In this note we exploit polynomial preconditioners for the Conjugate Gradient method to solve large symmetric positive definite linear systems in a parallel environment. We put in connection a specialized Newton method to solve the matrix equation X^{-1} = A and the Chebyshev polynomials for preconditioning. We propose a simple modification of one parameter which avoids clustering of extremal eigenvalues in order to speed-up convergence. We provide results on very large matrices (up to 8 billion unknowns) in a parallel environment showing the efficiency of the proposed class of preconditioners.
The Poisson pressure solve resulting from the spectral element discretization of the incompressible Navier-Stokes equation requires fast, robust, and scalable preconditioning. In the current work, a parallel scaling study of Chebyshev-accelerated Schwarz and Jacobi preconditioning schemes is presented, with special focus on GPU architectures, such as OLCF's Summit. Convergence properties of the Chebyshev-accelerated schemes are compared with alternative methods, such as low-order preconditioners combined with algebraic multigrid. Performance and scalability results are presented for a variety …
This note proposes an efficient preconditioner for solving linear and semi-linear parabolic equations. With the Crank-Nicholson time stepping method, the algebraic system of equations at each time step is solved with the conjugate gradient method, preconditioned with hierarchical interpolative factorization. Stiffness matrices arising in the discretization of parabolic equations typically have large condition numbers, and therefore preconditioning becomes essential, especially for large time steps. We propose to use the hierarchical interpolative factorization as the preconditioning for the co…
Solving systems of linear equations is a problem occuring frequently in water engineering applications. Usually the size of the problem is too large to be solved via direct factorization. One can resort to iterative approaches, in particular the conjugate gradients method if the matrix is symmetric positive definite. Preconditioners further enhance the rate of convergence but hitherto only handcrafted ones requiring expert knowledge have been used. We propose an innovative approach employing Machine Learning, in particular a Convolutional Neural Network, to unassistedly design preconditioning …
BandPC: Learning Residual-Band Preconditioner Combinations for Flexible Conjugate Gradient Solvers
Wuhan University of Technology（武汉理工大学）；Embodied Physical Intelligence Center in Civil Engineering, Wuhan University of Technology（武汉理工大学土木工程具身智能中心）；Key Laboratory of Low-Altitude Technology and Smart Urban Renewal, Department of Housing and Urban-Rural Development of Hubei Province, Wuhan University of Technology（湖北省住房和城乡建设厅低空技术与智慧城市更新重点实验室（武汉理工大学））；Sanya Science and Education Innovation Park of Wuhan University of Technology（武汉理工大学三亚科教创新园）
Learning from Linear Algebra: A Graph Neural Network Approach to Preconditioner Design for Conjugate Gradient Solvers
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术学院）；Sberbank of Russia（俄罗斯联邦储蓄银行）；Artificial Intelligence Research Institute (AIRI)（人工智能研究所（AIRI））；Fraunhofer Institute for Industrial Mathematics ITWM（弗劳恩霍夫工业数学研究所 ITWM）；Institute of Computational Mathematics and Mathematical Geophysics SB RAS（俄罗斯科学院西伯利亚分院计算数学与数学地球物理研究所）
In this work, we propose a multigrid preconditioner for Jacobian-free Newton-Krylov (JFNK) methods. Our multigrid method does not require knowledge of the Jacobian at any level of the multigrid hierarchy. As it is common in standard multigrid methods, the proposed method also relies on three building blocks: transfer operators, smoothers, and a coarse level solver. In addition to the restriction and prolongation operator, we also use a projection operator to transfer the current Newton iterate to a coarser level. The three-level Chebyshev semi-iterative method is employed as a smoother, as it …
In this paper preconditioners for the Conjugate Gradient method are studied to solve the Newton system with symmetric positive definite Jacobian. In particular, we define a sequence of preconditioners built by means of SR1 and BFGS low-rank updates. We develop conditions under which the SR1 update maintains the preconditioner SPD. Spectral analysis of the SR1 preconditioned Jacobians shows an improved eigenvalue distribution as the Newton iteration proceeds. A compact matrix formulation of the preconditioner update is developed which reduces the cost of its application and is more suitable for…
We develop a robust matrix-free, communication avoiding parallel, high-degree polynomial preconditioner for the Conjugate Gradient method for large and sparse symmetric positive definite linear systems. We discuss the selection of a scaling parameter aimed at avoiding unwanted clustering of eigenvalues of the preconditioned matrices at the extrema of the spectrum. We use this preconditioned framework to solve a $3 \times 3$ block system arising in the simulation of fluid flow in large-size discrete fractured networks. We apply our polynomial preconditioner to a suitable Schur complement relate…
