We describe preconditioned iterative methods for estimating the number of eigenvalues of a Hermitian matrix within a given interval. Such estimation is useful in a number of applications.In particular, it can be used to develop an efficient spectrum-slicing strategy to compute many eigenpairs of a Hermitian matrix. Our method is based on the Lanczos- and Arnoldi-type of iterations. We show that with a properly defined preconditioner, only a few iterations may be needed to obtain a good estimate of the number of eigenvalues within a prescribed interval. We also demonstrate that the number of iterations required by the proposed preconditioned schemes is independent of the size and condition number of the matrix. The efficiency of the methods is illustrated on several problems arising from density functional theory based electronic structure calculations.
This paper introduces and analyzes a preconditioned modified of the Hermitian and skew-Hermitian splitting (PMHSS). The large sparse continuous Sylvester equations are solved by PMHSS iterative algorithm based on nonHermitian, complex, positive definite/semidefinite, and symmetric matrices. We prove that the PMHSS is converged under suitable conditions. In addition, we propose an accelerated PMHSS method consisting of two preconditioned matrices and two iteration parameters α, \b{eta}. Theoretical analysis showed that the convergence speed of the accelerated PMHSS is faster compared to the PMH…
On the preconditioned AOR iterative method for Z-matrices
Faculty of Mathematical Sciences, University of Guilan（吉兰大学数学科学学院）；Faculty of Science, Department of Mathematics, Islamic Azad University, Shahrood（伊斯兰阿扎德大学沙赫鲁德理学院）；Department of Mathematics, Vali-e-Asr University of Rafsanjan（拉夫桑詹瓦利·埃斯尔大学数学系）
In this thesis, the numerical solution of three different classes of problems have been studied. Specifically, new techniques have been proposed and their theoretical analysis has been performed, accompanied by a wide set of numerical experiments, for investigating further and comparing the effectiveness and performance of the presented approach. The first two belong to the research area of numerical linear algebra and concern the spectral analysis and preconditioning for Krylov subspace methods of the coefficient matrix of large structured linear systems. The third concerns a problem from the…
We consider the iterative solution of large linear systems of equations in which the coefficient matrix is the sum of two terms, a sparse matrix $A$ and a possibly dense, rank deficient matrix of the form $γUU^T$, where $γ> 0$ is a parameter which in some applications may be taken to be 1. The matrix $A$ itself can be singular, but we assume that the symmetric part of $A$ is positive semidefinite and that $A+γUU^T$ is nonsingular. Linear systems of this form arise frequently in fields like optimization, fluid mechanics, computational statistics, and others. We investigate preconditioning strat…
We present a family of discretizations for the Variable Eddington Factor (VEF) equations that have high-order accuracy on curved meshes and efficient preconditioned iterative solvers. The VEF discretizations are combined with a high-order Discontinuous Galerkin transport discretization to form an effective high-order, linear transport method. The VEF discretizations are derived by extending the unified analysis of Discontinuous Galerkin methods for elliptic problems to the VEF equations. This framework is used to define analogs of the interior penalty, second method of Bassi and Rebay, minimal…
Solving Convolution-type Integral Equations using Preconditioned Neural Operators
Lingnan University（岭南大学）；Hong Kong Centre for Cerebro-Cardiovascular Health Engineering（香港脑心血管健康工程中心）；Hetao Institute of Mathematics and Interdisciplinary Sciences (Shenzhen)（河套数学与交叉科学研究院（深圳））
Solving evolutionary equations in a parallel-in-time manner is an attractive topic and many algorithms are proposed in recent two decades. The algorithm based on the block $α$-circulant preconditioning technique has shown promising advantages, especially for wave propagation problems. By fast Fourier transform for factorizing the involved circulant matrices, the preconditioned iteration can be computed efficiently via the so-called diagonalization technique, which yields a direct parallel implementation across all time levels. In recent years, considerable efforts have been devoted to explorin…
