Algorithms For High-performance Computational Science Research Articles

Numerical algorithms for high-performance computational science

A number of features of today's high-performance computers make it challenging to exploit these machines fully for computational science. These include increasing core counts but stagnant clock frequencies; the high cost of data movement; use of accelerators (GPUs, FPGAs, coprocessors), making architectures increasingly heterogeneous; and multi- ple precisions of floating-point arithmetic, including half-precision. Moreover, as well as maximizing speed and accuracy, minimizing energy consumption is an important criterion. New generations of algorithms are needed to tackle these challenges. We discuss some approaches that we can take to develop numerical algorithms for high-performance computational science, with a view to exploiting the next generation of supercomputers. This article is part of a discussion meeting issue 'Numerical algorithms for high-performance computational science'.

High-performance sampling of generic determinantal point processes.

Determinantal point processes (DPPs) were introduced by Macchi (Macchi 1975 Adv. Appl. Probab. 7, 83-122) as a model for repulsive (fermionic) particle distributions. But their recent popularization is largely due to their usefulness for encouraging diversity in the final stage of a recommender system (Kulesza & Taskar 2012 Found. Trends Mach. Learn. 5, 123-286). The standard sampling scheme for finite DPPs is a spectral decomposition followed by an equivalent of a randomly diagonally pivoted Cholesky factorization of an orthogonal projection, which is only applicable to Hermitian kernels and has an expensive set-up cost. Researchers Launay et al. 2018 (http://arxiv.org/abs/1802.08429); Chen & Zhang 2018 NeurIPS (https://papers.nips.cc/paper/7805-fast-greedy-map-inference-for-determinantal-point-process-to-improve-recommendation-diversity.pdf) have begun to connect DPP sampling to LDLH factorizations as a means of avoiding the initial spectral decomposition, but existing approaches have only outperformed the spectral decomposition approach in special circumstances, where the number of kept modes is a small percentage of the ground set size. This article proves that trivial modifications of LU and LDLH factorizations yield efficient direct sampling schemes for non-Hermitian and Hermitian DPP kernels, respectively. Furthermore, it is experimentally shown that even dynamically scheduled, shared-memory parallelizations of high-performance dense and sparse-direct factorizations can be trivially modified to yield DPP sampling schemes with essentially identical performance. The software developed as part of this research, Catamari (hodgestar.com/catamari) is released under the Mozilla Public License v.2.0. It contains header-only, C++14 plus OpenMP 4.0 implementations of dense and sparse-direct, Hermitian and non-Hermitian DPP samplers. This article is part of a discussion meeting issue 'Numerical algorithms for high-performance computational science'.

Big telescope, big data: towards exascale with the Square Kilometre Array.

Unlike optical telescopes, radio interferometers do not image the sky directly but require specialized image formation algorithms. For the Square Kilometre Array (SKA), the computational requirements of this image formation are extremely demanding due to the huge data rates produced by the telescope. This processing will be performed by the SKA Science Data Processor facilities and a network of SKA Regional Centres, which must not only deal with SKA-scale data volumes but also with stringent science-driven image fidelity requirements.This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.

The parallelism motifs of genomic data analysis.

Genomic datasets are growing dramatically as the cost of sequencing continues to decline and small sequencing devices become available. Enormous community databases store and share these data with the research community, but some of these genomic data analysis problems require large-scale computational platforms to meet both the memory and computational requirements. These applications differ from scientific simulations that dominate the workload on high-end parallel systems today and place different requirements on programming support, software libraries and parallel architectural design. For example, they involve irregular communication patterns such as asynchronous updates to shared data structures. We consider several problems in high-performance genomics analysis, including alignment, profiling, clustering and assembly for both single genomes and metagenomes. We identify some of the common computational patterns or ‘motifs’ that help inform parallelization strategies and compare our motifs to some of the established lists, arguing that at least two key patterns, sorting and hashing, are missing.This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.

The physics of numerical analysis: a climate modelling case study.

The case is made for a much closer synergy between climate science, numerical analysis and computer science. This article is part of a discussion meeting issue 'Numerical algorithms for high-performance computational science'.

A survey of algorithms for transforming molecular dynamics data into metadata for in situ analytics based on machine learning methods.

