FFTW is an implementation of the discrete Fourier transform (DFT) that adapts to the hardware in order to maximize performance. This paper shows that such an approach can yield an implementation that is competitive with hand-optimized libraries, and describes the software structure that makes our current FFTW3 version flexible and adaptive. We further discuss a new algorithm for real-data DFTs of prime size, a new way of implementing DFTs by means of machine-specific single-instruction, multiple-data (SIMD) instructions, and how a special-purpose compiler can derive optimized implementations of the discrete cosine and sine transforms automatically from a DFT algorithm.

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The Design and Implementation of FFTW3

Image processing pipelines combine the challenges of stencil computations and stream programs. They are composed of large graphs of different stencil stages, as well as complex reductions, and stages with global or data-dependent access patterns. Because of their complex structure, the performance difference between a naive implementation of a pipeline and an optimized one is often an order of magnitude. Efficient implementations require optimization of both parallelism and locality, but due to the nature of stencils, there is a fundamental tension between parallelism, locality, and introducing redundant recomputation of shared values.We present a systematic model of the tradeoff space fundamental to stencil pipelines, a schedule representation which describes concrete points in this space for each stage in an image processing pipeline, and an optimizing compiler for the Halide image processing language that synthesizes high performance implementations from a Halide algorithm and a schedule. Combining this compiler with stochastic search over the space of schedules enables terse, composable programs to achieve state-of-the-art performance on a wide range of real image processing pipelines, and across different hardware architectures, including multicores with SIMD, and heterogeneous CPU+GPU execution. From simple Halide programs written in a few hours, we demonstrate performance up to 5x faster than hand-tuned C, intrinsics, and CUDA implementations optimized by experts over weeks or months, for image processing applications beyond the reach of past automatic compilers.

Halide: a language and compiler for optimizing parallelism, locality, and recomputation in image processing pipelines

Recent advances in computing have led to an explosion in the amount of data being generated. Processing the ever-growing data in a timely manner has made throughput computing an important aspect for emerging applications. Our analysis of a set of important throughput computing kernels shows that there is an ample amount of parallelism in these kernels which makes them suitable for today's multi-core CPUs and GPUs. In the past few years there have been many studies claiming GPUs deliver substantial speedups (between 10X and 1000X) over multi-core CPUs on these kernels. To understand where such large performance difference comes from, we perform a rigorous performance analysis and find that after applying optimizations appropriate for both CPUs and GPUs the performance gap between an Nvidia GTX280 processor and the Intel Core i7-960 processor narrows to only 2.5x on average. In this paper, we discuss optimization techniques for both CPU and GPU, analyze what architecture features contributed to performance differences between the two architectures, and recommend a set of architectural features which provide significant improvement in architectural efficiency for throughput kernels.

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Debunking the 100X GPU vs. CPU myth: an evaluation of throughput computing on CPU and GPU

Heterogeneous multiprocessors are increasingly important in the multi-core era due to their potential for high performance and energy efficiency. In order for software to fully realize this potential, the step that maps computations to processing elements must be as automated as possible. However, the state-of-the-art approach is to rely on the programmer to specify this mapping manually and statically. This approach is not only labor intensive but also not adaptable to changes in runtime environments like problem sizes and hardware/software configurations. In this study, we propose adaptive mapping, a fully automatic technique to map computations to processing elements on a CPU+GPU machine. We have implemented it in our experimental heterogeneous programming system called Qilin. Our results show that, by judiciously distributing works over the CPU and GPU, automatic adaptive mapping achieves a 25% reduction in execution time and a 20% reduction in energy consumption than static mappings on average for a set of important computation benchmarks. We also demonstrate that our technique is able to adapt to changes in the input problem size and system configuration.

