Novel Monte Carlo Methods for Large-Scale Linear Algebra Operations

Linear algebra operations play an important role in scientific computing and data analysis. With increasing data volume and complexity in the Big Data era, linear algebra operations are important tools to process massive datasets. On one hand, the advent of modern high-performance computing architectures with increasing computing power has greatly enhanced our capability to deal with a large volume of data. One the other hand, many classical, deterministic numerical linear algebra algorithms have difficulty to scale to handle large data sets. Monte Carlo methods, which are based on statistical sampling, exhibit many attractive properties in dealing with large volume of datasets, including fast approximated results, memory efficiency, reduced data accesses, natural parallelism, and inherent fault tolerance. In this dissertation, we present new Monte Carlo methods to accommodate a set of fundamental and ubiquitous large-scale linear algebra operations, including solving large-scale linear systems, constructing low-rank matrix approximation, and approximating the extreme eigenvalues/ eigenvectors, across modern distributed and parallel computing architectures. First of all, we revisit the classical Ulam-von Neumann Monte Carlo algorithm and derive the necessary and sufficient condition for its convergence. To support a broad family of linear systems, we develop Krylov subspace Monte Carlo solvers that go beyond the use of Neumann series. New algorithms used in the Krylov subspace Monte Carlo solvers include (1) a Breakdown-Free Block Conjugate Gradient algorithm to address the potential rank deficiency problem occurred in block Krylov subspace methods; (2) a Block Conjugate Gradient for Least Squares algorithm to stably approximate the least squares solutions of general linear systems; (3) a BCGLS algorithm with deflation to gain convergence acceleration; and (4) a Monte Carlo Generalized Minimal Residual algorithm based on sampling matrix-vector products to provide fast approximation of solutions. Secondly, we design a rank-revealing randomized Singular Value Decomposition (R3SVD) algorithm for adaptively constructing low-rank matrix approximations to satisfy application-specific accuracy. Thirdly, we study the block power method on Markov Chain Monte Carlo transition matrices and find that the convergence is actually depending on the number of independent vectors in the block. Correspondingly, we develop a sliding window power method to find stationary distribution, which has demonstrated success in modeling stochastic luminal Calcium release site. Fourthly, we take advantage of hybrid CPU-GPU computing platforms to accelerate the performance of the Breakdown-Free Block Conjugate Gradient algorithm and the randomized Singular Value Decomposition algorithm. Finally, we design a Gaussian variant of Freivalds’ algorithm to efficiently verify the correctness of matrix-matrix multiplication while avoiding undetectable fault patterns encountered in deterministic algorithms

Extensions of Task-based Runtime for High Performance Dense Linear Algebra Applications

On the road to exascale computing, the gap between hardware peak performance and application performance is increasing as system scale, chip density and inherent complexity of modern supercomputers are expanding. Even if we put aside the difficulty to express algorithmic parallelism and to efficiently execute applications at large scale, other open questions remain. The ever-growing scale of modern supercomputers induces a fast decline of the Mean Time To Failure. A generic, low-overhead, resilient extension becomes a desired aptitude for any programming paradigm. This dissertation addresses these two critical issues, designing an efficient unified linear algebra development environment using a task-based runtime, and extending a task-based runtime with fault tolerant capabilities to build a generic framework providing both soft and hard error resilience to task-based programming paradigm. To bridge the gap between hardware peak performance and application perfor- mance, a unified programming model is designed to take advantage of a lightweight task-based runtime to manage the resource-specific workload, and to control the data ow and parallel execution of tasks. Under this unified development, linear algebra tasks are abstracted across different underlying heterogeneous resources, including multicore CPUs, GPUs and Intel Xeon Phi coprocessors. Performance portability is guaranteed and this programming model is adapted to a wide range of accelerators, supporting both shared and distributed-memory environments. To solve the resilient challenges on large scale systems, fault tolerant mechanisms are designed for a task-based runtime to protect applications against both soft and hard errors. For soft errors, three additions to a task-based runtime are explored. The first recovers the application by re-executing minimum number of tasks, the second logs intermediary data between tasks to minimize the necessary re-execution, while the last one takes advantage of algorithmic properties to recover the data without re- execution. For hard errors, we propose two generic approaches, which augment the data logging mechanism for soft errors. The first utilizes non-volatile storage device to save logged data, while the second saves local logged data on a remote node to protect against node failure. Experimental results have confirmed that our soft and hard error fault tolerant mechanisms exhibit the expected correctness and efficiency

Using reconfigurable computing technology to accelerate matrix decomposition and applications

