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The solving of tridiagonal systems is one of the most computationally expensive parts in many applications, so that multiple studies have explored the use of NVIDIA GPUs to accelerate such computation. However, these studies have mainly focused on using parallel algorithms to compute such systems, which can efficiently exploit the shared memory and are able to saturate the GPUs capacity with a low number of systems, presenting a poor scalability when dealing with a relatively high number of systems. The gtsvStridedBatch routine in the cuSPARSE NVIDIA package is one of these examples, which is used as reference in this article. We propose a new implementation (cuThomasBatch) based on the Thomas algorithm. Unlike other algorithms, the Thomas algorithm is sequential, and so a coarse-grained approach is implemented where one CUDA thread solves a complete tridiagonal system instead of one CUDA block as in gtsvStridedBatch. To achieve a good scalability using this approach, it is necessary to carry out a transformation in the way that the inputs are stored in memory to exploit coalescence (contiguous threads access to contiguous memory locations). Different variants regarding the transformation of the data are explored in detail. We also explore some variants for the case of variable batch, when the size of the systems of the batch have different size (cuThomasVBatch). The results given in this study prove that the implementations carried out in this work are able to beat the reference code, being up to 5× (in double precision) and 6× (in single precision) faster using the latest NVIDIA GPU architecture, the Pascal P100.

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The simulation of the behavior of the Human Brain is one of the most important challenges in computing today. The main problem consists of finding efficient ways to manipulate and compute the huge volume of data that this kind of simulations need, using the current technology. In this sense, this work is focused on one of the main steps of such simulation, which consists of computing the Voltage on neurons’ morphology. This is carried out using the Hines Algorithm and, although this algorithm is the optimum method in terms of number of operations, it is in need of non-trivial modifications to be efficiently parallelized on GPUs. We proposed several optimizations to accelerate this algorithm on GPU-based architectures, exploring the limitations of both, method and architecture, to be able to solve efficiently a high number of Hines systems (neurons). Each of the optimizations are deeply analyzed and described. Two different approaches are studied, one for mono-morphology simulations (batch of neurons with the same shape) and one for multi-morphology simulations (batch of neurons where every neuron has a different shape). In mono-morphology simulations we obtain a good performance using just a single kernel to compute all the neurons. However this turns out to be inefficient on multi-morphology simulations. Unlike the previous scenario, in multi-morphology simulations a much more complex implementation is necessary to obtain a good performance. In this case, we must execute more than one single GPU kernel. In every execution (kernel call) one specific part of the batch of the neurons is solved. These parts can be seen as multiple and independent tridiagonal systems. Although the present paper is focused on the simulation of the behavior of the Human Brain, some of these techniques, in particular those related to the solving of tridiagonal systems, can be also used for multiple oil and gas simulations. Our studies have proven that the optimizations proposed in the present work can achieve high performance on those computations with a high number of neurons, being our GPU implementations about 4× and 8× faster than the OpenMP multicore implementation (16 cores), using one and two NVIDIA K80 GPUs respectively. Also, it is important to highlight that these optimizations can continue scaling, even when dealing with a very high number of neurons.

Many scientific applications are in need to solve a high number of small-size independent problems. However, these individual problems do not provide enough parallelism to efficiently exploit the current parallel architectures, and then, these must be computed as a batch in order to saturate the hardware. Today, vendors such as Intel and NVIDIA are developing their own suite of batch routines. Although most of the works focus on batches of fixed size, in real applications we can not assume a uniform size for all set of problems, so it is necessary the study of new efficient approaches to deal with batches of variable size, which is more challenging compared to batches of fixed size. We explore and analyze different strategies based on parallel for, task and taskloop OpenMP pragmas. These strategies are straightforward from programmer's point of view, however they present a very different impact on performance. We also analyze a new prototype provided by Intel (MKL), which deals with batch operations (cblas dgemm batch). Basically, these techniques attempt to distribute subgroups composed by the same number of problems without considering the differences among them. For this reason that we propose a new approach called grouping. It basically groups a set of problems until filling a limit in terms of memory occupancy or number of operations. In this way, groups composed by different number of problems are distributed on cores, achieving a more balanced distribution in terms of computational cost. However, these strategies must be tuned to achieve a good performance, and so we also explore and evaluate some off-line auto-tuning strategies on the proposed schedulers.