The Problem with the Linpack Benchmark Matrix Generator

We characterize the matrix sizes for which the Linpack Benchmark matrix generator constructs a matrix with identical columns

Analysis of a benchmark suite to evaluate mixed numeric and symbolic processing

The suite of programs that formed the benchmark for a proposed advanced computer is described and analyzed. The features of the processor and its operating system that are tested by the benchmark are discussed. The computer codes and the supporting data for the analysis are given as appendices

Random circuit block-encoded matrix and a proposal of quantum LINPACK benchmark

The LINPACK benchmark reports the performance of a computer for solving a system of linear equations with dense random matrices. Although this task was not designed with a real application directly in mind, the LINPACK benchmark has been used to define the list of TOP500 supercomputers since the debut of the list in 1993. We propose that a similar benchmark, called the quantum LINPACK benchmark, could be used to measure the whole machine performance of quantum computers. The success of the quantum LINPACK benchmark should be viewed as the minimal requirement for a quantum computer to perform a useful task of solving linear algebra problems, such as linear systems of equations. We propose an input model called the RAndom Circuit Block-Encoded Matrix (RACBEM), which is a proper generalization of a dense random matrix in the quantum setting. The RACBEM model is efficient to be implemented on a quantum computer, and can be designed to optimally adapt to any given quantum architecture, with relying on a black-box quantum compiler. Besides solving linear systems, the RACBEM model can be used to perform a variety of linear algebra tasks relevant to many physical applications, such as computing spectral measures, time series generated by a Hamiltonian simulation, and thermal averages of the energy. We implement these linear algebra operations on IBM Q quantum devices as well as quantum virtual machines, and demonstrate their performance in solving scientific computing problems.Comment: 22 pages, 18 figure

Automated Dynamic Error Analysis Methods for Optimization of Computer Arithmetic Systems

Computer arithmetic is one of the more important topics within computer science and engineering. The earliest implementations of computer systems were designed to perform arithmetic operations and cost if not all digital systems will be required to perform some sort of arithmetic as part of their normal operations. This reliance on the arithmetic operations of computers means the accurate representation of real numbers within digital systems is vital, and an understanding of how these systems are implemented and their possible drawbacks is essential in order to design and implement modern high performance systems. At present the most widely implemented system for computer arithmetic is the IEEE754 Floating Point system, while this system is deemed to the be the best available implementation it has several features that can result in serious errors of computation if not implemented correctly. Lack of understanding of these errors and their effects has led to real world disasters in the past on several occasions. Systems for the detection of these errors are highly important and fast, efficient and easy to use implementations of these detection systems is a high priority. Detection of floating point rounding errors normally requires run-time analysis in order to be effective. Several systems have been proposed for the analysis of floating point arithmetic including Interval Arithmetic, Affine Arithmetic and Monte Carlo Arithmetic. While these systems have been well studied using theoretical and software based approaches, implementation of systems that can be applied to real world situations has been limited due to issues with implementation, performance and scalability. The majority of implementations have been software based and have not taken advantage of the performance gains associated with hardware accelerated computer arithmetic systems. This is especially problematic when it is considered that systems requiring high accuracy will often require high performance. The aim of this thesis and associated research is to increase understanding of error and error analysis methods through the development of easy to use and easy to understand implementations of these techniques

Automated Dynamic Error Analysis Methods for Optimization of Computer Arithmetic Systems

Computer arithmetic is one of the more important topics within computer science and engineering. The earliest implementations of computer systems were designed to perform arithmetic operations and cost if not all digital systems will be required to perform some sort of arithmetic as part of their normal operations. This reliance on the arithmetic operations of computers means the accurate representation of real numbers within digital systems is vital, and an understanding of how these systems are implemented and their possible drawbacks is essential in order to design and implement modern high performance systems. At present the most widely implemented system for computer arithmetic is the IEEE754 Floating Point system, while this system is deemed to the be the best available implementation it has several features that can result in serious errors of computation if not implemented correctly. Lack of understanding of these errors and their effects has led to real world disasters in the past on several occasions. Systems for the detection of these errors are highly important and fast, efficient and easy to use implementations of these detection systems is a high priority. Detection of floating point rounding errors normally requires run-time analysis in order to be effective. Several systems have been proposed for the analysis of floating point arithmetic including Interval Arithmetic, Affine Arithmetic and Monte Carlo Arithmetic. While these systems have been well studied using theoretical and software based approaches, implementation of systems that can be applied to real world situations has been limited due to issues with implementation, performance and scalability. The majority of implementations have been software based and have not taken advantage of the performance gains associated with hardware accelerated computer arithmetic systems. This is especially problematic when it is considered that systems requiring high accuracy will often require high performance. The aim of this thesis and associated research is to increase understanding of error and error analysis methods through the development of easy to use and easy to understand implementations of these techniques

