Lecture 11: The Road to Exascale and Legacy Software for Dense Linear Algebra

In this talk, we will look at the current state of high performance computing and look at the next stage of extreme computing. With extreme computing, there will be fundamental changes in the character of floating point arithmetic and data movement. In this talk, we will look at how extreme-scale computing has caused algorithm and software developers to change their way of thinking on implementing and program-specific applications

Doctor of Philosophy

dissertationMemory access irregularities are a major bottleneck for bandwidth limited problems on Graphics Processing Unit (GPU) architectures. GPU memory systems are designed to allow consecutive memory accesses to be coalesced into a single memory access. Noncontiguous accesses within a parallel group of threads working in lock step may cause serialized memory transfers. Irregular algorithms may have data-dependent control flow and memory access, which requires runtime information to be evaluated. Compile time methods for evaluating parallelism, such as static dependence graphs, are not capable of evaluating irregular algorithms. The goals of this dissertation are to study irregularities within the context of unstructured mesh and sparse matrix problems, analyze the impact of vectorization widths on irregularities, and present data-centric methods that improve control flow and memory access irregularity within those contexts. Reordering associative operations has often been exploited for performance gains in parallel algorithms. This dissertation presents a method for associative reordering of stencil computations over unstructured meshes that increases data reuse through caching. This novel parallelization scheme offers considerable speedups over standard methods. Vectorization widths can have significant impact on performance in vectorized computations. Although the hardware vector width is generally fixed, the logical vector width used within a computation can range from one up to the width of the computation. Significant performance differences can occur due to thread scheduling and resource limitations. This dissertation analyzes the impact of vectorization widths on dense numerical computations such as 3D dG postprocessing. It is difficult to efficiently perform dynamic updates on traditional sparse matrix formats. Explicitly controlling memory segmentation allows for in-place dynamic updates in sparse matrices. Dynamically updating the matrix without rebuilding or sorting greatly improves processing time and overall throughput. This dissertation presents a new sparse matrix format, dynamic compressed sparse row (DCSR), which allows for dynamic streaming updates to a sparse matrix. A new method for parallel sparse matrix-matrix multiplication (SpMM) that uses dynamic updates is also presented

Aceleración de algoritmos de procesamiento de imágenes para el análisis de partículas individuales con microscopia electrónica

Tesis Doctoral inédita cotutelada por la Masaryk University (República Checa) y la Universidad Autónoma de Madrid, Escuela Politécnica Superior, Departamento de Ingeniería Informática. Fecha de Lectura: 24-10-2022Cryogenic Electron Microscopy (Cryo-EM) is a vital field in current structural biology. Unlike X-ray crystallography and Nuclear Magnetic Resonance, it can be used to analyze membrane proteins and other samples with overlapping spectral peaks. However, one of the significant limitations of Cryo-EM is the computational complexity. Modern electron microscopes can produce terabytes of data per single session, from which hundreds of thousands of particles must be extracted and processed to obtain a near-atomic resolution of the original sample. Many existing software solutions use high-Performance Computing (HPC) techniques to bring these computations to the realm of practical usability. The common approach to acceleration is parallelization of the processing, but in praxis, we face many complications, such as problem decomposition, data distribution, load scheduling, balancing, and synchronization. Utilization of various accelerators further complicates the situation, as heterogeneous hardware brings additional caveats, for example, limited portability, under-utilization due to synchronization, and sub-optimal code performance due to missing specialization. This dissertation, structured as a compendium of articles, aims to improve the algorithms used in Cryo-EM, esp. the SPA (Single Particle Analysis). We focus on the single-node performance optimizations, using the techniques either available or developed in the HPC field, such as heterogeneous computing or autotuning, which potentially needs the formulation of novel algorithms. The secondary goal of the dissertation is to identify the limitations of state-of-the-art HPC techniques. Since the Cryo-EM pipeline consists of multiple distinct steps targetting different types of data, there is no single bottleneck to be solved. As such, the presented articles show a holistic approach to performance optimization. First, we give details on the GPU acceleration of the specific programs. The achieved speedup is due to the higher performance of the GPU, adjustments of the original algorithm to it, and application of the novel algorithms. More specifically, we provide implementation details of programs for movie alignment, 2D classification, and 3D reconstruction that have been sped up by order of magnitude compared to their original multi-CPU implementation or sufficiently the be used on-the-fly. In addition to these three programs, multiple other programs from an actively used, open-source software package XMIPP have been accelerated and improved. Second, we discuss our contribution to HPC in the form of autotuning. Autotuning is the ability of software to adapt to a changing environment, i.e., input or executing hardware. Towards that goal, we present cuFFTAdvisor, a tool that proposes and, through autotuning, finds the best configuration of the cuFFT library for given constraints of input size and plan settings. We also introduce a benchmark set of ten autotunable kernels for important computational problems implemented in OpenCL or CUDA, together with the introduction of complex dynamic autotuning to the KTT tool. Third, we propose an image processing framework Umpalumpa, which combines a task-based runtime system, data-centric architecture, and dynamic autotuning. The proposed framework allows for writing complex workflows which automatically use available HW resources and adjust to different HW and data but at the same time are easy to maintainThe project that gave rise to these results received the support of a fellowship from the “la Caixa” Foundation (ID 100010434). The fellowship code is LCF/BQ/DI18/11660021. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 71367

