arXiv subjects

Chen-Song Zhang

Publications and source records attributed to Chen-Song Zhang.

23 records · Page 2Linked to original sources

arXiv2013

An Error-Resilient Redundant Subspace Correction Method

As we stride toward the exascale era, due to increasing complexity of supercomputers, hard and soft errors are causing more and more problems in high-performance scientific and engineering computation. In order to improve reliability (increase the mean time to failure) of computing systems, a lot of efforts have been devoted to developing techniques to forecast, prevent, and recover from errors at different levels, including architecture, application, and algorithm. In this paper, we focus on algorithmic error resilient iterative linear solvers and introduce a redundant subspace correction method. Using a general framework of redundant subspace corrections, we construct iterative methods, which have the following properties: (1) Maintain convergence when error occurs assuming it is detectable; (2) Introduce low computational overhead when no error occurs; (3) Require only small amount of local (point-to-point) communication compared to traditional methods and maintain good load balance; (4) Improve the mean time to failure. With the proposed method, we can improve reliability of many scientific and engineering applications. Preliminary numerical experiments demonstrate the efficiency and effectiveness of the new subspace correction method.

math.NA↗

arXiv2013

Combined Preconditioning with Applications in Reservoir Simulation

We develop a simple algorithmic framework to solve large-scale symmetric positive definite linear systems. At its core, the framework relies on two components: (1) a norm-convergent iterative method (i.e. smoother) and (2) a preconditioner. The resulting preconditioner, which we refer to as a combined preconditioner, is much more robust and efficient than the iterative method and preconditioner when used in Krylov subspace methods. We prove that the combined preconditioner is positive definite and show estimates on the condition number of the preconditioned system. We combine an algebraic multigrid method and an incomplete factorization preconditioner to test the proposed framework on problems in petroleum reservoir simulation. Our numerical experiments demonstrate noticeable speed-up when we compare our combined method with the standalone algebraic multigrid method or the incomplete factorization preconditioner.

math.NA↗

arXiv2013

Numerical Study of Geometric Multigrid Methods on CPU--GPU Heterogeneous Computers

The geometric multigrid method (GMG) is one of the most efficient solving techniques for discrete algebraic systems arising from elliptic partial differential equations. GMG utilizes a hierarchy of grids or discretizations and reduces the error at a number of frequencies simultaneously. Graphics processing units (GPUs) have recently burst onto the scientific computing scene as a technology that has yielded substantial performance and energy-efficiency improvements. A central challenge in implementing GMG on GPUs, though, is that computational work on coarse levels cannot fully utilize the capacity of a GPU. In this work, we perform numerical studies of GMG on CPU--GPU heterogeneous computers. Furthermore, we compare our implementation with an efficient CPU implementation of GMG and with the most popular fast Poisson solver, Fast Fourier Transform, in the cuFFT library developed by NVIDIA.

math.NA↗

arXiv2012

Self-sustaining oscillations of a falling sphere through Johnson-Segalman fluids

We confirm numerically that the Johnson-Segalman model is able to reproduce the continual oscillations of the falling sphere observed in some viscoelastic models. The empirical choice of parameters used in the Johnson-Segalman model is from the ones that show the non-monotone stress-strain relation of the steady shear flows of the model. The carefully chosen parameters yield continual, self-sustaining, (ir)regular and periodic oscillations of the speed for the falling sphere through the Johnson-Segalman fluids. In particular, our simulations reproduce the phenomena: the falling sphere settles slower and slower until a certain point at which the sphere suddenly accelerates and this pattern is repeated continually.

math-ph↗

arXiv2012

A Scalable Auxiliary Space Preconditioner for High-Order Finite Element Methods

In this paper, we revisit an auxiliary space preconditioning method proposed by Xu [Computing 56, 1996], in which low-order finite element spaces are employed as auxiliary spaces for solving linear algebraic systems arising from high-order finite element discretizations. We provide a new convergence rate estimate and parallel implementation of the proposed algorithm. We show that this method is user-friendly and can play an important role in a variety of Poisson-based solvers for more challenging problems such as the Navier--Stokes equation. We investigate the performance of the proposed algorithm using the Poisson equation and the Stokes equation on 3D unstructured grids. Numerical results demonstrate the advantages of the proposed algorithm in terms of efficiency, robustness, and parallel scalability.

math.NA↗