arXiv subjects

David E. Keyes

Publications and source records attributed to David E. Keyes.

30 records · Page 2Linked to original sources

arXiv2021

High Performance Multivariate Geospatial Statistics on Manycore Systems

Modeling and inferring spatial relationships and predicting missing values of environmental data are some of the main tasks of geospatial statisticians. These routine tasks are accomplished using multivariate geospatial models and the cokriging technique. The latter requires the evaluation of the expensive Gaussian log-likelihood function, which has impeded the adoption of multivariate geospatial models for large multivariate spatial datasets. However, this large-scale cokriging challenge provides a fertile ground for supercomputing implementations for the geospatial statistics community as it is paramount to scale computational capability to match the growth in environmental data coming from the widespread use of different data collection technologies. In this paper, we develop and deploy large-scale multivariate spatial modeling and inference on parallel hardware architectures. To tackle the increasing complexity in matrix operations and the massive concurrency in parallel systems, we leverage low-rank matrix approximation techniques with task-based programming models and schedule the asynchronous computational tasks using a dynamic runtime system. The proposed framework provides both the dense and the approximated computations of the Gaussian log-likelihood function. It demonstrates accuracy robustness and performance scalability on a variety of computer systems. Using both synthetic and real datasets, the low-rank matrix approximation shows better performance compared to exact computation, while preserving the application requirements in both parameter estimation and prediction accuracy. We also propose a novel algorithm to assess the prediction accuracy after the online parameter estimation. The algorithm quantifies prediction performance and provides a benchmark for measuring the efficiency and accuracy of several approximation techniques in multivariate spatial modeling.

cs.DC↗

arXiv2020

Exploiting Low Rank Covariance Structures for Computing High-Dimensional Normal and Student-$t$ Probabilities

We present a preconditioned Monte Carlo method for computing high-dimensional multivariate normal and Student-$t$ probabilities arising in spatial statistics. The approach combines a tile-low-rank representation of covariance matrices with a block-reordering scheme for efficient Quasi-Monte Carlo simulation. The tile-low-rank representation decomposes the high-dimensional problem into many diagonal-block-size problems and low-rank connections. The block-reordering scheme reorders between and within the diagonal blocks to reduce the impact of integration variables from right to left, thus improving the Monte Carlo convergence rate. Simulations up to dimension $65{,}536$ suggest that the new method can improve the run time by an order of magnitude compared with the non-reordered tile-low-rank Quasi-Monte Carlo method and two orders of magnitude compared with the dense Quasi-Monte Carlo method. Our method also forms a strong substitute for the approximate conditioning methods as a more robust estimation with error guarantees. An application study to wind stochastic generators is provided to illustrate that the new computational method makes the maximum likelihood estimation feasible for high-dimensional skew-normal random fields.

stat.CO↗

arXiv2020

Geostatistical Modeling and Prediction Using Mixed-Precision Tile Cholesky Factorization

Geostatistics represents one of the most challenging classes of scientific applications due to the desire to incorporate an ever increasing number of geospatial locations to accurately model and predict environmental phenomena. For example, the evaluation of the Gaussian log-likelihood function, which constitutes the main computational phase, involves solving systems of linear equations with a large dense symmetric and positive definite covariance matrix. Cholesky, the standard algorithm, requires O(n^3) floating point operators and has an O(n^2) memory footprint, where n is the number of geographical locations. Here, we present a mixed-precision tile algorithm to accelerate the Cholesky factorization during the log-likelihood function evaluation. Under an appropriate ordering, it operates with double-precision arithmetic on tiles around the diagonal, while reducing to single-precision arithmetic for tiles sufficiently far off. This translates into an improvement of the performance without any deterioration of the numerical accuracy of the application. We rely on the StarPU dynamic runtime system to schedule the tasks and to overlap them with data movement. To assess the performance and the accuracy of the proposed mixed-precision algorithm, we use synthetic and real datasets on various shared and distributed-memory systems possibly equipped with hardware accelerators. We compare our mixed-precision Cholesky factorization against the double-precision reference implementation as well as an independent block approximation method. We obtain an average of 1.6X performance speedup on massively parallel architectures while maintaining the accuracy necessary for modeling and prediction.

cs.DC↗

arXiv2019

On the robustness and performance of entropy stable discontinuous collocation methods for the compressible Navier-Stokes equations

In computational fluid dynamics, the demand for increasingly multidisciplinary reliable simulations, for both analysis and design optimization purposes, requires transformational advances in individual components of future solvers. At the algorithmic level, hardware compatibility and efficiency are of paramount importance in determining viability at exascale and beyond. However, equally important (if not more so) is algorithmic robustness with minimal user intervention, which becomes progressively more challenging to achieve as problem size and physics complexity increase. We numerically show that low and high order entropy stable discontinuous spatial discretizations based on summation-by-part operators and simultaneous-approximation-terms technique provides an essential step toward a truly enabling technology in terms of reliability and robustness for both under-resolved turbulent flow simulations and flows with discontinuities.