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Qilin: exploiting parallelism on heterogeneous multiprocessors with adaptive mapping

Determining the best set of optimizations to apply to a kernel to be executed on the graphics processing unit (GPU) is a challenging problem. There are large sets of possible optimization configurations that can be applied, and many applications have multiple kernels. Each kernel may require a specific configuration to achieve the best performance, and moving an application to new hardware often requires a new optimization configuration for each kernel. In this work, we apply optimizations to GPU code using HMPP, a high-level directive-based language and source-to-source compiler that can generate CUDA / OpenCL code. However, programming with high-level languages may mean a loss of performance compared to using low-level languages. Our work shows that it is possible to improve the performance of a high-level language by using auto-tuning. We perform auto-tuning on a large optimization space on GPU kernels, focusing on loop permutation, loop unrolling, tiling, and specifying which loop(s) to parallelize, and show results on convolution kernels, codes in the PolyBench suite, and an implementation of belief propagation for stereo vision. The results show that our auto-tuned HMPP-generated implementations are significantly faster than the default HMPP implementation and can meet or exceed the performance of manually coded CUDA / OpenCL implementations.

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Auto-tuning a high-level language targeted to GPU codes

Fast changing, increasingly complex, and diverse computing platforms pose central problems in scientific computing: How to achieve, with reasonable effort, portable optimal performance? We present SPIRAL, which considers this problem for the performance-critical domain of linear digital signal processing (DSP) transforms. For a specified transform, SPIRAL automatically generates high-performance code that is tuned to the given platform. SPIRAL formulates the tuning as an optimization problem and exploits the domain-specific mathematical structure of transform algorithms to implement a feedback-driven optimizer. Similar to a human expert, for a specified transform, SPIRAL "intelligently" generates and explores algorithmic and implementation choices to find the best match to the computer's microarchitecture. The "intelligence" is provided by search and learning techniques that exploit the structure of the algorithm and implementation space to guide the exploration and optimization. SPIRAL generates high-performance code for a broad set of DSP transforms, including the discrete Fourier transform, other trigonometric transforms, filter transforms, and discrete wavelet transforms. Experimental results show that the code generated by SPIRAL competes with, and sometimes outperforms, the best available human tuned transform library code.

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SPIRAL: Code Generation for DSP Transforms

We investigate performance tradeoffs for a class of filtered backprojection (FBP) image reconstruction algorithms. The recently developed fast hierarchical backprojection asymptotically achieves the same O(N/sup 2/ log N) cost as Fourier-based methods while retaining many advantages of the FBP technique. In this paper, we provide a detailed cost and performance analysis of the algorithm on a general purpose platform. Based on carefully tuned implementations of both the direct and the hierarchical backprojection, we explore the tradeoffs between distortion and runtime by varying several algorithm and implementation choices. Experimental results show that, given the desired performance, the choice of algorithm parameters is not obvious and largely depends on the image properties and the underlying computer platform.

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Performance analysis of the filtered backprojection image reconstruction algorithms

SPIRAL is a generator for platform-adapted libraries of DSP transform algorithms. SPIRAL represents and automatically generates fast algorithms as mathematical formulas and translates them into programs. Adaptation is achieved by searching in the space of algorithmic and coding alternatives for the fastest implementation. We extend SPIRAL to generate platform-adapted implementations of FIR filters. First, we present various filter algorithms and introduce the mathematical constructs needed to include them into SPIRAL's architecture. Then we use SPIRAL to find fast filter implementations. The results show runtime improvements to a standard loop implementation of up to 70% using different blocking techniques. Further, we show that the usefulness of frequency-domain methods is not determined by the number of operations.

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Fast automatic software implementations of FIR filters

A growing number of performance-critical DSP applications use the discrete wavelet transform (DWT), thus prompting the need for highly efficient DWT software implementations. Unfortunately, the rapid evolution of computing platforms and compiler technology makes carefully hand-tuned code obsolete almost as fast as it is written. In this paper, we describe our work on the automatic generation of DWT implementations that are tuned to a given platform. Our approach captures the various DWT algorithms in a concise mathematical framework that enables the integration of DWTs into the SPIRAL code generation system. Experiments show the quality of our automatically generated code and provide interesting insights; for example, the fastest code differs between platforms and is usually based on a non-obvious combination of DWT algorithms.

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A. Gacic

Papers

SPIRAL: Code Generation for DSP Transforms

Performance analysis of the filtered backprojection image reconstruction algorithms

Fast automatic software implementations of FIR filters

Automatically generated high-performance code for discrete wavelet transforms