Discrete Orthogonal Polynomials and Superlinear Convergence of Krylov Subspace Methods in Numerical Linear Algebra

2006 ◽  
pp. 119-185 ◽  
Author(s):  
Bernhard Beckermann
Keyword(s):  
Orthogonal Polynomials ◽  
Linear Algebra ◽  
Superlinear Convergence ◽  
Krylov Subspace ◽  
Krylov Subspace Methods ◽  
Numerical Linear Algebra ◽  
Subspace Methods ◽  
Discrete Orthogonal Polynomials ◽  
Discrete Orthogonal

scholarly journals Krylov Subspace Solvers and Preconditioners

2018 ◽  
Vol 63 ◽  
pp. 1-43
Author(s):  
C. Vuik
Keyword(s):  
Iterative Methods ◽  
Krylov Subspace ◽  
State Of The Art ◽  
Linear Equations ◽  
Iterative Solution ◽  
Krylov Subspace Methods ◽  
Numerical Linear Algebra ◽  
Subspace Methods ◽  
Starting Point ◽  
Lecture Notes

In these lecture notes an introduction to Krylov subspace solvers and preconditioners is presented. After a discretization of partial differential equations large, sparse systems of linear equations have to be solved. Fast solution of these systems is very urgent nowadays. The size of the problems can be 1013 unknowns and 1013 equations. Iterative solution methods are the methods of choice for these large linear systems. We start with a short introduction of Basic Iterative Methods. Thereafter preconditioned Krylov subspace methods, which are state of the art, are describeed. A distinction is made between various classes of matrices. At the end of the lecture notes many references are given to state of the art Scientific Computing methods. Here, we will discuss a number of books which are nice to use for an overview of background material. First of all the books of Golub and Van Loan [19] and Horn and Johnson [26] are classical works on all aspects of numerical linear algebra. These books also contain most of the material, which is used for direct solvers. Varga [50] is a good starting point to study the theory of basic iterative methods. Krylov subspace methods and multigrid are discussed in Saad [38] and Trottenberg, Oosterlee and Schüller [42]. Other books on Krylov subspace methods are [1, 6, 21, 34, 39].

scholarly journals Krylov subspace methods for estimating operator-vector multiplications in Hilbert spaces

2021 ◽  
Author(s):  
Yuka Hashimoto ◽  
Takashi Nodera
Keyword(s):  
Krylov Subspace ◽  
Time Series Data ◽  
Linear Equations ◽  
Transfer Operator ◽  
Krylov Subspace Methods ◽  
Krylov Subspace Method ◽  
Linear Operators ◽  
Series Data ◽  
Subspace Method ◽  
Subspace Methods

AbstractThe Krylov subspace method has been investigated and refined for approximating the behaviors of finite or infinite dimensional linear operators. It has been used for approximating eigenvalues, solutions of linear equations, and operator functions acting on vectors. Recently, for time-series data analysis, much attention is being paid to the Krylov subspace method as a viable method for estimating the multiplications of a vector by an unknown linear operator referred to as a transfer operator. In this paper, we investigate a convergence analysis for Krylov subspace methods for estimating operator-vector multiplications.

scholarly journals Krylov Subspace Methods in Dynamical Sampling

2016 ◽  
Vol 15 (1) ◽  
pp. 9-20
Author(s):  
Akram Aldroubi ◽  
Ilya Krishtal
Keyword(s):  
Krylov Subspace ◽  
Krylov Subspace Methods ◽  
Subspace Methods

POWER GRIDS ANALYSIS IN COMPRESSED KRYLOV-SUBSPACE METHODS

2008 ◽  
Vol 17 (03) ◽  
pp. 439-446
Author(s):  
HAOHANG SU ◽  
YIMEN ZHANG ◽  
YUMING ZHANG ◽  
JINCAI MAN
Keyword(s):  
Present Method ◽  
Large Scale ◽  
Krylov Subspace ◽  
Power Grid ◽  
Excellent Result ◽  
Krylov Subspace Methods ◽  
Power Grids ◽  
Improved Method ◽  
Subspace Methods ◽  
Transient Simulations

An improved method is proposed based on compressed and Krylov-subspace iterative approaches to perform efficient static and transient simulations for large-scale power grid circuits. It is implemented with CG and BiCGStab algorithms and an excellent result has been obtained. Extensive experimental results on large-scale power grid circuits show that the present method is over 200 times faster than SPICE3 and around 10–20 times faster than ICCG method in transient simulations. Furthermore, the presented algorithm saves the memory usage over 95% of SPICE3 and 75% of ICCG method, respectively while the accuracy is not compromised.

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