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Convergence Analysis of an Extended Krylov Subspace Method for the Approximation of Operator Functions in Exponential Integrators

2013
*
SIAM Journal on Numerical Analysis
*

We analyze

doi:10.1137/12089226x
fatcat:jmyjehf3vzh7xl352dadnbriu4
*the**convergence**of**an**extended**Krylov**subspace**method**for**the**approximation**of**operator**functions*that appear*in**exponential**integrators*. ...*For**operators*,*the*size*of**the*polynomial part*of**the**extended**Krylov**subspace*is restricted according to*the*smoothness*of**the*initial data. ... This work has been supported by*the*Deutsche Forschungsgemeinschaft (DFG) via GR 3787/1-1. ...##
###
Computing low-rank approximations of the Fréchet derivative of a matrix function using Krylov subspace methods
[article]

2020
*
arXiv
*
pre-print

We present several different

arXiv:2008.12926v1
fatcat:iaikyt3kmneahba4ul5fftezcy
*Krylov**subspace**methods**for*computing low-rank*approximations**of*L_f(A,E) when*the*direction term E is*of*rank one (which can easily be*extended*to general low-rank). ... We analyze*the**convergence**of**the*resulting*method**for**the*important special case that A is Hermitian and f is either*the**exponential*,*the*logarithm or a Stieltjes*function*. ... One*of**the*main tools we use-both*for**the*derivation*of*algorithms and*for*their*convergence**analysis*-is*an**integral*representation*of**the*Fréchet derivative, which can be derived*in*cases where*the**function*...##
###
On Krylov Subspace Approximations to the Matrix Exponential Operator

1997
*
SIAM Journal on Numerical Analysis
*

*Krylov*

*subspace*

*methods*

*for*

*approximating*

*the*action

*of*matrix

*exponentials*are analyzed

*in*this paper. ... A further open question concerns

*the*relationship between

*the*

*convergence*properties

*of*

*Krylov*

*subspace*

*methods*

*for*

*exponential*

*operators*and those

*for*

*the*linear systems

*of*equations arising

*in*implicit ... We are grateful to Peter Leinen and Harry Yserentant

*for*providing

*the*initial motivation

*for*this work. ...

##
###
Dynamic analysis of power delivery network with nonlinear components using matrix exponential method

2015
*
2015 IEEE Symposium on Electromagnetic Compatibility and Signal Integrity
*

*In*this work, we propose a matrix

*exponential*-based timeintegration algorithm

*for*dynamic

*analysis*

*of*power delivery network (PDN) with nonlinear components. ... Second,

*the*

*method*takes only one LU decomposition per time step while BENR requires at least two LU decompositions

*for*

*the*

*convergence*check

*of*solutions

*of*nonlinear system. ...

*The*drawbacks

*of*previous works is

*the*slow

*convergence*rate

*for*matrix

*exponential*and vector product (MEVP) by standard

*Krylov*

*subspace*[12] . ...

##
###
Page 5229 of Mathematical Reviews Vol. , Issue 98H
[page]

1998
*
Mathematical Reviews
*

Anal. 34 (1997), no. 5, 1911-1925
Summary: “

*Krylov**subspace**methods**for**approximating**the*ac- tion*of*matrix*exponentials*are analyzed*in*this paper. ... We therefore pro- pose a new class*of*time*integration**methods**for*large systems*of*nonlinear differential equations which use*Krylov*approxima- tions to*the**exponential**function**of**the*Jacobian instead ...##
###
Approximation of Semigroups and Related Operator Functions by Resolvent Series

2010
*
SIAM Journal on Numerical Analysis
*

We consider

doi:10.1137/090768084
fatcat:5o2neqihs5e3xhd6yakpnwkrfi
*the**approximation**of*semigroups e τ A and*of**the**functions*ϕ j (τ A) that appear*in**exponential**integrators*by resolvent series. ...*The**approximation**of**the**operator**functions*ϕ j (τ A)*in*a general strongly continuous semigroup setting has not been discussed*in**the*literature so far, while this is crucial*for**an*application*of*these ...*The*exp4*integrator*internally uses*the**Krylov**subspace**method*proposed*in*[11]*for**the**approximation**of**the*ϕ-*functions*. ...##
###
A new investigation of the extended Krylov subspace method for matrix function evaluations

2009
*
Numerical Linear Algebra with Applications
*

*In*this paper we investigate

*the*

*Extended*

*Krylov*

*subspace*

*method*, a technique that was recently proposed to

*approximate*f (A)v

*for*A symmetric. ...

*For*large square matrices A and

*functions*f ,

*the*numerical

*approximation*

*of*

*the*action

*of*f (A) to a vector v has received considerable attention

*in*

*the*last two decades. ... Moret

*for*discussions on [39] and T. Driscoll

*for*his help with

*the*use

*of*

*the*SCToolbox [10] . ...

##
###
Exponential Rosenbrock-Type Methods

2009
*
SIAM Journal on Numerical Analysis
*

*The*application

*of*

*the*required matrix

*functions*to vectors are computed by

*Krylov*

*subspace*

*approximations*. ...

*In*particular, we derive

*an*abstract stability and

*convergence*result

*for*variable step sizes. ... We implemented

*the*

*methods*

*in*Matlab, using

*Krylov*

*subspace*

*methods*to

*approximate*

*the*applications

*of*matrix

*functions*to vectors. ...

