On reserve and double covering problems for the sets with non-Euclidean metrics

The article is devoted to Circle covering problem for a bounded set in a two-dimensional metric space with a given amount of circles. Here we focus on a more complex problem of constructing reserve and multiply coverings. Besides that, we consider the case where covering set is a multiply-connected domain. The numerical algorithms based on fundamental physical principles, established by Fermat and Huygens, are suggested and implemented. This allows us to solve the problems for the cases of non-convex sets and non-Euclidean metrics. Preliminary results of numerical experiments are presented and discussed. Calculations show the applicability of the proposed approach.

A Parallel Interval Arithmetic-based Reliable Computing Method on a GPU

Video cards have now outgrown their purpose of being only a simple tool for graphic display. With their high speed video memories, lots of maths units and parallelism, they can be very powerful accessories for general purpose computing tasks. Our selected platform for testing is the CUDA (Compute Unified Device Architecture), which offers us direct access to the virtual instruction set of the video card, and we are able to run our computations on dedicated computing kernels. The CUDA development kit comes with a useful toolbox and a wide range of GPU-based function libraries. In this parallel environment, we implemented a reliable method based on the Branch-and-Bound algorithm. This algorithm will give us the opportunity to use node level (also called low-level or type 1) parallelization, since we do not modify the searching trajectories; nor do we modify the dimensions of the Branch-and-Bound tree [5]. For testing, we chose the circle covering problem. We then scaled the problem up to three dimensions, and ran tests with sphere covering problems as well.