In this paper, we investigate GPU based parallel triangular solvers systematically. The parallel triangular solvers are fundamental to incomplete LU factorization family preconditioners and algebraic multigrid solvers. We develop a new matrix format suitable for GPU devices. Parallel lower triangular solvers and upper triangular solvers are developed for this new data structure. With these solvers, ILU preconditioners and domain decomposition preconditioners are developed. Numerical results show that we can speed triangular solvers around seven times faster.
This research introduce our work on developing Krylov subspace and AMG solvers on NVIDIA GPUs. As SpMV is a crucial part for these iterative methods, SpMV algorithms for single GPU and multiple GPUs are implemented. A HEC matrix format and a communication mechanism are established. And also, a set of specific algorithms for solving preconditioned systems in parallel environments are designed, including ILU(k), RAS and parallel triangular solvers. Based on these work, several Krylov solvers and AMG solvers are developed. According to numerical experiments, favorable acceleration performance is …
Driven by the insatiable needs to process ever larger amount of data with more complex models, modern computer processors and accelerators are beginning to offer half precision floating point arithmetic support, and extremely optimized special units such as NVIDIA TensorCore on GPU and Google Tensor Processing Unit (TPU) that does half precision matrix-matrix multiplication exceptionally efficiently. In this paper we present a large scale mixed precision linear least square solver that achieves high accuracy using the low precision TensorCore GPU. The mixed precision system consists of both in…
In this paper we modify a fast heuristic solver for the Linear Sum Assignment Problem (LSAP) for use on Graphical Processing Units (GPUs). The motivating scenario is an industrial application for P2P live streaming that is moderated by a central node which is periodically solving LSAP instances for assigning peers to one another. The central node needs to handle LSAP instances involving thousands of peers in as near to real-time as possible. Our findings are generic enough to be applied in other contexts. Our main result is a parallel version of a heuristic algorithm called Deep Greedy Switchi…
This paper gives the first algorithm for finding a set of natural $ε$-clusters of complex zeros of a triangular system of polynomials within a given polybox in $\mathbb{C}^n$, for any given $ε>0$. Our algorithm is based on a recent near-optimal algorithm of Becker et al (2016) for clustering the complex roots of a univariate polynomial where the coefficients are represented by number oracles. Our algorithm is numeric, certified and based on subdivision. We implemented it and compared it with two well-known homotopy solvers on various triangular systems. Our solver always gives correct answers,…
This paper presents the implementation of a HLLC finite volume solver using GPU technology for the solution of shallow water problems in two dimensions. It compares both CPU and GPU approaches for implementing all the solver's steps. The technology of graphics and central processors is highlighted with a particular emphasis on the CUDA architecture of NVIDIA. The simple and well-documented Application Programming Interface (CUDA API) facilitates the use of the display card workstation as an additional computer unit to the central processor. Four professional solutions of the NVIDIA Quadro line…
The acceleration of sparse matrix computations on modern many-core processors, such as the graphics processing units (GPUs), has been recognized and studied over a decade. Significant performance enhancements have been achieved for many sparse matrix computational kernels such as sparse matrix-vector products and sparse matrix-matrix products. Solving linear systems with sparse triangular structured matrices is another important sparse kernel as demanded by a variety of scientific and engineering applications such as sparse linear solvers. However, the development of efficient parallel algorit…
Gauss-Seidel (GS) relaxation is often employed as a preconditioner for a Krylov solver or as a smoother for Algebraic Multigrid (AMG). However, the requisite sparse triangular solve is difficult to parallelize on many-core architectures such as graphics processing units (GPUs). In the present study, the performance of the traditional GS relaxation based on a triangular solve is compared with two-stage variants, replacing the direct triangular solve with a fixed number of inner Jacobi-Richardson (JR) iterations. When a small number of inner iterations is sufficient to maintain the Krylov conver…
Physics-Informed Neural Networks promise to revolutionize science and engineering practice, by introducing domain-aware deep machine learning models into scientific computation. Several software suites have emerged to make the implementation and usage of these architectures available to the research and industry communities. Here we introduce\linebreak TensorDiffEq, built on Tensorflow 2.x, which presents an intuitive Keras-like interface for problem domain definition, model definition, and solution of forward and inverse problems using physics-aware deep learning methods. TensorDiffEq takes f…
A User Manual for cuHALLaR: A GPU Accelerated Low-Rank Semidefinite Programming Solver
H. M. Stewart School of Industrial and Systems Engineering, Georgia Tech（H.M. 斯图尔特工业与系统工程学院，佐治亚理工学院）；School of Applied Mathematics, FGV（应用数学学院，FGV）；Department of Combinatorics and Optimization, University of Waterloo（组合优化系，滑铁卢大学）