On the Efficient Evaluation of the Exchange Correlation Potential on Graphics Processing Unit Clusters

The predominance of Kohn-Sham density functional theory (KS-DFT) for the theoretical treatment of large experimentally relevant systems in molecular chemistry and materials science relies primarily on the existence of efficient software implementations which are capable of leveraging the latest advances in modern high performance computing (HPC). With recent trends in HPC leading towards in increasing reliance on heterogeneous accelerator based architectures such as graphics processing units (GPU), existing code bases must embrace these architectural advances to maintain the high-levels of performance which have come to be expected for these methods. In this work, we purpose a three-level parallelism scheme for the distributed numerical integration of the exchange-correlation (XC) potential in the Gaussian basis set discretization of the Kohn-Sham equations on large computing clusters consisting of multiple GPUs per compute node. In addition, we purpose and demonstrate the efficacy of the use of batched kernels, including batched level-3 BLAS operations, in achieving high-levels of performance on the GPU. We demonstrate the performance and scalability of the implementation of the purposed method in the NWChemEx software package by comparing to the existing scalable CPU XC integration in NWChem.Comment: 26 pages, 9 figure

Mixed-Precision Numerical Linear Algebra Algorithms: Integer Arithmetic Based LU Factorization and Iterative Refinement for Hermitian Eigenvalue Problem

Mixed-precision algorithms are a class of algorithms that uses low precision in part of the algorithm in order to save time and energy with less accurate computation and communication. These algorithms usually utilize iterative refinement processes to improve the approximate solution obtained from low precision to the accuracy we desire from doing all the computation in high precision. Due to the demand of deep learning applications, there are hardware developments offering different low-precision formats including half precision (FP16), Bfloat16 and integer operations for quantized integers, which uses integers with a shared scalar to represent a set of equally spaced numbers. As new hardware architectures focus on bringing performance in these formats, the mixed-precision algorithms have more potential leverage on them and outmatch traditional fixed-precision algorithms. This dissertation consists of two articles. In the first article, we adapt one of the most fundamental algorithms in numerical linear algebra---LU factorization with partial pivoting--- to use integer arithmetic. With the goal of obtaining a low accuracy factorization as the preconditioner of generalized minimal residual (GMRES) to solve systems of linear equations, the LU factorization is adapted to use two different fixed-point formats for matrices L and U. A left-looking variant is also proposed for matrices with unbounded column growth. Finally, GMRES iterative refinement has shown that it can work on matrices with condition numbers up to 10000 with the algorithm that uses int16 as input and int32 accumulator for the update step. The second article targets symmetric and Hermitian eigenvalue problems. In this section we revisit the SICE algorithm from Dongarra et al. By applying the Sherman-Morrison formula on the diagonally-shifted tridiagonal systems, we propose an updated SICE-SM algorithm. By incorporating the latest two-stage algorithms from the PLASMA and MAGMA software libraries for numerical linear algebra, we achieved up to 3.6x speedup using the mixed-precision eigensolver with the blocked SICE-SM algorithm for iterative refinement when compared with full double complex precision solvers for the cases with a portion of eigenvalues and eigenvectors requested