Optimizing the computation of a parallel 3D finite difference algorithm for graphics processing units

Abstract

This paper explores the possibilities of using a graphics processing unit for complex 3D finite difference computation via MUSTA-FORCE and WENO algorithms. We propose a novel algorithm based on the new properties of CUDA surface memory optimized for 2D spatial locality and compare it with 3D stencil computations carried out via shared memory, which is currently considered to be the best approach. A case study was performed for the extensive generation of a time series of 3D grids of arbitrary size used in the computation of collisions between heavy nuclei in terms of relativistic hydrodynamics. It proved that implementation based on surface memory is as much as 23% faster than an equivalent implementation using shared memory. 4 4 1592 J. PORTER-SOBIERAJ ET AL.heavy ion collisions create a highly dynamic system, hydrodynamic models can be used to describe an intermediate stage in the collisions, where the assumption is valid [1].Hydrodynamic simulations of heavy ion collisions include three major components:1. Initial conditions for the simulations; 2. Hydrodynamic evolution of the system; 3. Freeze-out, where a hydrodynamic picture is no longer valid and particles have to be treated as individual entities.At the beginning, nuclear matter is in the pre-equilibrium stage, and a hydro approach can be only used when the system reaches the local thermal equilibrium. For the initial conditions, another model has to be used. The Glauber model [2] is often used for this purpose, although there are other choices available. Freeze-out is another stage where a hydro-like description is not valid because the distance between particles no longer allows for treating them as a continuous medium. For this stage, a transport theory is more appropriate. Freeze-out usually occurs as a sudden freeze-out, where a particle decouples from a bulk when the energy density is below a certain threshold. Then, after the hydro stage, further propagation of particles is simulated using a transport model, UrQMD [3] for instance.To exploit the full potential of realistic hydrodynamics, fully (3 + 1)-dimensional simulations (the three spatial dimensions + time) are necessary to study the system's evolution without any assumptions regarding its symmetries. It is a compelling idea; however, such simulations are extremely expensive in terms of computing power. Initial conditions for the hydro models are also important. The fluctuations in the pre-equilibrium stage phase might have an impact on the dynamics of the nuclear medium; therefore, event-by-event studies (where an event is a single collision between heavy ions) with fluctuating initial conditions and with a large amount of statistics are of particular interest and require a fast and efficient computer code. Another important topic, from the physics perspective, is studies of jets. Jets are narrow beams of particles with a high momentum, and their propagation through the hot nuclear medium can provide information about the fundamental properties of the Quark-Gluon Plasma (transport coefficien...

Accelerating simulations of cardiac electrical dynamics through a multi‐GPU platform and an optimized data structure

“…[7][8][9]11,19,21 This happens because, although using row-major order enables some coalesced memory accesses for block warps, all blocks in the grid will compete for space in the cache. However, as each thread within a block will need its own data plus its six neighbors, this may lead to data access redundancy.…”

“…This means that the chance of a cache miss increases as the grid size grows. [7][8][9]11,21 In CUDA, threads of the same block can communicate with each other through shared memory, which is a low-latency memory, but limited in total storage. According to Equation (5), at the time thread k asks for data in position (x + 1, y, z), thread k + 1 had requested this same data already, so it should be in cache memory.…”

“…They evaluated not only the performance of adapted CPU implementations for GPUs, [33][34][35] but also different parallel strategies 11,20,[36][37][38][39] and numerical methods. This means they can process up to 9.96 × 10 10 domain points (cells) per second. The authors compared spiral waves simulated with their WebGL approach with spiral waves from experimental data.…”

“…[7][8][9] With this in mind, we propose in this work a novel and smart data structure suitable for problems that access data from nearest neighbors, such as the discretization of the Laplacian operator, and we emphasize the GPU memory architecture as well as the CPU features. Discretization of the Laplacian requires some data memory accesses in locations that are physically close in space.…”

“…As a result, 3D domains, eg, require several clock cycles to load all the necessary data, thereby increasing the latency of memory access. [7][8][9] With this in mind, we propose in this work a novel and smart data structure suitable for problems that access data from nearest neighbors, such as the discretization of the Laplacian operator, and we emphasize the GPU memory architecture as well as the CPU features. Although our solution may be generalized for several problems that use Laplacian discretizations, we focus this work on cardiac electrical simulations.…”