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At least 1,495 records · Page 83
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DOE OSTI2016

A Posteriori Error Analysis of Two-Stage Computation Methods with Application to Efficient Discretization and the Parareal Algorithm

We consider numerical methods for initial value problems that employ a two stage approach consisting of solution on a relatively coarse discretization followed by solution on a relatively fine discretization. Examples include adaptive error control, parallel-in-time solution schemes, and efficient solution of adjoint problems for computing a posteriori error estimates. We describe a general formulation of two stage computations then perform a general a posteriori error analysis based on computable residuals and solution of an adjoint problem. The analysis accommodates various variations in the two stage computation and in formulation of the adjoint problems. We apply the analysis to compute “dual-weighted” a posteriori error estimates, to develop novel algorithms for efficient solution that take into account cancellation of error, and to the Parareal Algorithm. We test the various results using several numerical examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

A Linear-Time Algorithm and Analysis of Graph Relative Hausdorff Distance

Graph similarity metrics serve far-ranging purposes across many domains in data science. As graph datasets grow in size, scientists need comparative tools that capture meaningful differences, yet are lightweight and scalable. Graph Relative Hausdorff (RH) distance is a promising, recently proposed measure for quantifying degree distribution similarity. In spite of recent interest in RH distance, little is known about its properties. Here, we conduct an algorithmic and analytic study of RH distance. In particular, we provide the first linear-time algorithm for computing RH distance, analyze examples of RH distance between families of graphs, and prove several analytic results concerning the range, density, and extremal behavior of RH distance values.

Relative Hausdorff distance, degree distribution, ↗
DOE OSTI2023

Online Adaptive Algorithm for Constraint Energy Minimizing Generalized Multiscale Discontinuous Galerkin Method

Here in this research, we propose an online basis enrichment strategy within the framework of a recently developed constraint energy minimizing generalized multiscale discontinuous Galerkin method. Combining the technique of oversampling, one makes use of the information of the current residuals to adaptively construct basis functions in the online stage to reduce the error of multiscale approximation. A complete analysis of the method is presented, which shows the proposed online enrichment leads to a fast convergence from multiscale approximation to the fine-scale solution. The error reduction can be made sufficiently large by suitably selecting oversampling regions and the number of oversampling layers. Further, the convergence rate of the enrichment algorithm depends on a factor of exponential decay regarding the number of oversampling layers and a user-defined parameter. Numerical results are provided to demonstrate the effectiveness and efficiency of the proposed online adaptive algorithm.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Space-Split Algorithm for Sensitivity Analysis of Discrete Chaotic Systems With Multidimensional Unstable Manifolds

Accurate approximations of the change of a system's output and its statistics with respect to the input are highly desired in computational dynamics. Ruelle's linear response theory provides breakthrough mathematical machinery for computing the linear response of chaotic dynamical systems. In this paper, we propose an algorithm for sensitivity analysis of discrete chaos with an arbitrary number of positive Lyapunov exponents. We combine the concept of perturbation space-splitting, which regularizes Ruelle's original expression, together with measure-based parameterization of the expanding subspace. We use these tools to rigorously derive trajectory-following recursive relations that converge exponentially fast, and construct a memory-efficient Monte Carlo scheme for derivatives of the output statistics. Thanks to the regularization and lack of simplifying assumptions on the system's behavior, our method is immune to the common problems of other popular methods such as the exploding tangent solutions and unphysical shadowing directions. Here, we provide a ready-to-use algorithm, analyze its complexity, and demonstrate several numerical examples of sensitivity computation using physically-inspired low-dimensional systems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Randomized Algorithms for Symmetric Nonnegative Matrix Factorization

Symmetric Nonnegative Matrix Factorization (SymNMF) is a technique in data analysis and machine learning that approximates a matrix with a product of a nonnegative, low-rank matrix and it transpose. To design faster and more scalable algorithms for SymNMF we develop two randomized algorithms for its computation. The first method uses randomized matrix sketching to compute an initial low-rank approximation to the input matrix and proceeds to uses this as a low-rank input to rapidly compute a SymNMF. The second methods uses randomized leverage score sampling to approximately solve constrained least squares problems. Many successful methods for SymNMF rely on (approximately) solving sequences of constrained least squares problems. Here, we prove theoretically that leverage score sampling can approximately solve constrained least squares problems to e-accuracy. Finally we demonstrate both methods work in practice by applying them to graph clustering tasks on large real world data sets. These experiments show that our methods approximately maintain solution quality and achieve significant speed ups for both large dense and large sparse problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Enhancing ZFP: A Statistical Approach to Understanding and Reducing Error Bias in a Lossy Floating-Point Compression Algorithm