math.NA↗

arXiv2019

Hierarchical Matrix Operations on GPUs: Matrix-Vector Multiplication and Compression

Hierarchical matrices are space and time efficient representations of dense matrices that exploit the low rank structure of matrix blocks at different levels of granularity. The hierarchically low rank block partitioning produces representations that can be stored and operated on in near-linear complexity instead of the usual polynomial complexity of dense matrices. In this paper, we present high performance implementations of matrix vector multiplication and compression operations for the $\mathcal{H}^2$ variant of hierarchical matrices on GPUs. This variant exploits, in addition to the hierarchical block partitioning, hierarchical bases for the block representations and results in a scheme that requires only $O(n)$ storage and $O(n)$ complexity for the mat-vec and compression kernels. These two operations are at the core of algebraic operations for hierarchical matrices, the mat-vec being a ubiquitous operation in numerical algorithms while compression/recompression represents a key building block for other algebraic operations, which require periodic recompression during execution. The difficulties in developing efficient GPU algorithms come primarily from the irregular tree data structures that underlie the hierarchical representations, and the key to performance is to recast the computations on flattened trees in ways that allow batched linear algebra operations to be performed. This requires marshaling the irregularly laid out data in a way that allows them to be used by the batched routines. Marshaling operations only involve pointer arithmetic with no data movement and as a result have minimal overhead. Our numerical results on covariance matrices from 2D and 3D problems from spatial statistics show the high efficiency our routines achieve---over 550GB/s for the bandwidth-limited mat-vec and over 850GFLOPS/s in sustained performance for the compression on the P100 Pascal GPU.

cs.DS↗

arXiv2018

O(N) Hierarchical algorithm for computing the expectations of truncated multi-variate normal distributions in N dimensions

In this paper, we study the $N$-dimensional integral $ϕ(a,b; A) = \int_{a}^{b} H(x) f(x | A) \text{d} x$ representing the expectation of a function $H(X)$ where $f(x | A)$ is the truncated multi-variate normal (TMVN) distribution with zero mean, $x$ is the vector of integration variables for the $N$-dimensional random vector $X$, $A$ is the inverse of the covariance matrix $Σ$, and $a$ and $b$ are constant vectors. We present a new hierarchical algorithm which can evaluate $ϕ(a,b; A)$ using asymptotically optimal $O(N)$ operations when $A$ has "low-rank" blocks with "low-dimensional" features and $H(x)$ is "low-rank". We demonstrate the divide-and-conquer idea when $A$ is a symmetric positive definite tridiagonal matrix, and present the necessary building blocks and rigorous potential theory based algorithm analysis when $A$ is given by the exponential covariance model. Numerical results are presented to demonstrate the algorithm accuracy and efficiency for these two cases. We also briefly discuss how the algorithm can be generalized to a wider class of covariance models and its limitations.

math.NA↗

arXiv2018

ExaGeoStat: A High Performance Unified Software for Geostatistics on Manycore Systems

We present ExaGeoStat, a high performance framework for geospatial statistics in climate and environment modeling. In contrast to simulation based on partial differential equations derived from first-principles modeling, ExaGeoStat employs a statistical model based on the evaluation of the Gaussian log-likelihood function, which operates on a large dense covariance matrix. Generated by the parametrizable Matern covariance function, the resulting matrix is symmetric and positive definite. The computational tasks involved during the evaluation of the Gaussian log-likelihood function become daunting as the number n of geographical locations grows, as O(n2) storage and O(n3) operations are required. While many approximation methods have been devised from the side of statistical modeling to ameliorate these polynomial complexities, we are interested here in the complementary approach of evaluating the exact algebraic result by exploiting advances in solution algorithms and many-core computer architectures. Using state-of-the-art high performance dense linear algebra libraries associated with various leading edge parallel architectures (Intel KNLs, NVIDIA GPUs, and distributed-memory systems), ExaGeoStat raises the game for statistical applications from climate and environmental science. ExaGeoStat provides a reference evaluation of statistical parameters, with which to assess the validity of the various approaches based on approximation. The framework takes a first step in the merger of large-scale data analytics and extreme computing for geospatial statistical applications, to be followed by additional complexity reducing improvements from the solver side that can be implemented under the same interface. Thus, a single uncompromised statistical model can ultimately be executed in a wide variety of emerging exascale environments.

cs.DC↗

arXiv2018

Parallel Approximation of the Maximum Likelihood Estimation for the Prediction of Large-Scale Geostatistics Simulations

