- A block rational Krylov method for three-dimensional time-domain marine controlled-source electromagnetic modeling

*with C. Qiu et al.*
We introduce a novel block rational Krylov method to accelerate three-dimensional time-domain marine controlled-source electromagnetic modeling with multiple sources. The method is tested on a layered seafloor model and a 3D hydrocarbon reservoir model with seafloor bathymetry.

Submitted, 2018.
@techreport{QG18,
title = {A block rational {K}rylov method for three-dimensional time-domain
marine controlled-source electromagnetic modeling},
author = {Qiu, Changkai and G\"{u}ttel, Stefan and Ren, Xiuyan and Yin, Changchun
and Liu, Yunhe and Zhang, Bo and Egbert, Gary},
year = {2018},
number = {2018.22},
pages = {54},
institution = {Manchester Institute for Mathematical Sciences, The University of Manchester},
address = {UK},
type = {MIMS EPrint},
url = {http://eprints.maths.manchester.ac.uk/2652/}
}

- Conversions between barycentric, RKFUN, and Newton representations of rational interpolants

*with S. Elsworth.*
We derive explicit formulas for converting between rational interpolants in barycentric, rational Krylov (RKFUN), and Newton form. We show applications of these conversions when working with rational approximants produced by the AAA algorithm [Y. Nakatsukasa et al., arXiv:1612.00337, 2016] within the RKToolbox and for the solution of nonlinear eigenvalue problems.

Submitted, 2017.
@techreport{GGP17,
title = {Conversions between barycentric, {RKFUN}, and {N}ewton representations of
rational interpolants},
author = {Elsworth, Steven and G{\"u}ttel, Stefan},
year = {2017},
number = {2017.40},
pages = {10},
institution = {Manchester Institute for Mathematical Sciences, The University of Manchester},
address = {UK},
type = {MIMS EPrint},
url = {http://eprints.maths.manchester.ac.uk/2632/}
}

- Compressing variable-coefficient exterior Helmholtz problems via RKFIT

*with V. Druskin and L. Knizhnerman.*
The efficient discretization of Helmholtz problems on unbounded domains is a challenging task, in particular, when the wave medium is nonhomogeneous. We present a new numerical approach for compressing finite difference discretizations of such problems, thereby giving rise to efficient perfectly matched layers (PMLs) for nonhomogeneous media.

Submitted, 2016.
@techreport{DGK16b,
title = {Compressing variable-coefficient exterior {H}elmholtz problems via {RKFIT}},
author = {Druskin, Vladimir and G{\"u}ttel, Stefan and Knizhnerman, Leonid},
year = {2016},
number = {2016.53},
pages = {21},
institution = {Manchester Institute for Mathematical Sciences, The University of Manchester},
address = {UK},
type = {MIMS EPrint},
url = {http://eprints.ma.man.ac.uk/2512/}
}

- A nonlinear ParaExp algorithm

*with M. J. Gander and M. Petcu.*
We propose and analyse a variant of the ParaExp algorithm for nonlinear initial value problems. We show that this algorithm is mathematically equivalent to a parareal iteration where the coarse integrator solves linear subproblems on overlapping time intervals. A numerical example with a nonlinear wave equation illustrates the convergence behaviour.

Accepted for publication in Domain Decomposition Methods in Science and Engineering, Lecture Notes in Computational Science and Engineering, Springer-Verlag, 2018.
@techreport{GGP17,
title = {A nonlinear {ParaExp} algorithm},
author = {Gander, Martin J. and G{\"u}ttel, Stefan and Petcu, Madalina},
year = {2017},
number = {2017.17},
pages = {8},
institution = {Manchester Institute for Mathematical Sciences, The University of Manchester},
address = {UK},
type = {MIMS EPrint},
url = {http://eprints.ma.man.ac.uk/2550/}
}

- A rational deferred correction approach to parabolic optimal control problems

*with J. W. Pearson.*
We devise a new deferred correction method for coupled systems of time-dependent PDEs, allowing one to iteratively improve the accuracy of low-order time stepping schemes. We test our approach on a number of PDE-constrained optimization problems and obtain solution accuracies far superior to that achieved when solving a single discretized problem, in particular in cases where the accuracy is limited by the time discretization.