The amount of data generated and gathered in scientific simulations and data collection applications is continuously growing, putting mounting pressure on storage and bandwidth concerns. A means of reducing such issues is data compression; but, lossless data compression is typically ineffective when applied to floating-point data. Thus, users tend to apply a lossy data compressor, which allows for small deviations from the original data. It is essential to understand how the error from lossy compression impacts the accuracy of the data analytics. Thus, we must analyze not only the compression properties but the error as well. In this paper, we provide a statistical analysis of the error caused by ZFP compression, a state-of-the-art, lossy compression algorithm explicitly designed for floating-point data. We show that the error is indeed biased and propose simple modifications to the algorithm to neutralize the bias and further reduce the resulting error.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Fast and Scalable FFT-Based GPU-Accelerated Algorithms for Block-Triangular Toeplitz Matrices with Application to Linear Inverse Problems Governed by Autonomous Dynamical Systems

In this work, we present an efficient and scalable algorithm for performing matrix-vector multiplications (matvecs) for block Toeplitz matrices. Such matrices, which are shift-invariant with respect to their blocks, arise in the context of solving inverse problems governed by autonomous systems, and time-invariant systems in particular. In this article, we consider inverse problems that infer unknown parameters from observational data of a linear time-invariant dynamical system given in the form of partial differential equations (PDEs). Matrix-free Newton-conjugate-gradient methods are often the gold standard for solving these inverse problems, but they require numerous actions of the Hessian on a vector. Matrix-free adjoint-based Hessian matvecs require solution of a pair of linearized forward/adjoint PDE solves per Hessian action, which may be prohibitive for large-scale inverse problems. Time invariance of the forward PDE problem leads to a block Toeplitz structure of the discretized parameter-to-observable (p2o) map defining the mapping from inputs (parameters) to outputs (observables) of the PDEs. This block Toeplitz structure enables us to exploit two key properties: (1) compact storage of the p2o map and its adjoint, and (2) efficient fast Fourier transform–based Hessian matvecs. The proposed algorithm is mapped onto large multi-GPU clusters and achieves more than 80% of peak bandwidth on NVIDIA A100 GPUs. Excellent weak scaling is shown for up to 48 A100 GPUs. For the targeted problems, the implementation executes Hessian matvecs within fractions of a second, which is orders of magnitude faster than can be achieved by conventional matrix-free Hessian matvecs via forward/adjoint PDE solves.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

ATLAS flavour-tagging algorithms for the LHC Run 2 $pp$ collision dataset

The flavour-tagging algorithms developed by the ATLAS Collaboration and used to analyse its dataset of $\sqrt{s}$ = 13 TeV $pp$ collisions from Run 2 of the Large Hadron Collider are presented. These new tagging algorithms are based on recurrent and deep neural networks, and their performance is evaluated in simulated collision events. These developments yield considerable improvements over previous jet-flavour identification strategies. At the 77% $b$-jet identification efficiency operating point, light-jet (charm-jet) rejection factors of 170 (5) are achieved in a sample of simulated Standard Model $t\bar{t}$ events; similarly, at a $c$-jet identification efficiency of 30%, a light-jet ($b$-jet) rejection factor of 70 (9) is obtained.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

Session Introduction: Graph Representations and Algorithms in Biomedicine

Connectivity is a fundamental property of biological systems: on the cellular level, proteins interact with each other to form protein-protein interaction networks (PPIs); on the organism level, neurons are arranged in a network; and on a community-level, species can have complex relationships with one another that drive the development and balance of an ecosystem. Graphs, representations of systems consisting of entities as vertices and their connections as edges, are a useful structure to characterize many such systems. Such models can be used to understand biological systems that naturally have a network structure, including PPIs, biological neurons, and ecosystems. In today’s information age, graph representations and algorithms (often in combination with machine learning techniques) are used to organize massive amounts of related data, much of which may be heterogeneous or unstructured, and identify patterns that represent novel biological insights. PSB’s 2023 session “Graph Representations and algorithms in Biomedicine,” encompasses modern developments in graph theory and its applications to various fields of biomedicine. This session includes a wide range of research - knowledge graphs built from text-mined health data, heterogeneous networks using multi-omic databases, and graphs refined to represent uncertainty or improve memory usage.