##
###
Efficient matrix exponential method based on extended Krylov subspace for transient simulation of large-scale linear circuits

2014
*
2014 19th Asia and South Pacific Design Automation Conference (ASP-DAC)
*

*In*this work we explore

*the*use

*of*

*extended*

*Krylov*

*subspace*to generate more accurate and efficient

*approximation*

*for*MEXP. ... Matrix

*exponential*(MEXP)

*method*has been demonstrated to be a competitive candidate

*for*transient simulation

*of*very large-scale

*integrated*circuits. ... Using

*the*

*extended*

*Krylov*

*subspace*

*for*evaluating matrix

*functions*was first proposed

*in*[3] , which proved that, when A is symmetric,

*the*

*approximation*quality

*of*

*the*

*exponential*

*function*

*in*K 2m (A, ...

##
###
Rational Krylov for Stieltjes matrix functions: convergence and pole selection
[article]

2020
*
arXiv
*
pre-print

We see how to leverage tensorized

arXiv:1908.02032v4
fatcat:xecmvmlj3rgglg4dwa67pu6ela
*Krylov**subspaces*to exploit*the*Kronecker structure and we introduce*an*error*analysis**for**the*numerical*approximation**of*x. ... Evaluating*the*action*of*a matrix*function*on a vector, that is x=f( M)v, is*an*ubiquitous task*in*applications. When M is large, one usually relies on*Krylov*projection*methods*. ...*The*author wish to thank Paul Van Dooren and André Ran*for*fruitful discussions about Lemma 3.5. ...##
###
Computing low‐rank approximations of the Fréchet derivative of a matrix function using Krylov subspace methods

2021
*
Numerical Linear Algebra with Applications
*

We present several different

doi:10.1002/nla.2401
fatcat:qch5zw4pzfhs3o2lngbq2bhzhu
*Krylov**subspace**methods**for*computing low-rank*approximations**of*L f (A, E) when*the*direction term E is*of*rank one (which can easily be*extended*to general low rank). ...*The*Fréchet derivative L f (A, E)*of**the*matrix*function*f (A) plays*an*important role*in*many different applications, including condition number estimation and network*analysis*. ...*The*work*of*Marcel Schweitzer was partly supported by*the*SNSF research project Low-rank updates*of*matrix*functions*and fast eigenvalue solvers. ...##
###
Uniform Approximation of $\varphi$-Functions in Exponential Integrators by a Rational Krylov Subspace Method with Simple Poles

2014
*
SIAM Journal on Matrix Analysis and Applications
*

We consider

doi:10.1137/140964655
fatcat:b4id5qekijerza5hfvb7e4b7mi
*the**approximation**of**the*matrix ϕ-*functions*that appear*in**exponential**integrators**for*stiff systems*of*differential equations. ...*In*order to obtain*an*efficient*method*uniformly*for*all matrices with a field-*of*-values*in**the*left complex half-plane, we consider*the**approximation*by a rational*Krylov**subspace**method*with equidistant ... This work has been supported by*the*Deutsche Forschungsgemeinschaft (DFG) via GR 3787/1-1. ...##
###
Residual, Restarting, and Richardson Iteration for the Matrix Exponential

2013
*
SIAM Journal on Scientific Computing
*

*In*particular, a variant

*of*

*the*Richardson

*method*

*for*

*the*new residual appears to provide

*an*efficient way to restart

*Krylov*

*subspace*

*methods*

*for*evaluating

*the*matrix

*exponential*. ...

*An*important matrix

*function*

*for*which this is

*the*case is

*the*matrix

*exponential*. Suppose

*the*matrix

*exponential*

*of*a given matrix times a given vector has to be computed. ...

*The*first author would like to thank anonymous referees and a number

*of*colleagues,

*in*particular, Michael Saunders, Jan Verwer, and Julien Langou

*for*valuable comments on

*an*earlier version

*of*this paper ...

##
###
Residual, restarting and Richardson iteration for the matrix exponential, revised
[article]

2011
*
arXiv
*
pre-print

*In*particular, a variant

*of*

*the*Richardson

*method*

*for*

*the*new residual appears to provide

*an*efficient way to restart

*Krylov*

*subspace*

*methods*

*for*evaluating

*the*matrix

*exponential*. ...

*An*important matrix

*function*

*for*which this is

*the*case is

*the*matrix

*exponential*. Suppose

*the*matrix

*exponential*

*of*a given matrix times a given vector has to be computed. ...

*The*author would like to thank anonymous referees and a number

*of*colleagues,

*in*particular, Michael Saunders, Jan Verwer and Julien Langou

*for*valuable comments and Marlis Hochbruck

*for*explaining

*the*...

##
###
Rational Krylov for Stieltjes matrix functions: convergence and pole selection

2020
*
BIT Numerical Mathematics
*

We see how to leverage tensorized

doi:10.1007/s10543-020-00826-z
fatcat:tbl76bynqjdh7hyf44tzzrrlxy
*Krylov**subspaces*to exploit*the*Kronecker structure and we introduce*an*error*analysis**for**the*numerical*approximation**of*x. ... Evaluating*the*action*of*a matrix*function*on a vector, that is x = f (M)v, is*an*ubiquitous task*in*applications. When M is large, one usually relies on*Krylov*projection*methods*. ... Acknowledgements*The*author wish to thank Paul Van Dooren and André Ran*for*fruitful discussions about Lemma 2. ...
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