This paper presents the survey of three algorithms to transform atomic-level molecular snapshots from molecular dynamics (MD) simulations into metadata representations that are suitable for in situ analytics based on machine learning methods. MD simulations studying the classical time evolution of a molecular system at atomic resolution are widely recognized in the fields of chemistry, material sciences, molecular biology and drug design; these simulations are one of the most common simulations on supercomputers. Next-generation supercomputers will have a dramatically higher performance than current systems, generating more data that needs to be analysed (e.g. in terms of number and length of MD trajectories). In the future, the coordination of data generation and analysis can no longer rely on manual, centralized analysis traditionally performed after the simulation is completed or on current data representations that have been defined for traditional visualization tools. Powerful data preparation phases (i.e. phases in which original row data is transformed to concise and still meaningful representations) will need to proceed data analysis phases. Here, we discuss three algorithms for transforming traditionally used molecular representations into concise and meaningful metadata representations. The transformations can be performed locally. The new metadata can be fed into machine learning methods for runtime in situ analysis of larger MD trajectories supported by high-performance computing. In this paper, we provide an overview of the three algorithms and their use for three different applications: protein-ligand docking in drug design; protein folding simulations; and protein engineering based on analytics of protein functions depending on proteins' three-dimensional structures. This article is part of a discussion meeting issue 'Numerical algorithms for high-performance computational science'.

The future of computing beyond Moore’s Law

Moore's Law is a techno-economic model that has enabled the information technology industry to double the performance and functionality of digital electronics roughly every 2 years within a fixed cost, power and area. Advances in silicon lithography have enabled this exponential miniaturization of electronics, but, as transistors reach atomic scale and fabrication costs continue to rise, the classical technological driver that has underpinned Moore's Law for 50 years is failing and is anticipated to flatten by 2025. This article provides an updated view of what a post-exascale system will look like and the challenges ahead, based on our most recent understanding of technology roadmaps. It also discusses the tapering of historical improvements, and how it affects options available to continue scaling of successors to the first exascale machine. Lastly, this article covers the many different opportunities and strategies available to continue computing performance improvements in the absence of historical technology drivers. This article is part of a discussion meeting issue 'Numerical algorithms for high-performance computational science'.

Rethinking arithmetic for deep neural networks.

We consider efficiency in the implementation of deep neural networks. Hardware accelerators are gaining interest as machine learning becomes one of the drivers of high-performance computing. In these accelerators, the directed graph describing a neural network can be implemented as a directed graph describing a Boolean circuit. We make this observation precise, leading naturally to an understanding of practical neural networks as discrete functions, and show that the so-called binarized neural networks are functionally complete. In general, our results suggest that it is valuable to consider Boolean circuits as neural networks, leading to the question of which circuit topologies are promising. We argue that continuity is central to generalization in learning, explore the interaction between data coding, network topology, and node functionality for continuity and pose some open questions for future research. As a first step to bridging the gap between continuous and Boolean views of neural network accelerators, we present some recent results from our work on LUTNet, a novel Field-Programmable Gate Array inference approach. Finally, we conclude with additional possible fruitful avenues for research bridging the continuous and discrete views of neural networks.This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.

Exascale applications: skin in the game.

As noted in Wikipedia, skin in the game refers to having ‘incurred risk by being involved in achieving a goal’, where ‘skin is a synecdoche for the person involved, and game is the metaphor for actions on the field of play under discussion’. For exascale applications under development in the US Department of Energy Exascale Computing Project, nothing could be more apt, with the skin being exascale applications and the game being delivering comprehensive science-based computational applications that effectively exploit exascale high-performance computing technologies to provide breakthrough modelling and simulation and data science solutions. These solutions will yield high-confidence insights and answers to the most critical problems and challenges for the USA in scientific discovery, national security, energy assurance, economic competitiveness and advanced healthcare.This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.

Machine learning and big scientific data.

This paper reviews some of the challenges posed by the huge growth of experimental data generated by the new generation of large-scale experiments at UK national facilities at the Rutherford Appleton Laboratory (RAL) site at Harwell near Oxford. Such ‘Big Scientific Data’ comes from the Diamond Light Source and Electron Microscopy Facilities, the ISIS Neutron and Muon Facility and the UK's Central Laser Facility. Increasingly, scientists are now required to use advanced machine learning and other AI technologies both to automate parts of the data pipeline and to help find new scientific discoveries in the analysis of their data. For commercially important applications, such as object recognition, natural language processing and automatic translation, deep learning has made dramatic breakthroughs. Google's DeepMind has now used the deep learning technology to develop their AlphaFold tool to make predictions for protein folding. Remarkably, it has been able to achieve some spectacular results for this specific scientific problem. Can deep learning be similarly transformative for other scientific problems? After a brief review of some initial applications of machine learning at the RAL, we focus on challenges and opportunities for AI in advancing materials science. Finally, we discuss the importance of developing some realistic machine learning benchmarks using Big Scientific Data coming from several different scientific domains. We conclude with some initial examples of our ‘scientific machine learning’ benchmark suite and of the research challenges these benchmarks will enable.This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.