Accepted for publication in IMA J. Numer. Anal., 2017.
@techreport{GP17,
title = {A rational deferred correction approach to parabolic optimal control problems},
author = {G{\"u}ttel, Stefan and Pearson, John W.},
year = {2017},
number = {2016.11},
pages = {23},
institution = {Manchester Institute for Mathematical Sciences, The University of Manchester},
address = {UK},
type = {MIMS EPrint},
url = {http://eprints.ma.man.ac.uk/2559/},
note = {Accepted 2017 for publication in IMA J. Numer. Anal.}
}

- Scabies in residential care homes: Modelling, inference and interventions for well-connected population sub-units

*with T. Kinyanjui, J. Middleton, J. Cassell, J. Ross and T. House.*
We present and test a modelling methodology based on the efficient solution of a large-scale system of linear differential equations that allows statistical calibration of individual-based random models to real data on scabies in residential care homes.

PLoS Computational Biology, 14(3):1--24, 2018.
@article{KMG18,
title = {Scabies in residential care homes: {M}odelling, inference and interventions
for well-connected population sub-units},
author = {Kinyanjui, Timothy and Middleton, Jo and G{\"u}ttel, Stefan and Cassell, Jackie
and Ross, Joshua},
journal = {PLoS Comput. Biol.},
volume = {14},
number = {3},
pages = {1--24},
year = {2018}
}

- The RKFIT algorithm for nonlinear rational approximation

*with M. Berljafa.*
The RKFIT algorithm is a Krylov-based approach for solving nonlinear rational least squares problems. RKFIT can compute nondiagonal rational approximants and families of approximants sharing a common denominator. We also present methods for the degree reduction of the approximants, conversion to partial fraction form, efficient evaluation, and root-finding.

SIAM J. Sci. Comput., 39(5):A2049--A2071, 2017.
@article{BG17b,
title = {The {RKFIT} algorithm for nonlinear rational approximation},
author = {Berljafa, Mario and G{\"u}ttel, Stefan},
journal = {SIAM J. Sci. Comput.},
volume = {39},
number = {5},
pages = {A2049--A2071},
year = {2017}
}

- Parallelization of the rational Arnoldi algorithm

*with M. Berljafa.*
The rational Arnoldi algorithm is a commonly used procedure for computing an orthonormal basis of a rational Krylov space.
Our analysis of this algorithm allows to control the growth of the condition number of the nonorthogonal basis being implicitly computed. As a consequence we obtain a significantly more accurate and reliable parallel rational Arnoldi algorithm.

SIAM J. Sci. Comput., 39(5):S197--S221, 2017.
@article{BG17a,
title = {Parallelization of the rational {A}rnoldi algorithm},
author = {Berljafa, Mario and G{\"u}ttel, Stefan},
journal = {SIAM J. Sci. Comput.},
volume = {39},
number = {5},
pages = {S197--S221},
year = {2017}
}

- The nonlinear eigenvalue problem

*with F. Tisseur.*
This article surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties and available numerical solution techniques. Solvers based on Newton's method, contour integration, and sampling via rational interpolation are reviewed.

Acta Numer., 26:1--94, 2017.
@article{GT17,
title = {The nonlinear eigenvalue problem},
author = {G{\"u}ttel, Stefan and Tisseur, Fran\c{c}oise},
journal = {Acta Numer.},
volume = {26},
pages = {1--94},
year = {2017}
}

- Near-optimal perfectly matched layers for indefinite Helmholtz problems

*with V. Druskin and L. Knizhnerman.*
A construction of a perfectly matched layer (PML) for indefinite Helmholtz problems on unbounded domains is presented. The accuracy of this PML converges exponentially fast in the number of grid layers, with the convergence rate being asymptotically optimal for both propagative and evanescent wave modes.