Generalized preconditioned conjugate gradients for adaptive FEM with optimal complexity
Sorbonne Université, Université Paris Cité, CNRS, Laboratoire Jacques-Louis Lions（索邦大学，巴黎 Cité 大学，法国国家科学研究中心，雅克-路易·利翁斯实验室）；TU Wien, Institute of Analysis and Scientific Computing（维也纳工业大学，分析与科学计算研究所）
We show that the mass matrix derived from finite elements can be effectively used as a preconditioner for iteratively solving the linear system arising from finite-difference discretization of the Poisson equation, using the conjugate gradient method. We derive analytically the condition number of the preconditioned operator. Theoretical analysis shows that the ratio of the condition number of the Laplacian to the preconditioned operator is $8/3$ in one dimension, $9/2$ in two dimensions, and $2^9/3^4 \approx 6.3$ in three dimensions. From this it follows that the expected iteration count for …
The aim of this paper is to solve large-and-sparse linear Semidefinite Programs (SDPs) with low-rank solutions. We propose to use a preconditioned conjugate gradient method within second-order SDP algorithms and introduce a new efficient preconditioner fully utilizing the low-rank information. We demonstrate that the preconditioner is universal, in the sense that it can be efficiently used within a standard interior-point algorithm, as well as a newly developed primal-dual penalty method. The efficiency is demonstrated by numerical experiments using the truss topology optimization problems of …
Contingency analysis (CA) plays a critical role to guarantee operation security in the modern power systems. With the high penetration of renewable energy, a real-time and comprehensive N-1 CA is needed as a power system analysis tool to ensure system security. In this paper, a graph-based preconditioning conjugate gradient (GPCG) approach is proposed for the nodal parallel computing in N-1 CA. To pursue a higher performance in the practical application, the coefficient matrix of the base case is used as the incomplete LU (ILU) preconditioner for each N-1 scenario. Additionally, the re-dispatc…
This work investigates a variant of the conjugate gradient (CG) method and embeds it into the context of high-order finite-element schemes with fast matrix-free operator evaluation and cheap preconditioners like the matrix diagonal. Relying on a data-dependency analysis and appropriate enumeration of degrees of freedom, we interleave the vector updates and inner products in a CG iteration with the matrix-vector product with only minor organizational overhead. As a result, around 90% of the vector entries of the three active vectors of the CG method are transferred from slow RAM memory exactly …
A fundamental task in numerical computation is the solution of large linear systems. The conjugate gradient method is an iterative method which offers rapid convergence to the solution, particularly when an effective preconditioner is employed. However, for more challenging systems a substantial error can be present even after many iterations have been performed. The estimates obtained in this case are of little value unless further information can be provided about the numerical error. In this paper we propose a novel statistical model for this numerical error set in a Bayesian framework. Our…
The adaptive $s$-step CG algorithm is a solver for sparse, symmetric positive definite linear systems designed to reduce the synchronization cost per iteration while still achieving a user-specified accuracy requirement. In this work, we improve the adaptive $s$-step conjugate gradient algorithm by use of iteratively updated estimates of the largest and smallest Ritz values, which give approximations of the largest and smallest eigenvalues of $A$, using a technique due to Meurant and Tich{\' y} [G. Meurant and P. Tich{\' y}, Numer. Algs. (2018), pp.~1--32]. The Ritz value estimates are used to…
An important class of nonlinear weighted least-squares problems arises from the assimilation of observations in atmospheric and ocean models. In variational data assimilation, inverse error covariance matrices define the weighting matrices of the least-squares problem. For observation errors, a diagonal matrix (i.e., uncorrelated errors) is often assumed for simplicity even when observation errors are suspected to be correlated. While accounting for observationerror correlations should improve the quality of the solution, it also affects the convergence rate of the minimization algorithms used…
Efficient spectral computation of the stationary states of rotating Bose-Einstein condensates by the preconditioned nonlinear conjugate gradient method
Université de Lorraine（洛林大学）；Inria Nancy-Grand Est（法国国家信息与自动化研究院南锡-大东部中心）；Inria Paris（法国国家信息与自动化研究院巴黎中心）；Université Paris-Est, CERMICS (ENPC)（巴黎东大学，CERMICS（国立路桥学校））；Laboratoire de Mathématiques Raphaël Salem, Université de Rouen（鲁昂大学拉斐尔·萨莱姆数学实验室）
