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Register a new userVectorization and Parallelization of the Adaptive Mesh Refinement N-body Code
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Hideki Yahagi
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In this paper, we describe our vectorized and parallelized adaptive mesh refinement (AMR) N-body code with shared time steps, and report its performance on a Fujitsu VPP5000 vector-parallel supercomputer. Our AMR N-body code puts hierarchical meshes recursively where higher resolution is required and the time step of all particles are the same. The parts which are the most difficult to vectorize are loops that access the mesh data and particle data. We vectorized such parts by changing the loop structure, so that the innermost loop steps through the cells instead of the particles in each cell, in other words, by changing the loop order from the depth-first order to the breadth-first order. Mass assignment is also vectorizable using this loop order exchange and splitting the loop into $2^{N_{dim}}$ loops, if the cloud-in-cell scheme is adopted. Here, $N_{dim}$ is the number of dimension. These vectorization schemes which eliminate the unvectorized loops are applicable to parallelization of loops for shared-memory multiprocessors. We also parallelized our code for distributed memory machines. The important part of parallelization is data decomposition. We sorted the hierarchical mesh data by the Morton order, or the recursive N-shaped order, level by level and split and allocated the mesh data to the processors. Particles are allocated to the processor to which the finest refined cells including the particles are also assigned. Our timing analysis using the $Lambda$-dominated cold dark matter simulations shows that our parallel code speeds up almost ideally up to 32 processors, the largest number of processors in our test.
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We have developed a simulation code with the techniques which enhance both spatial and time resolution of the PM method for which the spatial resolution is restricted by the spacing of structured mesh. The adaptive mesh refinement (AMR) technique subdivides the cells which satisfy the refinement criterion recursively. The hierarchical meshes are maintained by the special data structure and are modified in accordance with the change of particle distribution. In general, as the resolution of the simulation increases, its time step must be shortened and more computational time is required to complete the simulation. Since the AMR enhances the spatial resolution locally, we reduce the time step locally also, instead of shortening it globally. For this purpose we used a technique of hierarchical time steps (HTS) which changes the time step, from particle to particle, depending on the size of the cell in which particles reside. Some test calculations show that our implementation of AMR and HTS is successful. We have performed cosmological simulation runs based on our code and found that many of halo objects have density profiles which are well fitted to the universal profile proposed by Navarro, Frenk, & White (1996) over the entire range of their radius.
This paper describes the open-source code Enzo, which uses block-structured adaptive mesh refinement to provide high spatial and temporal resolution for modeling astrophysical fluid flows. The code is Cartesian, can be run in 1, 2, and 3 dimensions, and supports a wide variety of physics including hydrodynamics, ideal and non-ideal magnetohydrodynamics, N-body dynamics (and, more broadly, self-gravity of fluids and particles), primordial gas chemistry, optically-thin radiative cooling of primordial and metal-enriched plasmas (as well as some optically-thick cooling models), radiation transport, cosmological expansion, and models for star formation and feedback in a cosmological context. In addition to explaining the algorithms implemented, we present solutions for a wide range of test problems, demonstrate the codes parallel performance, and discuss the Enzo collaborations code development methodology.
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.
We present GAMER-2, a GPU-accelerated adaptive mesh refinement (AMR) code for astrophysics. It provides a rich set of features, including adaptive time-stepping, several hydrodynamic schemes, magnetohydrodynamics, self-gravity, particles, star formation, chemistry and radiative processes with GRACKLE, data analysis with yt, and memory pool for efficient object allocation. GAMER-2 is fully bitwise reproducible. For the performance optimization, it adopts hybrid OpenMP/MPI/GPU parallelization and utilizes overlapping CPU computation, GPU computation, and CPU-GPU communication. Load balancing is achieved using a Hilbert space-filling curve on a level-by-level basis without the need to duplicate the entire AMR hierarchy on each MPI process. To provide convincing demonstrations of the accuracy and performance of GAMER-2, we directly compare with Enzo on isolated disk galaxy simulations and with FLASH on galaxy cluster merger simulations. We show that the physical results obtained by different codes are in very good agreement, and GAMER-2 outperforms Enzo and FLASH by nearly one and two orders of magnitude, respectively, on the Blue Waters supercomputers using $1-256$ nodes. More importantly, GAMER-2 exhibits similar or even better parallel scalability compared to the other two codes. We also demonstrate good weak and strong scaling using up to 4096 GPUs and 65,536 CPU cores, and achieve a uniform resolution as high as $10{,}240^3$ cells. Furthermore, GAMER-2 can be adopted as an AMR+GPUs framework and has been extensively used for the wave dark matter ($psi$DM) simulations. GAMER-2 is open source (available at https://github.com/gamer-project/gamer) and new contributions are welcome.
Large-scale finite element simulations of complex physical systems governed by partial differential equations crucially depend on adaptive mesh refinement (AMR) to allocate computational budget to regions where higher resolution is required. Existing scalable AMR methods make heuristic refinement decisions based on instantaneous error estimation and thus do not aim for long-term optimality over an entire simulation. We propose a novel formulation of AMR as a Markov decision process and apply deep reinforcement learning (RL) to train refinement policies directly from simulation. AMR poses a new problem for RL in that both the state dimension and available action set changes at every step, which we solve by proposing new policy architectures with differing generality and inductive bias. The model sizes of these policy architectures are independent of the mesh size and hence scale to arbitrarily large and complex simulations. We demonstrate in comprehensive experiments on static function estimation and the advection of different fields that RL policies can be competitive with a widely-used error estimator and generalize to larger, more complex, and unseen test problems.
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