Chrisman, Brianna S.↗
DOE OSTI2018

A 3D Parallel Algorithm for QR Decomposition

Interprocessor communication often dominates the runtime of large matrix computations. Here, we present a parallel algorithm for computing QR decompositions whose bandwidth cost (communication volume) can be decreased at the cost of increasing its latency cost (number of messages). By varying a parameter to navigate the bandwidth/latency tradeoff, we can tune this algorithm for machines with different communication costs.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Abstractions and Directives for Adapting Wavefront Algorithms to Future Architectures

Architectures are rapidly evolving, and exascale machines are expected to offer billion-way concurrency. We need to rethink algorithms, languages and programming models among other components in order to migrate large scale applications and explore parallelism on these machines. Although directive-based programming models allow programmers to worry less about programming and more about science, expressing complex parallel patterns in these models can be a daunting task especially when the goal is to match the performance that the hardware platforms can offer. One such pattern is wavefront. This paper extensively studies a wavefront-based miniapplication for Denovo, a production code for nuclear reactor modeling. We parallelize the Koch-Baker-Alcouffe (KBA) parallel-wavefront sweep algorithm in the main kernel of Minisweep (the miniapplication) using CUDA, OpenMP and OpenACC. Our OpenACC implementation running on NVIDIA's next-generation Volta GPU boasts an 85.06x speedup over serial code, which is larger than CUDA's 83.72x speedup over the same serial implementation. Our experimental platform includes SummitDev, an ORNL representative architecture of the upcoming Summit supercomputer. Our parallelization effort across platforms also motivated us to define an abstract parallelism model that is architecture independent, with a goal of creating software abstractions that can be used by applications employing the wavefront sweep motif.

Searles, Robert↗
DOE OSTI2020

waveSZ: A Hardware-Algorithm Co-Design of Efficient Lossy Compression for Scientific Data

Error-bounded lossy compression is critical to the success of extreme-scale scientific research because of ever-increasing volumes of data produced by today's high-performance computing (HPC) applications. Not only can error-controlled lossy compressors significantly reduce the I/O and storage burden but they can retain high data fidelity for post analysis. Existing state-of-the-art lossy compressors, however, generally suffer from relatively low compression and decompression throughput (up to hundreds of megabytes per second on a single CPU core), which considerably restrict the adoption of lossy compression by many HPC applications especially those with a fairly high data production rate. In this paper, we propose a highly efficient lossy compression approach based on field programmable gate arrays (FPGAs) under the state-of-the-art lossy compression model SZ. Our contributions are fourfold. (1) We adopt a wavefront memory layout to alleviate the data dependency during the prediction for higher-dimensional predictors, such as the Lorenzo predictor. (2) We propose a co-design framework named WAVESZ based on the wavefront memory layout and the characteristics of SZ algorithm and carefully implement it by using high-level synthesis. (3) We propose a hardware-algorithm co-optimization method to improve the performance. (4) We evaluate our proposed WAVESZ on three real-world HPC simulation datasets from the Scientific Data Reduction Benchmarks and compare it with other state-of-the-art methods on both CPUs and FPGAs. Experiments show that our WAVESZ can improve SZ's compression throughput by 6.9x similar to 8.7x over the production version running on a state-of-the-art CPU and improve the compression ratio and throughput by 2.1x and 5.8x on average, respectively, respectively, compared with the state-of-the-art FPGA design.

Compression Ratio↗
DOE OSTI2020

Neuromorphic Graph Algorithms: Extracting Longest Shortest Paths and Minimum Spanning Trees

Neuromorphic computing is poised to become a promising computing paradigm in the post Moore's law era due to its extremely low power usage and inherent parallelism. Traditionally speaking, a majority of the use cases for neuromorphic systems have been in the field of machine learning. In order to expand their usability, it is imperative that neuromorphic systems be used for non-machine learning tasks as well. The structural aspects of neuromorphic systems (i.e., neurons and synapses) are similar to those of graphs (i.e., nodes and edges), However, it is not obvious how graph algorithms would translate to their neuromorphic counterparts. In this work, we propose a preprocessing technique that introduces fractional offsets on the synaptic delays of neuromorphic graphs in order to break ties. This technique, in turn, enables two graph algorithms: longest shortest path extraction and minimum spanning trees.

Kay, Bill↗
DOE OSTI2020

Programming Strategies for Irregular Algorithms on the Emu Chick

The Emu Chick prototype implements migratory memory-side processing in a novel hardware system. Rather than transferring large amounts of data across the system interconnect, the Emu Chick moves lightweight thread contexts to near-memory cores before the beginning of each remote memory read. Previous work has characterized the performance of the Chick prototype in terms of memory bandwidth and programming differences from more typical, non-migratory platforms, but there has not yet been an analysis of algorithms on this system. This work evaluates irregular algorithms that could benefit from the lightweight, memory-side processing of the Chick and demonstrates techniques and optimization strategies for achieving performance in sparse matrix-vector multiply operation (SpMV), breadth-first search (BFS), and graph alignment across up to eight distributed nodes encompassing 64 nodelets in the Chick system. We also define and justify relative metrics to compare prototype FPGA-based hardware with established ASIC architectures. The Chick currently supports up to 68x scaling for graph alignment, 80 MTEPS for BFS on balanced graphs, and 50% of measured STREAM bandwidth for SpMV.

