Algorithm Architecture Co-design for Dense and Sparse Matrix Computations

abstract: With the end of Dennard scaling and Moore's law, architects have moved towards heterogeneous designs consisting of specialized cores to achieve higher performance and energy efficiency for a target application domain. Applications of linear algebra are ubiquitous in the field of scientific computing, machine learning, statistics, etc. with matrix computations being fundamental to these linear algebra based solutions. Design of multiple dense (or sparse) matrix computation routines on the same platform is quite challenging. Added to the complexity is the fact that dense and sparse matrix computations have large differences in their storage and access patterns and are difficult to optimize on the same architecture. This thesis addresses this challenge and introduces a reconfigurable accelerator that supports both dense and sparse matrix computations efficiently. The reconfigurable architecture has been optimized to execute the following linear algebra routines: GEMV (Dense General Matrix Vector Multiplication), GEMM (Dense General Matrix Matrix Multiplication), TRSM (Triangular Matrix Solver), LU Decomposition, Matrix Inverse, SpMV (Sparse Matrix Vector Multiplication), SpMM (Sparse Matrix Matrix Multiplication). It is a multicore architecture where each core consists of a 2D array of processing elements (PE). The 2D array of PEs is of size 4x4 and is scheduled to perform 4x4 sized matrix updates efficiently. A sequence of such updates is used to solve a larger problem inside a core. A novel partitioned block compressed sparse data structure (PBCSC/PBCSR) is used to perform sparse kernel updates. Scalable partitioning and mapping schemes are presented that map input matrices of any given size to the multicore architecture. Design trade-offs related to the PE array dimension, size of local memory inside a core and the bandwidth between on-chip memories and the cores have been presented. An optimal core configuration is developed from this analysis. Synthesis results using a 7nm PDK show that the proposed accelerator can achieve a performance of upto 32 GOPS using a single core.Dissertation/ThesisMasters Thesis Computer Engineering 201

GPUにおける拡張精度浮動小数点演算を用いた線形計算の研究

筑波大学 (University of Tsukuba)201

Variable-size batched Gauss-Jordan elimination for block-Jacobi preconditioning on graphics processors

[EN] In this work, we address the efficient realization of block-Jacobi preconditioning on graphics processing units (GPUs). This task requires the solution of a collection of small and independent linear systems. To fully realize this implementation, we develop a variablesize batched matrix inversion kernel that uses Gauss-Jordan elimination (GJE) along with a variable-size batched matrix-vector multiplication kernel that transforms the linear systems' right-hand sides into the solution vectors. Our kernels make heavy use of the increased register count and the warp-local communication associated with newer GPU architectures. Moreover, in the matrix inversion, we employ an implicit pivoting strategy that migrates the workload (i.e., operations) to the place where the data resides instead of moving the data to the executing cores. We complement the matrix inversion with extraction and insertion strategies that allow the block-Jacobi preconditioner to be set up rapidly. The experiments on NVlDlA's K40 and P100 architectures reveal that our variable-size batched matrix inversion routine outperforms the CUDA basic linear algebra subroutine (cuBLAS) library functions that provide the same (or even less) functionality. We also show that the preconditioner setup and preconditioner application cost can be somewhat offset by the faster convergence of the iterative solver. (C) 2018 Elsevier B.V. All rights reserved.This material is based upon work supported by the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-SC-0010042. H. Anzt was supported by the "Impuls and Vernetzungsfond of the Helmholtz Association" under grant VH-NG-1241. G. Flegar and E. S. Quintana-Orti were supported by project TIN2014-53495-R of the MINECO-FEDER; and project OPRECOMP (http://oprecomp.eu) with the financial support of the Future and Emerging Technologies (FET) programme within the European Union's Horizon 2020 research and innovation programme, under grant agreement No 732631. The authors would also like to acknowledge the Swiss National Computing Centre (CSCS) for granting computing resources in the Small Development Project entitled "Energy-Efficient preconditioning for iterative linear solvers" (#d65).Anzt, H.; Dongarra, J.; Flegar, G.; Quintana Ortí, ES. (2019). Variable-size batched Gauss-Jordan elimination for block-Jacobi preconditioning on graphics processors. Parallel Computing. 81:131-146. https://doi.org/10.1016/j.parco.2017.12.006S1311468

Towards Closing the Programmability-Efficiency Gap using Software-Defined Hardware