SIAM Rev., 58(1):90--116, 2016.
@article{DGK16a,
title = {Near-optimal perfectly matched layers for indefinite {H}elmholtz problems},
author = {Druskin, Vladimir and G{\"u}ttel, Stefan and Knizhnerman, Leonid},
journal = {SIAM Rev.},
volume = {58},
number = {1},
pages = {90--116},
year = {2016}
}

- Scaled and squared subdiagonal Padé approximation for the matrix exponential

*with Y. Nakatsukasa.*
We introduce an efficient variant of scaling and squaring that uses a much smaller squaring factor when ||A||>>1 and a subdiagonal Padé approximant of low degree, thereby significantly reducing the overall cost and avoiding the potential instability caused by overscaling, while giving forward error of the same magnitude as the standard algorithm.

SIAM J. Matrix Anal. Appl., 37(1):145--170, 2016.
@article{GN16,
title = {Scaled and squared subdiagonal {P}ad\'{e} approximation for the matrix exponential},
author = {G{\"u}ttel, Stefan and Nakatsukasa, Yuji},
journal = {SIAM J. Matrix Anal. Appl.},
volume = {37},
number = {1},
pages = {145--170},
year = {2016}
}

- Automatic real-time fault detection for industrial assets using metasensors

*with T. D. Butters, J. L. Shapiro, and T. J. Sharpe.*
We present a method to construct so-called metasensors, virtual sensors that compress the information from several sensors in industrial plants in an optimal manner. The metasensors are used as inputs to a novel anomaly detection system that automatically alerts operators to abnormal behaviour.

Proceedings of the 2015 Asset Management Conference, The Institute of Engineering and Technology, 2015.
@inproceedings{BGSS15,
author = {Butters, Timothy D. and G{\"u}ttel, Stefan and Shapiro, Jonathan L. and Sharpe, Tim J.},
title = {Automatic real-time fault detection for industrial assets using metasensors},
year = {2015},
booktitle = {Proceedings of the 2015 Asset Management Conference},
publisher = {The Institute of Engineering and Technology},
pages = {1--5}
}

- Generalized rational Krylov decompositions with an application to rational approximation

*with M. Berljafa.*
We study the algebraic properties of generalized Krylov decompositions and present an implicit Q theorem for rational Krylov spaces. Transformations on rational Krylov decompositions allow for changing the poles of a rational Krylov space without recomputation and this allows for the development of an algorithm RKFIT for rational least squares approximation.

SIAM J. Matrix Anal. Appl., 36(2):894--916, 2015.
@article{BG15a,
title = {Generalized rational {K}rylov decompositions with an application to
rational approximation},
author = {Berljafa, Mario and G{\"u}ttel, Stefan},
journal = {SIAM J. Matrix Anal. Appl.},
volume = {36},
number = {2},
pages = {894--916},
year = {2015}
}

- Detecting and Reducing Redundancy in Alarm Networks

*with T. D. Butters and J. L. Shapiro.*
We present a new approach to alarm system optimization through the identification of redundant alarms. Our approach is based on a ranking of alarms by their connectivity in the alarm network. We also propose an overall alarm redundancy measure which can be used to monitor performance improvements after redundant alarms have been removed.

Proceedings of the IEEE International Conference on Automation Science and Engineering (CASE), pp. 1224--1229, 2015.
@inproceedings{BGS15,
author = {Butters, Timothy D. and G{\"u}ttel, Stefan and Shapiro, Jonathan L.},
title = {Detecting and reducing redundancy in alarm networks},
year = {2015},
booktitle = {Proceedings of the IEEE International Conference on Automation Science and Engineering (CASE)},
publisher = {IEEE},
pages = {1224--1229}
}

- Zolotarev quadrature rules and load balancing for the FEAST eigensolver

*with E. Polizzi, P. Tang, and G. Viaud.*
We propose improved quadrature rules leading to FEAST variants with faster convergence, in particular, when using rational approximants based on the work of Zolotarev. Numerical experiments demonstrate the possible computational savings especially for pencils whose eigenvalues are not well separated and when the dimension of the search space is only slightly larger than the number of wanted eigenvalues. The new approach improves both convergence robustness and load balancing when FEAST runs on multiple search intervals in parallel.

