An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks

We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of points. First, the amount of purely symbolic operations is significantly reduced, that is, only resultant computation and square-free factorization is still needed. Second, our algorithm neither assumes generic position of the input system nor demands for any change of the coordinate system. The latter is due to a novel inclusion predicate to certify that a certain region is isolating for a solution. Our implementation exploits graphics hardware to expedite the resultant computation. Furthermore, we integrate a number of filtering techniques to improve the overall performance. Efficiency of the proposed method is proven by a comparison of our implementation with two state-of-the-art implementations, that is, LPG and Maple's isolate. For a series of challenging benchmark instances, experiments show that our implementation outperforms both contestants.Comment: 16 pages with appendix, 1 figure, submitted to ALENEX 201

A Review on Software Architectures for Heterogeneous Platforms

The increasing demands for computing performance have been a reality regardless of the requirements for smaller and more energy efficient devices. Throughout the years, the strategy adopted by industry was to increase the robustness of a single processor by increasing its clock frequency and mounting more transistors so more calculations could be executed. However, it is known that the physical limits of such processors are being reached, and one way to fulfill such increasing computing demands has been to adopt a strategy based on heterogeneous computing, i.e., using a heterogeneous platform containing more than one type of processor. This way, different types of tasks can be executed by processors that are specialized in them. Heterogeneous computing, however, poses a number of challenges to software engineering, especially in the architecture and deployment phases. In this paper, we conduct an empirical study that aims at discovering the state-of-the-art in software architecture for heterogeneous computing, with focus on deployment. We conduct a systematic mapping study that retrieved 28 studies, which were critically assessed to obtain an overview of the research field. We identified gaps and trends that can be used by both researchers and practitioners as guides to further investigate the topic

Fast Reliable Ray-tracing of Procedurally Defined Implicit Surfaces Using Revised Affine Arithmetic

Fast and reliable rendering of implicit surfaces is an important area in the field of implicit modelling. Direct rendering, namely ray-tracing, is shown to be a suitable technique for obtaining good-quality visualisations of implicit surfaces. We present a technique for reliable ray-tracing of arbitrary procedurally defined implicit surfaces by using a modification of Affine Arithmetic called Revised Affine Arithmetic. A wide range of procedurally defined implicit objects can be rendered using this technique including polynomial surfaces, constructive solids, pseudo-random objects, procedurally defined microstructures, and others. We compare our technique with other reliable techniques based on Interval and Affine Arithmetic to show that our technique provides the fastest, while still reliable, ray-surface intersections and ray-tracing. We also suggest possible modifications for the GPU implementation of this technique for real-time rendering of relatively simple implicit models and for near real-time for complex implicit models

Experiences with porting and modelling wavefront algorithms on many-core architectures

We are currently investigating the viability of many-core architectures for the acceleration of wavefront applications and this report focuses on graphics processing units (GPUs) in particular. To this end, we have implemented NASA’s LU benchmark – a real world production-grade application – on GPUs employing NVIDIA’s Compute Unified Device Architecture (CUDA). This GPU implementation of the benchmark has been used to investigate the performance of a selection of GPUs, ranging from workstation-grade commodity GPUs to the HPC "Tesla” and "Fermi” GPUs. We have also compared the performance of the GPU solution at scale to that of traditional high perfor- mance computing (HPC) clusters based on a range of multi- core CPUs from a number of major vendors, including Intel (Nehalem), AMD (Opteron) and IBM (PowerPC). In previous work we have developed a predictive “plug-and-play” performance model of this class of application running on such clusters, in which CPUs communicate via the Message Passing Interface (MPI). By extending this model to also capture the performance behaviour of GPUs, we are able to: (1) comment on the effects that architectural changes will have on the performance of single-GPU solutions, and (2) make projections regarding the performance of multi-GPU solutions at larger scale

On the acceleration of wavefront applications using distributed many-core architectures

In this paper we investigate the use of distributed graphics processing unit (GPU)-based architectures to accelerate pipelined wavefront applications—a ubiquitous class of parallel algorithms used for the solution of a number of scientific and engineering applications. Specifically, we employ a recently developed port of the LU solver (from the NAS Parallel Benchmark suite) to investigate the performance of these algorithms on high-performance computing solutions from NVIDIA (Tesla C1060 and C2050) as well as on traditional clusters (AMD/InfiniBand and IBM BlueGene/P). Benchmark results are presented for problem classes A to C and a recently developed performance model is used to provide projections for problem classes D and E, the latter of which represents a billion-cell problem. Our results demonstrate that while the theoretical performance of GPU solutions will far exceed those of many traditional technologies, the sustained application performance is currently comparable for scientific wavefront applications. Finally, a breakdown of the GPU solution is conducted, exposing PCIe overheads and decomposition constraints. A new k-blocking strategy is proposed to improve the future performance of this class of algorithm on GPU-based architectures

Efficient transfer entropy analysis of non-stationary neural time series

Information theory allows us to investigate information processing in neural systems in terms of information transfer, storage and modification. Especially the measure of information transfer, transfer entropy, has seen a dramatic surge of interest in neuroscience. Estimating transfer entropy from two processes requires the observation of multiple realizations of these processes to estimate associated probability density functions. To obtain these observations, available estimators assume stationarity of processes to allow pooling of observations over time. This assumption however, is a major obstacle to the application of these estimators in neuroscience as observed processes are often non-stationary. As a solution, Gomez-Herrero and colleagues theoretically showed that the stationarity assumption may be avoided by estimating transfer entropy from an ensemble of realizations. Such an ensemble is often readily available in neuroscience experiments in the form of experimental trials. Thus, in this work we combine the ensemble method with a recently proposed transfer entropy estimator to make transfer entropy estimation applicable to non-stationary time series. We present an efficient implementation of the approach that deals with the increased computational demand of the ensemble method's practical application. In particular, we use a massively parallel implementation for a graphics processing unit to handle the computationally most heavy aspects of the ensemble method. We test the performance and robustness of our implementation on data from simulated stochastic processes and demonstrate the method's applicability to magnetoencephalographic data. While we mainly evaluate the proposed method for neuroscientific data, we expect it to be applicable in a variety of fields that are concerned with the analysis of information transfer in complex biological, social, and artificial systems.Comment: 27 pages, 7 figures, submitted to PLOS ON