The past decade has seen the breakdown of two important trends in the computing industry: Moore’s law, an observation that the number of transistors in a chip roughly doubles every eighteen months, and Dennard scaling, that enabled the use of these transistors within a constant power budget. This has caused a surge in domain-specific accelerators, i.e. specialized hardware that deliver significantly better energy efficiency than general-purpose processors, such as CPUs. While the performance and efficiency of such accelerators are highly desirable, the fast pace of algorithmic innovation and non-recurring engineering costs have deterred their widespread use, since they are only programmable across a narrow set of applications. This has engendered a programmability-efficiency gap across contemporary platforms. A practical solution that can close this gap is thus lucrative and is likely to engender broad impact in both academic research and the industry. This dissertation proposes such a solution with a reconfigurable Software-Defined Hardware (SDH) system that morphs parts of the hardware on-the-fly to tailor to the requirements of each application phase. This system is designed to deliver near-accelerator-level efficiency across a broad set of applications, while retaining CPU-like programmability. The dissertation first presents a fixed-function solution to accelerate sparse matrix multiplication, which forms the basis of many applications in graph analytics and scientific computing. The solution consists of a tiled hardware architecture, co-designed with the outer product algorithm for Sparse Matrix-Matrix multiplication (SpMM), that uses on-chip memory reconfiguration to accelerate each phase of the algorithm. A proof-of-concept is then presented in the form of a prototyped 40 nm Complimentary Metal-Oxide Semiconductor (CMOS) chip that demonstrates energy efficiency and performance per die area improvements of 12.6x and 17.1x over a high-end CPU, and serves as a stepping stone towards a full SDH system. The next piece of the dissertation enhances the proposed hardware with reconfigurability of the dataflow and resource sharing modes, in order to extend acceleration support to a set of common parallelizable workloads. This reconfigurability lends the system the ability to cater to discrete data access and compute patterns, such as workloads with extensive data sharing and reuse, workloads with limited reuse and streaming access patterns, among others. Moreover, this system incorporates commercial cores and a prototyped software stack for CPU-level programmability. The proposed system is evaluated on a diverse set of compute-bound and memory-bound kernels that compose applications in the domains of graph analytics, machine learning, image and language processing. The evaluation shows average performance and energy-efficiency gains of 5.0x and 18.4x over the CPU. The final part of the dissertation proposes a runtime control framework that uses low-cost monitoring of hardware performance counters to predict the next best configuration and reconfigure the hardware, upon detecting a change in phase or nature of data within the application. In comparison to prior work, this contribution targets multicore CGRAs, uses low-overhead decision tree based predictive models, and incorporates reconfiguration cost-awareness into its policies. Compared to the best-average static (non-reconfiguring) configuration, the dynamically reconfigurable system achieves a 1.6x improvement in performance-per-Watt in the Energy-Efficient mode of operation, or the same performance with 23% lower energy in the Power-Performance mode, for SpMM across a suite of real-world inputs. The proposed reconfiguration mechanism itself outperforms the state-of-the-art approach for dynamic runtime control by up to 2.9x in terms of energy-efficiency.PHDComputer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169859/1/subh_1.pd

EFFICIENTLY ACCELERATING SPARSE PROBLEMS BY ENABLING STREAM ACCESSES TO MEMORY USING HARDWARE/SOFTWARE TECHNIQUES

The objective of this research is to improve the performance of sparse problems that have a wide range of applications but still, suffer from serious challenges when running on modern computers. In summary, the challenges include the underutilization of available memory bandwidth because of lack of spatial locality, dependencies in computation, or slow mechanisms for decompressing the sparse data, and the underutilization of concurrent compute engines because of the distribution of non-zero values in sparse data. Our key insight to address the aforementioned challenges is that based on the type of the problem, we either use an intelligent reduction tree near memory to process data while gathering them from random locations of memory, transform the computations mathematically to extract more parallelism, modify the distribution of non-zero elements, or change the representation of sparse data. By applying such techniques, the execution adapts more effectively to given hardware resources. To this end, this research introduces hardware/software techniques to enable stream accesses to memory for accelerating four main categories of sparse problems including the inference of recommendation systems, iterative solvers of partial differential equations (PDEs), deep neural networks (DNNs), and graph algorithms.Ph.D

High Performance Implementation of Support Vector Machines Using OpenCL

Support Vector Machines are a machine learning approach that is well studied, thoroughly vetted and effective in a large number of applications. The objective of this thesis is to accelerate an implementation of Support Vector Machines (SVM) using a heterogeneous computing system programmed using OpenCL in C/C++. LIBSVM, a widely-available, popular and open source implementation of SVM is chosen, allowing the presented work to be integrated seamlessly into existing systems. The proposed framework is evaluated in terms of speed and accuracy when performing training and classification on a number of standard data sets. Testing was based on two work station GPUs, the NVIDIA GTX 480 and Tesla K20, and a modern, work station CPU (Intel i5 Quad Core, 3 GHz). We find that, for large data sets, training is accelerated by a factor ranging from 9 to 22. In general, speedup increases with the total number of training samples in the data set until the GPU device is fully utilized. While these gains in speedup are significant, they do not match the ideal parallel speedup, that is the total number of cores in the parallel system. Our findings indicate that performance is hampered by the portions of the SVM training algorithm that are sequential. In addition, we find that the classification phase of the SVM system is accelerated by a factor of up to 12. During classification only a relatively small number of samples are classified compared to the typical number of training samples, and the computational complexity of classification grows only linearly with the number of samples processed, as opposed to the training phase where it grows quadratically. The contri- butions of this thesis include the use of OpenCL for accelerating SVM training and testing on heterogeneous systems, and the performance analysis of the acceleration of SVM

Power Aware Computing on GPUs

Energy and power density concerns in modern processors have led to significant computer architecture research efforts in power-aware and temperature-aware computing. With power dissipation becoming an increasingly vexing problem, power analysis of Graphical Processing Unit (GPU) and its components has become crucial for hardware and software system design. Here, we describe our technique for a coordinated measurement approach that combines real total power measurement and per-component power estimation. To identify power consumption accurately, we introduce the Activity-based Model for GPUs (AMG), from which we identify activity factors and power for microarchitectures on GPUs that will help in analyzing power tradeoffs of one component versus another using microbenchmarks. The key challenge addressed in this thesis is real-time power consumption, which can be accurately estimated using NVIDIA\u27s Management Library (NVML) through Pthreads. We validated our model using Kill-A-Watt power meter and the results are accurate within 10\%. The resulting Performance Application Programming Interface (PAPI) NVML component offers real-time total power measurements for GPUs. This thesis also compares a single NVIDIA C2075 GPU running MAGMA (Matrix Algebra on GPU and Multicore Architectures) kernels, to a 48 core AMD Istanbul CPU running LAPACK