Abstractions and performance optimisations for finite element methods

Finding numerical solutions to partial differential equations (PDEs) is an essential task in the discipline of scientific computing. In designing software tools for this task, one of the ultimate goals is to balance the needs for generality, ease to use and high performance. Domain-specific systems based on code generation techniques, such as Firedrake, attempt to address this problem with a design consisting of a hierarchy of abstractions, where the users can specify the mathematical problems via a high-level, descriptive interface, which is progressively lowered through the intermediate abstractions. Well-designed abstraction layers are essential to enable performing code transformations and optimisations robustly and efficiently, generating high-performance code without user intervention. This thesis discusses several topics on the design of the abstraction layers of Firedrake, and presents the benefit of its software architecture by providing examples of various optimising code transformations at the appropriate abstraction layers. In particular, we discuss the advantage of describing the local assembly stage of a finite element solver in an intermediate representation based on symbolic tensor algebra. We successfully lift specific loop optimisations, previously implemented by rewriting ASTs of the local assembly kernels, to this higher-level tensor language, improving the compilation speed and optimisation effectiveness. The global assembly phase involves the application of local assembly kernels on a collection of entities of an unstructured mesh. We redesign the abstraction to express the global assembly loop nests using tools and concepts based on the polyhedral model. This enables us to implement the cross-element vectorisation algorithm that delivers stable vectorisation performance on CPUs automatically. This abstraction also improves the portability of Firedrake, as we demonstrate targeting GPU devices transparently from the same software stack.Open Acces

Quantum Monte Carlo for large chemical systems: Implementing efficient strategies for petascale platforms and beyond

Various strategies to implement efficiently QMC simulations for large chemical systems are presented. These include: i.) the introduction of an efficient algorithm to calculate the computationally expensive Slater matrices. This novel scheme is based on the use of the highly localized character of atomic Gaussian basis functions (not the molecular orbitals as usually done), ii.) the possibility of keeping the memory footprint minimal, iii.) the important enhancement of single-core performance when efficient optimization tools are employed, and iv.) the definition of a universal, dynamic, fault-tolerant, and load-balanced computational framework adapted to all kinds of computational platforms (massively parallel machines, clusters, or distributed grids). These strategies have been implemented in the QMC=Chem code developed at Toulouse and illustrated with numerical applications on small peptides of increasing sizes (158, 434, 1056 and 1731 electrons). Using 10k-80k computing cores of the Curie machine (GENCI-TGCC-CEA, France) QMC=Chem has been shown to be capable of running at the petascale level, thus demonstrating that for this machine a large part of the peak performance can be achieved. Implementation of large-scale QMC simulations for future exascale platforms with a comparable level of efficiency is expected to be feasible