Stochastic rounding and reduced-precision fixed-point arithmetic for solving neural ordinary differential equations.

Although double-precision floating-point arithmetic currently dominates high-performance computing, there is increasing interest in smaller and simpler arithmetic types. The main reasons are potential improvements in energy efficiency and memory footprint and bandwidth. However, simply switching to lower-precision types typically results in increased numerical errors. We investigate approaches to improving the accuracy of reduced-precision fixed-point arithmetic types, using examples in an important domain for numerical computation in neuroscience: the solution of ordinary differential equations (ODEs). The Izhikevich neuron model is used to demonstrate that rounding has an important role in producing accurate spike timings from explicit ODE solution algorithms. In particular, fixed-point arithmetic with stochastic rounding consistently results in smaller errors compared to single-precision floating-point and fixed-point arithmetic with round-to-nearest across a range of neuron behaviours and ODE solvers. A computationally much cheaper alternative is also investigated, inspired by the concept of dither that is a widely understood mechanism for providing resolution below the least significant bit in digital signal processing. These results will have implications for the solution of ODEs in other subject areas, and should also be directly relevant to the huge range of practical problems that are represented by partial differential equations.This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.

Optimal memory-aware backpropagation of deep join networks.

Deep learning training memory needs can prevent the user from considering large models and large batch sizes. In this work, we propose to use techniques from memory-aware scheduling and automatic differentiation (AD) to execute a backpropagation graph with a bounded memory requirement at the cost of extra recomputations. The case of a single homogeneous chain, i.e. the case of a network whose stages are all identical and form a chain, is well understood and optimal solutions have been proposed in the AD literature. The networks encountered in practice in the context of deep learning are much more diverse, both in terms of shape and heterogeneity. In this work, we define the class of backpropagation graphs, and extend those on which one can compute in polynomial time a solution that minimizes the total number of recomputations. In particular, we consider join graphs which correspond to models such as siamese or cross-modal networks. This article is part of a discussion meeting issue 'Numerical algorithms for high-performance computational science'.

Preparing sparse solvers for exascale computing.

Sparse solvers provide essential functionality for a wide variety of scientific applications. Highly parallel sparse solvers are essential for continuing advances in high-fidelity, multi-physics and multi-scale simulations, especially as we target exascale platforms. This paper describes the challenges, strategies and progress of the US Department of Energy Exascale Computing project towards providing sparse solvers for exascale computing platforms. We address the demands of systems with thousands of high-performance node devices where exposing concurrency, hiding latency and creating alternative algorithms become essential. The efforts described here are works in progress, highlighting current success and upcoming challenges. This article is part of a discussion meeting issue 'Numerical algorithms for high-performance computational science'.

Hierarchical algorithms on hierarchical architectures.

A traditional goal of algorithmic optimality, squeezing out flops, has been superseded by evolution in architecture. Flops no longer serve as a reasonable proxy for all aspects of complexity. Instead, algorithms must now squeeze memory, data transfers, and synchronizations, while extra flops on locally cached data represent only small costs in time and energy. Hierarchically low-rank matrices realize a rarely achieved combination of optimal storage complexity and high-computational intensity for a wide class of formally dense linear operators that arise in applications for which exascale computers are being constructed. They may be regarded as algebraic generalizations of the fast multipole method. Methods based on these hierarchical data structures and their simpler cousins, tile low-rank matrices, are well proportioned for early exascale computer architectures, which are provisioned for high processing power relative to memory capacity and memory bandwidth. They are ushering in a renaissance of computational linear algebra. A challenge is that emerging hardware architecture possesses hierarchies of its own that do not generally align with those of the algorithm. We describe modules of a software toolkit, hierarchical computations on manycore architectures, that illustrate these features and are intended as building blocks of applications, such as matrix-free higher-order methods in optimization and large-scale spatial statistics. Some modules of this open-source project have been adopted in the software libraries of major vendors.This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.

On the cost of iterative computations.

With exascale-level computation on the horizon, the art of predicting the cost of computations has acquired a renewed focus. This task is especially challenging in the case of iterative methods, for which convergence behaviour often cannot be determined with certainty a priori (unless we are satisfied with potentially outrageous overestimates) and which typically suffer from performance bottlenecks at scale due to synchronization cost. Moreover, the amplification of rounding errors can substantially affect the practical performance, in particular for methods with short recurrences. In this article, we focus on what we consider to be key points which are crucial to understanding the cost of iteratively solving linear algebraic systems. This naturally leads us to questions on the place of numerical analysis in relation to mathematics, computer science and sciences, in general. This article is part of a discussion meeting issue 'Numerical algorithms for high-performance computational science'.