As an entry for the 2006 Gordon Bell price/performance prize, we report an astrophysical N-body simulation performed with reconfigurable computing systems and a hierarchical tree algorithm. Each of our systems consists of a host PC and a reconfigurable add-in card attached via high-speed interface, PCI Express or PCI-X. The reconfigurable add-in card houses one FPGA chip, into which we integrated 16 pipeline processors specialized for gravitational force calculation. Other operations, such as tree construction, tree traverse and time integration, are performed on the host PC. On our system, we performed a cosmological N-body simulation with 2.1 million particles, which sustained a performance of 15.39 Gflops averaged over 4.33 hours. The price of our system is $2,384 in total and, price per performance obtained is $158/Gflops, which is 44 times better than that obtained by GRAPE-5 special-purpose hardware, the 1999 Gordon Bell winner.

In this paper, we describe the performance characteristics of GRAPE-6, the sixth-generation special-purpose computer for gravitational many-body problems. GRAPE-6 consists of 2048 custom pipeline chips, each of which integrates six pipeline processors specialized for the calculation of gravitational interaction between particles. The GRAPE hardware performs the evaluation of the interaction. The frontend processors perform all other operations, such as the time integration of the orbits of particles, I/O, on-the-fly analysis etc. The theoretical peak speed of GRAPE-6 is 63.4 Tflops. We present the result of benchmark runs, and discuss the performance characteristics. We also present the measured performance for a few real scientific applications. The best performance so far achieved with real applications is 35.3 Tflops.

HMCS (Heterogeneous Multi-Computer System) is a new parallel processing platform combining massively parallel processors for continuum simulation and particle simulation to realize multi-scale computational physics simulations. We are constructing a prototype system of HMCS with a general purpose scientific parallel processor CP-PACS and a gravity calculation parallel processor GRAPE-6 connecting them via commodity-base parallel network.On the prototype of HMCS, a microscopic gravity calculation on GRAPE-6 and a macroscopic radiation hydrodynamic calculation on CP-PACS are performed simultaneously to realize very detailed simulation on computational astrophysics. Both systems are connected via parallel network controlling system named PIO (Parallel I/O System).In this paper, we report the overall concept, design and implementation of HMCS, and our application result for radiative transfer smoothed particle hydrodynamics with self gravity for analysis of galaxy generation.

As an entry for the 2002 Gordon Bell performance prize, we report the performance achieved on the GRAPE-6 system for a simulation of the early evolution of the protoplanet-planetesimal system of the Uranus-Neptune region. GRAPE-6 is a special-purpose computer for astrophysical N-body calculations. The present configuration has 2048 custom pipeline chips, each containing six pipeline processors for the calculation of gravitational interactions between particles. Its theoretical peak performance is 63.4 Tflops. The actual performance obtained was 29.5 Tflops, for a simulation of the early evolution of outer Solar system with 1.8 million planetesimals and two massive protoplanets.

As an entry for the 2000 Gordon Bell performance prize, we report the performance achieved on a prototype GRAPE-6 system. GRAPE-6 is a special-purpose computer for as-trophysical N-body calculations. The present configuration has 96 custom pipeline processors, each containing six pipeline processors for the calculation of gravitational interactions between particles. Its theoretical peak performance is 2.889 Tflops. The complete GRAPE-6 system will consist of 3072 pipeline chips and will achieve a peak speed of 100 Tflops. The actual performance obtained on the present 96-chip system was 1.349 Tflops, for a simulation of massive black holes embedded in the core of a galaxy with 786,432 stars. For a short benchmark run with 1,400,000 particles, the average speed was 1.640 Tflops.

As an entry for the 1999 Gordon Bell price/performance prize, we report an astrophysical N -body simulation performed with a treecode on GRAPE-5 (Gravity Pipe 5) system, a special-purpose computer for astrophysical N -body simulations.The GRAPE-5 system has 32 pipeline processors specialized for the gravitational force calculation.Other operations, such as tree construction, tree traverse and time integration, are performed on a general purpose workstation.The total cost for the GRAPE-5 system is 40,900 dollars.We performed a cosmological N -body simulation with 2.1 million particles, which sustained a performance of 5.92 Gflops averaged over 8.37 hours.The price per performance obtained is 7.0 dollars per Mflops.

We have completed PROGRAPE-1, a programmable special-purpose computer for many-body simulations using FPGA (Field Programmable Gate Array). It has pipelines specialized for computations of interactions between particles. The peak performance of calculating gravitational/Coulomb force results in 2.4 Gflops.

We report on resent N-body simulations of galaxy formation performed on the GRAPE-4 (GRAvity PipE 4) system, a special-purpose computer for astrophysical N-body simulations. We review the astrophysical motivation, the algorithm, the actual performance, and the price per performance. The performance obtained is 332 Gflops averaged over 185 hours for a simulation of a galaxy formation with 786,400 particles. The price per performance obtained is 4,600 dollars per Gflops. The configuration used for the simulation consists of 1,269 pipeline processors and has a peak speed of 663 Gflops.