the task of approximating its vertex coordinates by floating point numbers while preserving mesh structure. Geometric rounding allows algorithms of computational geometry to interface with numerical algorithms. We present a practical geometric rounding algorithm for 3D triangle meshes that preserves the topology of the mesh. The basis of the algorithm is a novel strategy: 1) modify the mesh to achieve a feature separation that prevents topology changes when the coordinates change by the rounding unit; and 2) round each vertex coordinate to the closest floating point number. Feature separation is also useful on its own, for example for satisfying minimum separation rules in CAD models. We demonstrate a robust, accurate implementation