A new method for generating initial solutions of capacitated vehicle routing problems

In vehicle routing problems, the initial solutions of the routes are important for improving the quality and solution time of the algorithm. For a better route construction algorithm, the obtained initial solutions must be basic, fast, and flexible with reasonable accuracy. In this study, initial solutions improvement for CVRP is introduced based on a method that is introduced in the literature. Using a different formula for addressing the gravitational forces, a new method is introduced and compared with the previous physics inspired algorithm. By using the initial solutions of the proposed method and using them as RTR and SA initial routes, it is seen that better results are obtained when compared with various algorithms from the literature. Also, in order to fairly compare the algorithms executed on different machines, a new comparison scale for the solution quality of vehicle routing problems is proposed that depends on the solution time and the deviation from the best known solution. The obtained initial solutions are then input to Record-to-Record and Simulated Annealing algorithms to obtain final solutions. Various test instances and CVRP solutions from the literature are used for comparison. The comparisons with the proposed method have shown promising results

Direct transcription of low-thrust trajectories with finite trajectory elements

This paper presents a novel approach to the design of Low-Thrust trajectories, based on a first order approximated analytical solution of Gauss planetary equations. This analytical solution is shown to have a better accuracy than a second-order explicit numerical integrator and at a lower computational cost. Hence, it can be employed for the fast propagation of perturbed Keplerian motion when moderate accuracy is required. The analytical solution was integrated in a direct transcription method based on a decomposition of the trajectory into direct finite perturbative elements (DFPET). DFPET were applied to the solution of two-point boundary transfer problems. Furthermore the paper presents an example of the use of DFPET for the solution of a multiobjective trajectory optimisation problem in which both the total ∆V and transfer time are minimized with respect to departure and arrival dates. Two transfer problems were used as test cases: a direct transfer from Earth to Mars and a spiral from a low Earth orbit to the International Space Station

GAMER: a GPU-Accelerated Adaptive Mesh Refinement Code for Astrophysics

We present the newly developed code, GAMER (GPU-accelerated Adaptive MEsh Refinement code), which has adopted a novel approach to improve the performance of adaptive mesh refinement (AMR) astrophysical simulations by a large factor with the use of the graphic processing unit (GPU). The AMR implementation is based on a hierarchy of grid patches with an oct-tree data structure. We adopt a three-dimensional relaxing TVD scheme for the hydrodynamic solver, and a multi-level relaxation scheme for the Poisson solver. Both solvers have been implemented in GPU, by which hundreds of patches can be advanced in parallel. The computational overhead associated with the data transfer between CPU and GPU is carefully reduced by utilizing the capability of asynchronous memory copies in GPU, and the computing time of the ghost-zone values for each patch is made to diminish by overlapping it with the GPU computations. We demonstrate the accuracy of the code by performing several standard test problems in astrophysics. GAMER is a parallel code that can be run in a multi-GPU cluster system. We measure the performance of the code by performing purely-baryonic cosmological simulations in different hardware implementations, in which detailed timing analyses provide comparison between the computations with and without GPU(s) acceleration. Maximum speed-up factors of 12.19 and 10.47 are demonstrated using 1 GPU with 4096^3 effective resolution and 16 GPUs with 8192^3 effective resolution, respectively.Comment: 60 pages, 22 figures, 3 tables. More accuracy tests are included. Accepted for publication in ApJ

Systolic and Hyper-Systolic Algorithms for the Gravitational N-Body Problem, with an Application to Brownian Motion

A systolic algorithm rhythmically computes and passes data through a network of processors. We investigate the performance of systolic algorithms for implementing the gravitational N-body problem on distributed-memory computers. Systolic algorithms minimize memory requirements by distributing the particles between processors. We show that the performance of systolic routines can be greatly enhanced by the use of non-blocking communication, which allows particle coordinates to be communicated at the same time that force calculations are being carried out. Hyper-systolic algorithms reduce the communication complexity at the expense of increased memory demands. As an example of an application requiring large N, we use the systolic algorithm to carry out direct-summation simulations using 10^6 particles of the Brownian motion of the supermassive black hole at the center of the Milky Way galaxy. We predict a 3D random velocity of 0.4 km/s for the black hole.Comment: 33 pages, 10 postscript figure

NBODY6++GPU: Ready for the gravitational million-body problem

Accurate direct $N$-body simulations help to obtain detailed information about the dynamical evolution of star clusters. They also enable comparisons with analytical models and Fokker-Planck or Monte-Carlo methods. NBODY6 is a well-known direct $N$-body code for star clusters, and NBODY6++ is the extended version designed for large particle number simulations by supercomputers. We present NBODY6++GPU, an optimized version of NBODY6++ with hybrid parallelization methods (MPI, GPU, OpenMP, and AVX/SSE) to accelerate large direct $N$-body simulations, and in particular to solve the million-body problem. We discuss the new features of the NBODY6++GPU code, benchmarks, as well as the first results from a simulation of a realistic globular cluster initially containing a million particles. For million-body simulations, NBODY6++GPU is $400-2000$ times faster than NBODY6 with 320 CPU cores and 32 NVIDIA K20X GPUs. With this computing cluster specification, the simulations of million-body globular clusters including $5\%$ primordial binaries require about an hour per half-mass crossing time.Comment: 13 pages, 9 figures, 3 table