Maximum likelihood estimation is an important statistical technique for estimating missing data, for example in climate and environmental applications, which are usually large and feature data points that are irregularly spaced. In particular, the Gaussian log-likelihood function is the \emph{de facto} model, which operates on the resulting sizable dense covariance matrix. The advent of high performance systems with advanced computing power and memory capacity have enabled full simulations only for rather small dimensional climate problems, solved at the machine precision accuracy. The challenge for high dimensional problems lies in the computation requirements of the log-likelihood function, which necessitates ${\mathcal O}(n^2)$ storage and ${\mathcal O}(n^3)$ operations, where $n$ represents the number of given spatial locations. This prohibitive computational cost may be reduced by using approximation techniques that not only enable large-scale simulations otherwise intractable but also maintain the accuracy and the fidelity of the spatial statistics model. In this paper, we extend the Exascale GeoStatistics software framework (i.e., ExaGeoStat) to support the Tile Low-Rank (TLR) approximation technique, which exploits the data sparsity of the dense covariance matrix by compressing the off-diagonal tiles up to a user-defined accuracy threshold. The underlying linear algebra operations may then be carried out on this data compression format, which may ultimately reduce the arithmetic complexity of the maximum likelihood estimation and the corresponding memory footprint. Performance results of TLR-based computations on shared and distributed-memory systems attain up to 13X and 5X speedups, respectively, compared to full accuracy simulations using synthetic and real datasets (up to 2M), while ensuring adequate prediction accuracy.

math.NA↗

arXiv2018

Research and Education in Computational Science and Engineering

Over the past two decades the field of computational science and engineering (CSE) has penetrated both basic and applied research in academia, industry, and laboratories to advance discovery, optimize systems, support decision-makers, and educate the scientific and engineering workforce. Informed by centuries of theory and experiment, CSE performs computational experiments to answer questions that neither theory nor experiment alone is equipped to answer. CSE provides scientists and engineers of all persuasions with algorithmic inventions and software systems that transcend disciplines and scales. Carried on a wave of digital technology, CSE brings the power of parallelism to bear on troves of data. Mathematics-based advanced computing has become a prevalent means of discovery and innovation in essentially all areas of science, engineering, technology, and society; and the CSE community is at the core of this transformation. However, a combination of disruptive developments---including the architectural complexity of extreme-scale computing, the data revolution that engulfs the planet, and the specialization required to follow the applications to new frontiers---is redefining the scope and reach of the CSE endeavor. This report describes the rapid expansion of CSE and the challenges to sustaining its bold advances. The report also presents strategies and directions for CSE research and education for the next decade.

cs.CE↗

arXiv2017

Batched QR and SVD Algorithms on GPUs with Applications in Hierarchical Matrix Compression

We present high performance implementations of the QR and the singular value decomposition of a batch of small matrices hosted on the GPU with applications in the compression of hierarchical matrices. The one-sided Jacobi algorithm is used for its simplicity and inherent parallelism as a building block for the SVD of low rank blocks using randomized methods. We implement multiple kernels based on the level of the GPU memory hierarchy in which the matrices can reside and show substantial speedups against streamed cuSOLVER SVDs. The resulting batched routine is a key component of hierarchical matrix compression, opening up opportunities to perform H-matrix arithmetic efficiently on GPUs.

cs.MS↗

arXiv2015

Optimization of an electromagnetics code with multicore wavefront diamond blocking and multi-dimensional intra-tile parallelization

Understanding and optimizing the properties of solar cells is becoming a key issue in the search for alternatives to nuclear and fossil energy sources. A theoretical analysis via numerical simulations involves solving Maxwell's Equations in discretized form and typically requires substantial computing effort. We start from a hybrid-parallel (MPI+OpenMP) production code that implements the Time Harmonic Inverse Iteration Method (THIIM) with Finite-Difference Frequency Domain (FDFD) discretization. Although this algorithm has the characteristics of a strongly bandwidth-bound stencil update scheme, it is significantly different from the popular stencil types that have been exhaustively studied in the high performance computing literature to date. We apply a recently developed stencil optimization technique, multicore wavefront diamond tiling with multi-dimensional cache block sharing, and describe in detail the peculiarities that need to be considered due to the special stencil structure. Concurrency in updating the components of the electric and magnetic fields provides an additional level of parallelism. The dependence of the cache size requirement of the optimized code on the blocking parameters is modeled accurately, and an auto-tuner searches for optimal configurations in the remaining parameter space. We were able to completely decouple the execution from the memory bandwidth bottleneck, accelerating the implementation by a factor of three to four compared to an optimal implementation with pure spatial blocking on an 18-core Intel Haswell CPU.

cs.CE↗

arXiv2012

Optimizing the Performance of Streaming Numerical Kernels on the IBM Blue Gene/P PowerPC 450 Processor

Several emerging petascale architectures use energy-efficient processors with vectorized computational units and in-order thread processing. On these architectures the sustained performance of streaming numerical kernels, ubiquitous in the solution of partial differential equations, represents a challenge despite the regularity of memory access. Sophisticated optimization techniques are required to fully utilize the Central Processing Unit (CPU). We propose a new method for constructing streaming numerical kernels using a high-level assembly synthesis and optimization framework. We describe an implementation of this method in Python targeting the IBM Blue Gene/P supercomputer's PowerPC 450 core. This paper details the high-level design, construction, simulation, verification, and analysis of these kernels utilizing a subset of the CPU's instruction set. We demonstrate the effectiveness of our approach by implementing several three-dimensional stencil kernels over a variety of cached memory scenarios and analyzing the mechanically scheduled variants, including a 27-point stencil achieving a 1.7x speedup over the best previously published results.

cs.PF↗