Matrix decomposition plays an increasingly significant role in many scientific and engineering applications. Among numerous techniques, Singular Value Decomposition (SVD) and Eigenvalue Decomposition (EVD) are widely used as factorization tools to perform Principal Component Analysis for dimensionality reduction and pattern recognition in image processing, text mining and wireless communications, while QR Decomposition (QRD) and sparse LU Decomposition (LUD) are employed to solve the dense or sparse linear system of equations in bioinformatics, power system and computer vision. Matrix decompositions are computationally expensive and their sequential implementations often fail to meet the requirements of many time-sensitive applications. The emergence of reconfigurable computing has provided a flexible and low-cost opportunity to pursue high-performance parallel designs, and the use of FPGAs has shown promise in accelerating this class of computation. In this research, we have proposed and implemented several highly parallel FPGA-based architectures to accelerate matrix decompositions and their applications in data mining and signal processing. Specifically, in this dissertation we describe the following contributions: • We propose an efficient FPGA-based double-precision floating-point architecture for EVD, which can efficiently analyze large-scale matrices. • We implement a floating-point Hestenes-Jacobi architecture for SVD, which is capable of analyzing arbitrary sized matrices. • We introduce a novel deeply pipelined reconfigurable architecture for QRD, which can be dynamically configured to perform either Householder transformation or Givens rotation in a manner that takes advantage of the strengths of each. • We design a configurable architecture for sparse LUD that supports both symmetric and asymmetric sparse matrices with arbitrary sparsity patterns. • By further extending the proposed hardware solution for SVD, we parallelize a popular text mining tool-Latent Semantic Indexing with an FPGA-based architecture. • We present a configurable architecture to accelerate Homotopy l1-minimization, in which the modification of the proposed FPGA architecture for sparse LUD is used at its core to parallelize both Cholesky decomposition and rank-1 update. Our experimental results using an FPGA-based acceleration system indicate the efficiency of our proposed novel architectures, with application and dimension-dependent speedups over an optimized software implementation that range from 1.5ÃÂ to 43.6ÃÂ in terms of computation time

The fast multipole method at exascale

This thesis presents a top to bottom analysis on designing and implementing fast algorithms for current and future systems. We present new analysis, algorithmic techniques, and implementations of the Fast Multipole Method (FMM) for solving N- body problems. We target the FMM because it is broadly applicable to a variety of scientific particle simulations used to study electromagnetic, fluid, and gravitational phenomena, among others. Importantly, the FMM has asymptotically optimal time complexity with guaranteed approximation accuracy. As such, it is among the most attractive solutions for scalable particle simulation on future extreme scale systems. We specifically address two key challenges. The first challenge is how to engineer fast code for today’s platforms. We present the first in-depth study of multicore op- timizations and tuning for FMM, along with a systematic approach for transforming a conventionally-parallelized FMM into a highly-tuned one. We introduce novel opti- mizations that significantly improve the within-node scalability of the FMM, thereby enabling high-performance in the face of multicore and manycore systems. The second challenge is how to understand scalability on future systems. We present a new algorithmic complexity analysis of the FMM that considers both intra- and inter- node communication costs. Using these models, we present results for choosing the optimal algorithmic tuning parameter. This analysis also yields the surprising prediction that although the FMM is largely compute-bound today, and therefore highly scalable on current systems, the trajectory of processor architecture designs, if there are no significant changes could cause it to become communication-bound as early as the year 2015. This prediction suggests the utility of our analysis approach, which directly relates algorithmic and architectural characteristics, for enabling a new kind of highlevel algorithm-architecture co-design. To demonstrate the scientific significance of FMM, we present two applications namely, direct simulation of blood which is a multi-scale multi-physics problem and large-scale biomolecular electrostatics. MoBo (Moving Boundaries) is the infrastruc- ture for the direct numerical simulation of blood. It comprises of two key algorithmic components of which FMM is one. We were able to simulate blood flow using Stoke- sian dynamics on 200,000 cores of Jaguar, a peta-flop system and achieve a sustained performance of 0.7 Petaflop/s. The second application we propose as future work in this thesis is biomolecular electrostatics where we solve for the electrical potential using the boundary-integral formulation discretized with boundary element methods (BEM). The computational kernel in solving the large linear system is dense matrix vector multiply which we propose can be calculated using our scalable FMM. We propose to begin with the two dielectric problem where the electrostatic field is cal- culated using two continuum dielectric medium, the solvent and the molecule. This is only a first step to solving biologically challenging problems which have more than two dielectric medium, ion-exclusion layers, and solvent filled cavities. Finally, given the difficulty in producing high-performance scalable code, productivity is a key concern. Recently, numerical algorithms are being redesigned to take advantage of the architectural features of emerging multicore processors. These new classes of algorithms express fine-grained asynchronous parallelism and hence reduce the cost of synchronization. We performed the first extensive performance study of a recently proposed parallel programming model, called Concurrent Collections (CnC). In CnC, the programmer expresses her computation in terms of application-specific operations, partially-ordered by semantic scheduling constraints. The CnC model is well-suited to expressing asynchronous-parallel algorithms, so we evaluate CnC using two dense linear algebra algorithms in this style for execution on state-of-the-art mul- ticore systems. Our implementations in CnC was able to match and in some cases even exceed competing vendor-tuned and domain specific library codes. We combine these two distinct research efforts by expressing FMM in CnC, our approach tries to marry performance with productivity that will be critical on future systems. Looking forward, we would like to extend this to distributed memory machines, specifically implement FMM in the new distributed CnC, distCnC to express fine-grained paral- lelism which would require significant effort in alternative models.Ph.D

