SCIENTIFIC COMPUTING ON COMMODITY GRAPHICS HARDWARE

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1 SCIENTIFIC COMPUTING ON COMMODITY GRAPHICS HARDWARE RUIGANG YANG Department of Computer Science, University of Kentucky, One Quality Street, Suite 854, Lexington, KY 40507, USA Driven by the need for interactive entertainment, modern PCs are equipped with specialized graphics processors (GPUs) for creation and display of images. These GPUs have become increasingly programmable, to the point that they now are capable of efficiently executing a significant number of computational kernels from non-graphical applications. In this introductory paper we first present a highlevel overview of modern graphics hardware s architecture, then introduce several applications in scientific computing that can be efficiently accelerated by GPUs. Finally we list programming tools available for application development on GPUs. 1. Introduction As the mass-market emphasis in computing has shifted from word processing and spreadsheets to interactive entertainment, computer hardware has evolved to better support these new applications. Most of the performancelimiting processing today involves creation and display of images; thus, a new entity has appeared within most computer systems. Between the system s general-purpose processor and the video frame buffer, there is now a specialized Graphic Processing Unit (GPU). Early GPUs were not really processors, but hardwired pipelines for each of the most common rendering tasks. As more complex 3D-transformations have become common in a wide range of applications, GPUs have become increasingly programmable, to the point that they now are capable of efficiently executing a significant number of computational kernels from non-graphical applications. A GPU is simpler and more efficient than a conventional PC processor (CPU) because a GPU only needs to perform a relatively simple set of array processing operations (but at a very high speed). Many problems in scientific computing, such as physically-based simulation, information retrieval, and data mining, can boil down to relatively simple matrix operations. This characteristic makes these problems ideal candidates for GPU acceleration. In this introductory paper we first present a high-level overview of modern 1

2 2 graphics hardware s architecture and its phenomenal development in recent years. Then we introduce a large array of non-graphical computational tasks, in particular, linear algebra operations, that have been successfully implemented on GPUs and obtained significant performance improvements. Finally we list programming tools available for application development on GPUs. Some of them are designed to allow programming GPUs with familiar C-like constructs and syntax, without worrying about the details of the hardware. They hold the promise of bringing the vast computational power in GPUs to the broad scientific computing community. 2. A Brief Overview of GPUs In this section, we will explain the basic architecture of GPUs and the potential advantages of using GPUs to solve scientific problems The Rendering Pipeline GPUs are dedicated processors designed specifically to handle the intense computational requirements of display graphics, i.e., rendering texts or images over 30 frames per second. As depicted in Figure 1, a modern GPU can be abstracted as a rendering pipeline for 3D computer graphics (2D graphics is just a special case) 20. Geom etric prim itives Vertex Processing Rasterization Fragm ent Processing Fram e buffer Figure 1. Rendering Pipeline The inputs to the pipeline are geometric primitives, i.e., points, lines, polygons; and the output is the framebuffer a two-dimensional array of pixels that will be displayed on screen. The first stage operates on geometric primitives described by vertices. In this vertex-processing stage vertices are transformed and lit, and primitives are clipped to a viewing volume in preparation for the next stage, rasterization. The rasterizer produces a series of framebuffer addresses and color values, each is called a fragment that represents a portion of a primitive that corresponds to a pixel in the framebuffer. Each fragment is fed to the next fragment processing stage before it finally alters the framebuffer. Operations in this stage include texture mapping, depth test, alpha blending, etc.

3 Recent Trend in GPUs Until a few years ago, commercial GPUs, such as the RealityEngine from SGI 2, implement in hardware a fixed rendering pipeline with configurable parameters. As a result their applications are restricted to graphical computations. Driven by the market demand for better realism, the current generation of commercial GPUs such as the NVIDIA GeForce FX 19 and the ATI Radeon added significant programmable functionalities in both the vertex and the fragment processing stage(stages with double-lines in Figure 1). They allow developers to write a sequence of instructions to modify the vertex or fragment output. These programs are directly executed on the GPUs to achieve comparable performance to fixed-function GPUs. In addition to programable functionalities in modern GPUs, their support for floating point output has been improving. GPUs on the market today support up to 32-bit floating point output. Such a precision is usable for many diverse applications other than computer graphics Radeon Spec int200 Benchmark P3-733Mhz Voodoo3 GeForce 256 P4-1.7Ghz Radeon 8500 P4-3.2Ghz CPU GPU Millions of Triangles per Second 1 Jul-98 Feb-99 Aug-99 Mar-00 Oct-00 Apr-01 Nov-01 May-02 Dec-02 Jun-03 Jan-04 Date Introduced Figure 2. A graph of performance increase over time for CPUs and GPUs. GPU performance has increased at a faster rate than CPUs. (Data courtesy of Anselmo Lastra). GPUs have also demonstrated a rapid improvement in performance during the past few years. In Figure 2, we plot the performance increase of both GPUs and commodity Central Processor Units (CPUs). Similar to the number of integer operations per second for CPUs, a typical benchmark to gauge a GPU s performance is the number of triangles it can process every second. We can see that GPUs have maintained a performance improvement rate of approximately 3X/year, which exceeds the performance improvement of CPUs at 1.6X/year. This is because CPUs are designed for low latency computations, while GPUs are optimized for high throughput of vertices and fragments 14 ). Low latency on memory-intensive applications typically requires large caches, 1