A PROBABILISTIC MACHINE LEARNING FRAMEWORK FOR CLOUD RESOURCE SELECTION ON THE CLOUD

The execution of the scientific applications on the Cloud comes with great flexibility, scalability, cost-effectiveness, and substantial computing power. Market-leading Cloud service providers such as Amazon Web service (AWS), Azure, Google Cloud Platform (GCP) offer various general purposes, memory-intensive, and compute-intensive Cloud instances for the execution of scientific applications. The scientific community, especially small research institutions and undergraduate universities, face many hurdles while conducting high-performance computing research in the absence of large dedicated clusters. The Cloud provides a lucrative alternative to dedicated clusters, however a wide range of Cloud computing choices makes the instance selection for the end-users. This thesis aims to simplify Cloud instance selection for end-users by proposing a probabilistic machine learning framework to allow to users select a suitable Cloud instance for their scientific applications. This research builds on the previously proposed A2Cloud-RF framework that recommends high-performing Cloud instances by profiling the application and the selected Cloud instances. The framework produces a set of objective scores called the A2Cloud scores, which denote the compatibility level between the application and the selected Cloud instances. When used alone, the A2Cloud scores become increasingly unwieldy with an increasing number of tested Cloud instances. Additionally, the framework only examines the raw application performance and does not consider the execution cost to guide resource selection. To improve the usability of the framework and assist with economical instance selection, this research adds two Naïve Bayes (NB) classifiers that consider both the application’s performance and execution cost. These NB classifiers include: 1) NB with a Random Forest Classifier (RFC) and 2) a standalone NB module. Naïve Bayes with a Random Forest Classifier (RFC) augments the A2Cloud-RF framework\u27s final instance ratings with the execution cost metric. In the training phase, the classifier builds the frequency and probability tables. The classifier recommends a Cloud instance based on the highest posterior probability for the selected application. The standalone NB classifier uses the generated A2Cloud score (an intermediate result from the A2Cloud-RF framework) and execution cost metric to construct an NB classifier. The NB classifier forms a frequency table and probability (prior and likelihood) tables. For recommending a Cloud instance for a test application, the classifier calculates the highest posterior probability for all of the Cloud instances. The classifier recommends a Cloud instance with the highest posterior probability. This study performs the execution of eight real-world applications on 20 Cloud instances from AWS, Azure, GCP, and Linode. We train the NB classifiers using 80% of this dataset and employ the remaining 20% for testing. The testing yields more than 90% recommendation accuracy for the chosen applications and Cloud instances. Because of the imbalanced nature of the dataset and multi-class nature of classification, we consider the confusion matrix (true positive, false positive, true negative, and false negative) and F1 score with above 0.9 scores to describe the model performance. The final goal of this research is to make Cloud computing an accessible resource for conducting high-performance scientific executions by enabling users to select an effective Cloud instance from across multiple providers

Enabling high performance dynamic language programming for micro-core architectures

Micro-core architectures are intended to deliver high performance at a low overall power consumption by combining many simple central processing unit (CPU) cores, with an associated small amount of memory, onto a single chip. This technology is not only of great interest for embedded, Edge and IoT applications but also for High-Performance Computing (HPC) accelerators. However, micro-core architectures are difficult to program and exploit, not only because each technology is different, with its own idiosyncrasies, but also because they each present a different low-level interface to the programmer. Furthermore, micro-cores have very constrained amounts of on-chip, scratchpad memory (often around 32KB), further hampering programmer productivity by requiring the programmer to manually manage the regular loading and unloading of data from the host to the device during program execution. To help address these issues, dynamic languages such as Python have been ported to several micro-core architectures but these are often delivered as interpreters with the associated performance penalty over natively compiled languages, such as C. The research questions for this thesis target four areas of concern for dynamic programming languages on micro-core architectures: (RQ1) how to manage the limited on-chip memory for data, (RQ2) how to manage the limited on-chip memory for code, (RQ3) how to address the low runtime performance of virtual machines and (RQ4) how to manage the idiosyncratic architectures of micro-core architectures. The focus of this work is to address these concerns whilst maintaining the programmer productivity benefits of dynamic programming languages, using ePython as the research vehicle. Therefore, key areas of design (such as abstractions for offload) and implementation (novel compiler and runtime techniques for these technologies) are considered, resulting in a number of approaches that are not only applicable to the compilation of Python codes but also more generally to other dynamic languages on micro-cores architectures. RQ1 was addressed by providing support for kernels with arbitrary data size through high-level programming abstractions that enable access to the memory hierarchies of micro-core devices, allowing the deployment of real-world applications, such as a machine learning code to detect cancer cells in full-sized scan images. A new abstract machine, Olympus, addressed RQ2 by supporting the compilation of dynamic languages, such as Python, to micro-core native code. Olympus enables ePython to close the kernel runtime performance gap with native C, matching C for the LINPACK and an iterative Fibonacci benchmark, and to provide, on average, around 75\% of native C runtime performance across four benchmarks running on a set of eight CPU architectures. Olympus also addresses RQ3 by providing dynamic function loading, supporting kernel codes larger than the on-chip memory, whilst still retaining the runtime performance benefits of native code generation. Finally, RQ4 was addressed by the Eithne benchmarking framework which not only enabled a single benchmarking code to be deployed, unchanged, across different CPU architectures, but also provided the underlying communications framework for Olympus. The portability of end-user ePython codes and the underlying Olympus abstract machine were validated by running a set of four benchmarks on eight different CPU architectures, from a single codebase