Hein, Eric↗
DOE OSTI2022

CEAZ: Accelerating Parallel I/O Via Hardware-Algorithm Co-Designed Adaptive Lossy Compression

As supercomputers continue to grow to exa-scale, the amount of data that needs to be saved or transmitted is exploding. To this end, many previous works have studied using error-bounded lossy compressors to reduce the data size and improve the I/O performance. However, little work has been done for effectively offloading lossy compression onto FPGA-based SmartNICs to reduce the compression overhead. In this paper, we propose a hardware-algorithm co-design of efficient and adaptive lossy compressor for scientific data on FPGAs (called CEAZ) to accelerate parallel I/O. Our contribution is fourfold: (1) We propose an efficient Huffman coding approach that can adaptively update Huffman codewords online based on codewords generated offline (from a variety of representative scientific datasets). (2) We derive a theoretical analysis to support a precise control of compression ratio under an error-bounded compression mode, enabling accurate offline Huffman codewords generation. This also help us create a fixed-ratio compression mode for consistent throughput. (3) We develop an efficient compression pipeline by adopting cuSZ’s dual-quantization algorithm to our hardware use case. (4) We evaluate CEAC on five real-world datasets with both a single FPGA board and 256 nodes from Bridges2 supercomputer. Experiments show that CEAZ outperforms the second-best FPGA-based lossy compressor by 2× of throughput and 9.6× of compression ratio. It also improves MPI_File_write and MPI_Gather throughputs by up to 32.7× and 31.4×, respectively.

Zhang, Chengming↗
DOE OSTI2022

Training reinforcement learning models via an adversarial evolutionary algorithm

When training for control problems, more episodes used in training usually leads to better generalizability, but more episodes also requires significantly more training time. There are a variety of approaches for selecting the way that training episodes are chosen, including fixed episodes, uniform sampling, and stochastic sampling, but they can all leave gaps in the training landscape. In this work, we describe an approach that leverages an adversarial evolutionary algorithm to identify the worst performing states for a given model. We then use information about these states in the next cycle of training; this process can be repeated until the desired level of model performance is met. We demonstrate this approach with the OpenAI Gym cart-pole problem. With this problem, we show that the adversarial evolutionary algorithm did not reduce the number of episodes required in training needed to attain model generalizability when compared with stochastic sampling, and actually performed slightly worse.

Coletti, Mark↗
DOE OSTI2024

Algorithm 1049: The Delaunay Density Diagnostic

Accurate approximation of a real-valued function depends on two aspects of the available data: the density of inputs within the domain of interest and the variation of the outputs over that domain. There are few methods for assessing whether the density of inputs is sufficient to identify the relevant variations in outputs—i.e., the “geometric scale” of the function—despite the fact that sampling density is closely tied to the success or failure of an approximation method. In this article, we introduce a general purpose, computational approach to detecting the geometric scale of real-valued functions over a fixed domain using a deterministic interpolation technique from computational geometry. The algorithm is intended to work on scalar data in moderate dimensions (2–10). Our algorithm is based on the observation that a sequence of piecewise linear interpolants will converge to a continuous function at a quadratic rate (in L 2 norm) if and only if the data are sampled densely enough to distinguish the feature from noise (assuming sufficiently regular sampling). We present numerical experiments demonstrating how our method can identify feature scale, estimate uncertainty in feature scale, and assess the sampling density for fixed (i.e., static) datasets of input–output pairs. Finally, we include analytical results in support of our numerical findings and have released lightweight code that can be adapted for use in a variety of data science settings.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Communication Lower Bounds and Optimal Algorithms for Symmetric Matrix Computations

In this article, we focus on the communication costs of three symmetric matrix computations: (i) multiplying a matrix with its transpose, known as a symmetric rank-k update (SYRK) (ii) adding the result of the multiplication of a matrix with the transpose of another matrix and the transpose of that result, known as a symmetric rank-2k update (SYR2K) (iii) performing matrix multiplication with a symmetric input matrix (SYMM). All three computations appear in the Level 3 Basic Linear Algebra Subroutines (BLAS) and have wide use in applications involving symmetric matrices. We establish communication lower bounds for these kernels using sequential and distributed-memory parallel computational models, and we show that our bounds are tight by presenting communication-optimal algorithms for each setting. Our lower bound proofs rely on applying a geometric inequality for symmetric computations and analytically solving constrained nonlinear optimization problems. As a result, the symmetric matrix and its corresponding computations are accessed and performed according to a triangular block partitioning scheme in the optimal algorithms.

Al Daas, Hussam [Rutherford Appleton Laboratory, D↗