4 4 which use a large silicon area. Additional transistors are used to greater effect in GPU architectures because they are applied to additional functional units that increase throughput Applications of GPUs for General-Purpose Computation With the wide deployment of inexpensive yet powerful GPUs in the last several years, we have seen a surge of experimental research in using GPUs for tasks other than rendering. For example, Yang et. al. have experimented with using GPUs to solve computer visions problems 24,23 ; Holzschuch and Alonso to speed visibility queries 7 ; Hoff et. al. to compute generalized Voronoi Diagrams 8 and proximity information 9 ; and Lok to reconstruct an object s visual hull given live video from multiple cameras 15. Each of these applications obtained significant performance improvements by exploiting the speed and the inherent parallelism in modern graphics hardware. For the scope of this paper, we introduce several representative approaches to accelerate linear algebra operations on GPUs. Larsen and McAllister present a technique for large matrix-matrix multiplies using low cost graphics hardware 12. The method is an adaptation of the technique from parallel computing of distributing the computation over a logically cube-shaped lattice of processors and performing a portion of the computation at each processor. Graphics hardware is specialized in a manner that makes it well suited to this particular problem, giving faster results in some cases than using a general-purpose processor. A more complete and up-to-date implementation of dense matrix algebra is presented by Moravánszky 1. The paper of Bolz et al. shows two basic, broadly useful, computational kernels implemented on GPUs: a sparse matrix conjugate gradient solver, and a regular-grid multigrid solver 4. Performance analysis with realistic applications shows that a GPU-based implementation compares favorable over its CPU counterpart. A similar framework for implementation of linear algebra operators on GPUs is by Krüger and Westermann 11, which focuses on sparse and banded matrices. There are many other algorithms for scientific computing that have been implemented on GPUs, including FFT 17,level set 22,13, and various types of physically-based simulations 6,10,21. Interested readers are referred to for other general-purpose applications on GPUs. 4. GPU Programming Languages While many non-graphical applications on GPUs have obtained encouraging results by exploiting GPU s fast speed and high bandwidth, the development process is not trivial. Many of the existing applications are written using low level assemble languages that are directly executed on the GPU. There-

5 5 fore, novice developers are faced with a steep learning curve to master a thorough understanding of the graphics hardware and its programming interfaces, namely OpenGL 20 and DirectX 16. Fortunately, this is rapidly changing with several high-level languages available. The first is Cg a system for programming graphics hardware in a C-like language 18. It is, however, still a programming language geared towards rendering tasks and tightly coupled with graphics hardware. There are other high-level languages, such as Brook for GPUs and Sh, which allow programming GPUs with familiar constructs and syntax, without worrying about the details of the hardware. Brook extends C to include simple data-parallel constructs, enabling the use of the GPU as a streaming coprocessor. Sh is a metaprogramming language that offers the convenient syntax of C++ and takes the burden of register allocation and other low-level issues away from the programmer. While these languages are not fully mature yet, they are the most promising ones to allow non-graphics researchers or developers to tap into the vast computational power in GPUs. 5. Conclusion The versatile programmability and improved floating-point precisions now available in GPUs make them useful coprocessors for scientific computing. Many non-trivial computational kernels have been successfully implemented on GPUs to receive significant acceleration. As graphics hardware continues to evolve at a faster speed than CPUs and more user-friendly high-level programming languages are becoming available, we believe communities outside computer graphics can also benefit from the fast processing speed and high bandwidth that GPUs offer. We hope this introductory paper will encourage further thinking along this direction. Acknowledgments The author would like to thank Hank Dietz for providing some of the materials in this paper. This work is supported in part by fund from the office of research at the University of Kentucky and Kentucky Science & Engineering Foundation (RDE-005). References 1. Ádám Moravánszky. Dense Matrix Algebra on the GPU. In Shaderx2: Shader Programming Tips & Tricks With Directx 9. Wordware, K. Akeley. Realityengine graphics. In Proceedings of SIGGRAPH, ATI Technologies Inc. ATI Radeon 9800, Jeff Bolz, Ian Farmer, Eitan Grinspun, and Peter Schrder. Sparse Matrix Solvers on the GPU: Conjugate Gradients and Multigrid. ACM Transactions on Graphics (SIGGRAPH 2003), 22(3), 2003.

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