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

Tanja Göckler, Volker Grimm
2013 SIAM Journal on Numerical Analysis  
We analyze 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.  ... 
doi:10.1137/12089226x fatcat:jmyjehf3vzh7xl352dadnbriu4

Computing low-rank approximations of the Fréchet derivative of a matrix function using Krylov subspace methods [article]

Peter Kandolf and Antti Koskela and Samuel D. Relton and Marcel Schweitzer
2020 arXiv   pre-print
We present several different 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  ... 
arXiv:2008.12926v1 fatcat:iaikyt3kmneahba4ul5fftezcy

On Krylov Subspace Approximations to the Matrix Exponential Operator

Marlis Hochbruck, Christian Lubich
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.  ... 
doi:10.1137/s0036142995280572 fatcat:xoq2ybrvl5dk5eoamm5puigbdq

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

Hao Zhuang, Ilgweon Kang, Xinan Wang, Jeng-Hau Lin, Chung-Kuan Cheng
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] .  ... 
doi:10.1109/emcsi.2015.7107694 fatcat:j7tzfglcizayxhswalkvvaqx2i

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

Volker Grimm, Martin Gugat
2010 SIAM Journal on Numerical Analysis  
We consider 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.  ... 
doi:10.1137/090768084 fatcat:5o2neqihs5e3xhd6yakpnwkrfi

A new investigation of the extended Krylov subspace method for matrix function evaluations

L. Knizhnerman, V. Simoncini
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] .  ... 
doi:10.1002/nla.652 fatcat:wnwa3qe3xjg63axcsnz4ulbe5m

Exponential Rosenbrock-Type Methods

Marlis Hochbruck, Alexander Ostermann, Julia Schweitzer
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.  ... 
doi:10.1137/080717717 fatcat:vag5jtvx7bbenhss44c3qw77ee

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

Quan Chen, Wenhui Zhao, Ngai Wong
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,  ... 
doi:10.1109/aspdac.2014.6742900 dblp:conf/aspdac/ChenZW14 fatcat:h57wgsltq5gs5gu7cxqkqt3ur4

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

Stefano Massei, Leonardo Robol
2020 arXiv   pre-print
We see how to leverage tensorized 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.  ... 
arXiv:1908.02032v4 fatcat:xecmvmlj3rgglg4dwa67pu6ela

Computing low‐rank approximations of the Fréchet derivative of a matrix function using Krylov subspace methods

Peter Kandolf, Antti Koskela, Samuel D. Relton, Marcel Schweitzer
2021 Numerical Linear Algebra with Applications  
We present several different 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.  ... 
doi:10.1002/nla.2401 fatcat:qch5zw4pzfhs3o2lngbq2bhzhu

Uniform Approximation of $\varphi$-Functions in Exponential Integrators by a Rational Krylov Subspace Method with Simple Poles

Tanja Göckler, Volker Grimm
2014 SIAM Journal on Matrix Analysis and Applications  
We consider 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.  ... 
doi:10.1137/140964655 fatcat:b4id5qekijerza5hfvb7e4b7mi

Residual, Restarting, and Richardson Iteration for the Matrix Exponential

Mike A. Botchev, Volker Grimm, Marlis Hochbruck
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  ... 
doi:10.1137/110820191 fatcat:yoixb3kn3rcjzigdasjfqh7yj4

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

Mike A. Botchev
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  ... 
arXiv:1112.5670v1 fatcat:locyesfd6vdk3nbfi22hdh6com

Rational Krylov for Stieltjes matrix functions: convergence and pole selection

Stefano Massei, Leonardo Robol
2020 BIT Numerical Mathematics  
We see how to leverage tensorized 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.  ... 
doi:10.1007/s10543-020-00826-z fatcat:tbl76bynqjdh7hyf44tzzrrlxy
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