Randomized Rank-Revealing QLP for Low-Rank Matrix Decomposition

The pivoted QLP decomposition is computed through two consecutive pivoted QR decompositions, and provides an approximation to the singular value decomposition. This work is concerned with a partial QLP decomposition of low-rank matrices computed through randomization, termed Randomized Unpivoted QLP (RU-QLP). Like pivoted QLP, RU-QLP is rank-revealing and yet it utilizes random column sampling and the unpivoted QR decomposition. The latter modifications allow RU-QLP to be highly parallelizable on modern computational platforms. We provide an analysis for RU-QLP, deriving bounds in spectral and Frobenius norms on: i) the rank-revealing property; ii) principal angles between approximate subspaces and exact singular subspaces and vectors; and iii) low-rank approximation errors. Effectiveness of the bounds is illustrated through numerical tests. We further use a modern, multicore machine equipped with a GPU to demonstrate the efficiency of RU-QLP. Our results show that compared to the randomized SVD, RU-QLP achieves a speedup of up to 7.1 times on the CPU and up to 2.3 times with the GPU

Task-based Runtime Optimizations Towards High Performance Computing Applications

The last decades have witnessed a rapid improvement of computational capabilities in high-performance computing (HPC) platforms thanks to hardware technology scaling. HPC architectures benefit from mainstream advances on the hardware with many-core systems, deep hierarchical memory subsystem, non-uniform memory access, and an ever-increasing gap between computational power and memory bandwidth. This has necessitated continuous adaptations across the software stack to maintain high hardware utilization. In this HPC landscape of potentially million-way parallelism, task-based programming models associated with dynamic runtime systems are becoming more popular, which fosters developers’ productivity at extreme scale by abstracting the underlying hardware complexity. In this context, this dissertation highlights how a software bundle powered by a task-based programming model can address the heterogeneous workloads engendered by HPC applications., i.e., data redistribution, geospatial modeling and 3D unstructured mesh deformation here. Data redistribution aims to reshuffle data to optimize some objective for an algorithm, whose objective can be multi-dimensional, such as improving computational load balance or decreasing communication volume or cost, with the ultimate goal of increasing the efficiency and therefore reducing the time-to-solution for the algorithm. Geostatistical modeling, one of the prime motivating applications for exascale computing, is a technique for predicting desired quantities from geographically distributed data, based on statistical models and optimization of parameters. Meshing the deformable contour of moving 3D bodies is an expensive operation that can cause huge computational challenges in fluid-structure interaction (FSI) applications. Therefore, in this dissertation, Redistribute-PaRSEC, ExaGeoStat-PaRSEC and HiCMA-PaRSEC are proposed to efficiently tackle these HPC applications respectively at extreme scale, and they are evaluated on multiple HPC clusters, including AMD-based, Intel-based, Arm-based CPU systems and IBM-based multi-GPU system. This multidisciplinary work emphasizes the need for runtime systems to go beyond their primary responsibility of task scheduling on massively parallel hardware system for servicing the next-generation scientific applications

Evaluation of Distributed Programming Models and Extensions to Task-based Runtime Systems

High Performance Computing (HPC) has always been a key foundation for scientific simulation and discovery. And more recently, deep learning models\u27 training have further accelerated the demand of computational power and lower precision arithmetic. In this era following the end of Dennard\u27s Scaling and when Moore\u27s Law seemingly still holds true to a lesser extent, it is not a coincidence that HPC systems are equipped with multi-cores CPUs and a variety of hardware accelerators that are all massively parallel. Coupling this with interconnect networks\u27 speed improvements lagging behind those of computational power increases, the current state of HPC systems is heterogeneous and extremely complex. This was heralded as a great challenge to the software stacks and their ability to extract performance from these systems, but also as a great opportunity to innovate at the programming model level to explore the different approaches and propose new solutions. With usability, portability, and performance as the main factors to consider, this dissertation first evaluates some of the widely used parallel programming models (MPI, MPI+OpenMP, and task-based runtime systems) ability to manage the load imbalance among the processes computing the LU factorization of a large dense matrix stored in the Block Low-Rank (BLR) format. Next I proposed a number of optimizations and implemented them in PaRSEC\u27s Dynamic Task Discovery (DTD) model, including user-level graph trimming and direct Application Programming Interface (API) calls to perform data broadcast operation to further extend the limit of STF model. On the other hand, the Parameterized Task Graph (PTG) approach in PaRSEC is the most scalable approach for many different applications, which I then explored the possibility of combining both the algorithmic approach of Communication-Avoiding (CA) and the communication-computation overlapping benefits provided by runtime systems using 2D five-point stencil as the test case. This broad programming models evaluation and extension work highlighted the abilities of task-based runtime system in achieving scalable performance and portability on contemporary heterogeneous HPC systems. Finally, I summarized the profiling capability of PaRSEC runtime system, and demonstrated with a use case its important role in the performance bottleneck identification leading to optimizations