Context. Numerical solutions to transfer problems of polarized radiation in solar and stellar atmospheres commonly rely on stationary iterative methods, which often perform poorly when applied to large problems. In recent times, stationary iterative methods have been replaced by state-of-the-art preconditioned Krylov iterative methods for many applications. However, a general description and a convergence analysis of Krylov methods in the polarized radiative transfer context are still lacking. Aims. We describe the practical application of preconditioned Krylov methods to linear transfer probl…
In this paper, we focus on solving a sequence of linear systems with an identical (or similar) coefficient matrix. For this type of problems, we investigate the subspace correction and deflation methods, which use an auxiliary matrix (subspace) to accelerate the convergence of the iterative method. In practical simulations, these acceleration methods typically work well when the range of the auxiliary matrix contains eigenspaces corresponding to small eigenvalues of the coefficient matrix. We have developed a new algebraic auxiliary matrix construction method based on error vector sampling, in…
Efficient Solution of Generalized Sylvester Equations via Preconditioned Alternating Anderson Acceleration
Nanchang University（南昌大学）；Nanchang Hangkong University（南昌航空大学）；Fujian Agriculture and Forestry University（福建农林大学）；Dalian University of Technology（大连理工大学）
We introduce a novel eigenvalue algorithm for near-diagonal matrices inspired by Rayleigh-Schrödinger perturbation theory and termed Iterative Perturbative Theory (IPT). Contrary to standard eigenvalue algorithms, which are either 'direct' (to compute all eigenpairs) or 'iterative' (to compute just a few), IPT computes any number of eigenpairs with the same basic iterative procedure. Thanks to this perfect parallelism, IPT proves more efficient than classical methods (LAPACK or CUSOLVER for the full-spectrum problem, preconditioned Davidson solvers for extremal eigenvalues). We give sufficient…
We consider the numerical approximation of the radiative transfer equation using discontinuous angular and continuous spatial approximations for the even parts of the solution. The even-parity equations are solved using a diffusion synthetic accelerated source iteration. We provide a convergence analysis for the infinite-dimensional iteration as well as for its discretized counterpart. The diffusion correction is computed by a subspace correction, which leads to a convergence behavior that is robust with respect to the discretization. The proven theoretical contraction rate deteriorates for sc…
Convergence theory for preconditioned eigenvalue solvers in a nutshell
University Colorado Denver（科罗拉多大学丹佛分校）；Mitsubishi Electric Research Laboratories（三菱电机研究实验室）；Universität Rostock（罗斯托克大学）；STFC Rutherford Appleton Laboratory（STFC鲁瑟福阿普尔顿实验室）
Theoretical Analysis of Thermodynamic Matrix Inversion: First-order Equivalence to Preconditioned Gradient Descent and Implications for Analog Computing
The Legendre spectral Galerkin method of self-adjoint second order elliptic equations usually results in a linear system with a dense and ill-conditioned coefficient matrix. In this paper, the linear system is solved by a preconditioned conjugate gradient (PCG) method where the preconditioner $M$ is constructed by approximating the variable coefficients with a ($T$+1)-term Legendre series in each direction to a desired accuracy. A feature of the proposed PCG method is that the iteration step increases slightly with the size of the resulting matrix when reaching a certain approximation accuracy…
In this paper, we propose a simple sparse approximate inverse for triangular matrices (SAIT). Using the Jacobi iteration method, we obtain an expression of the exact inverse of triangular matrix, which is a finite series. The SAIT is constructed based on this series. We apply the SAIT matrices to iterative methods with ILU preconditioners. The two triangular solvers in the ILU preconditioning procedure are replaced by two matrix-vector multiplications, which can be fine-grained parallelized. We test this method by solving some linear systems and eigenvalue problems with preconditioned iterativ…
We propose a novel preconditioned inexact primal-dual interior point method for constrained convex quadratic programming problems. The algorithm we describe invokes the preconditioned conjugate gradient method on a new reduced Schur complement KKT system, in implicit form. In contrast to standard approaches, the Schur complement formulation we consider enables reuse of the factorization of the KKT matrix with rows and columns corresponding to inequality constraints excluded, across all interior point iterations. Further, two new preconditioners are presented for the resulting reduced system, t…
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 …