SIAM J. Sci. Comput., 37(4):A2100--A2122, 2015.
@article{GPTV15,
title = {Zolotarev quadrature rules and load balancing for the {FEAST} eigensolver},
author = {G{\"u}ttel, Stefan and Polizzi, Eric and Tang, Peter and Viaud, Gautier},
journal = {SIAM J. Sci. Comput.},
volume = {37},
number = {4},
pages = {A2100--A2122},
year = {2015}
}

- Three-dimensional transient electromagnetic modeling using rational Krylov methods

*with R.-U. Börner and O. G. Ernst.*
A computational method is given for solving the forward modeling problem for transient electromagnetic exploration in time domain.
Its key features are discretization of the quasi-static Maxwell’s equations in space using the first-kind family of curl-conforming
Nédélec elements combined with time integration using rational Krylov subspace methods.
We also propose a simple method for selecting the pole parameters of the rational Krylov subspace method which leads to convergence within
an a priori determined number of iterations independent of mesh size and conductivity structure.

Geophys. J. Int., 202(3):2025--2043, 2015.
@article{BGE15,
title = {Three-dimensional transient electromagnetic modeling using rational {K}rylov methods},
author = {B{\"o}rner, Ralph-Uwe and G{\"u}ttel, Stefan and Ernst, Oliver G.},
journal = {Geophys. J. Int.},
volume = {202},
number = {3},
pages = {2025--2043},
year = {2015}
}

- A Rational Krylov Toolbox for MATLAB

*with M. Berljafa.*
This is a short guide for the MATLAB Rational Krylov Toolbox. (Download the toolbox.)

MIMS EPrint 2014.56, 2014.
@techreport{BG14,
title = {A {R}ational {K}rylov {T}oolbox for {MATLAB}},
author = {Berljafa, Mario and G{\"u}ttel, Stefan},
year = {2014},
number = {2014.56},
pages = {8},
institution = {Manchester Institute for Mathematical Sciences, The University of Manchester},
address = {UK},
type = {MIMS EPrint},
url = {http://eprints.ma.man.ac.uk/2209/}
}

- Convergence of restarted Krylov subspace methods for Stieltjes functions of matrices

*with A. Frommer and M. Schweitzer.*
To approximate matrix functions by Krylov methods, restarts may become mandatory due to storage requirements or computational complexity.
However, the question under which circumstances convergence of these methods can
be guaranteed has remained largely unanswered. We prove convergence for the class of Stieltjes
functions of Hermitian positive definite matrices. We also propose
a modification of the Arnoldi method which guarantees convergence for positive real matrices.

SIAM J. Matrix Anal. Appl., 35(4):1602--1624, 2014.
@article{FGS14b,
title = {Convergence of restarted {K}rylov subspace methods for {S}tieltjes functions of matrices},
author = {Frommer, Andreas and G{\"u}ttel, Stefan and Schweitzer, Marcel},
journal = {SIAM J. Matrix Anal. Appl.},
volume = {35},
number = {4},
pages = {1602--1624},
year = {2014}
}

- Statistical cluster analysis and visualisation for alarm management configuration

*with T. D. Butters, J. L. Shapiro, and T. J. Sharpe.*
The effective performance of an alarm system is a key aspect of asset management for any industrial installation.
Here we present a novel method for the identification of redundant or bad actors in alarm systems through the application of statistical cluster analysis.

Proceedings of the 2014 Asset Management Conference, The Institute of Engineering and Technology, pages 1--6, 2014.
@inproceedings{BGSS14,
author = {Butters, Timothy D. and G{\"u}ttel, Stefan and Shapiro, Jonathan L. and Sharpe, Tim J.},
title = {Statistical cluster analysis and visualisation for alarm management configuration},
year = {2014},
booktitle = {Proceedings of the 2014 Asset Management Conference},
publisher = {The Institute of Engineering and Technology},
pages = {1--6}
}

- NLEIGS: A class of fully rational Krylov methods for nonlinear eigenvalue problems

*with R. Van Beeumen, K. Meerbergen, and W. Michiels.*
A new rational Krylov method, called NLEIGS, for the solution of nonlinear eigenvalue problems is proposed.
It is based on a novel companion-type linearization obtained from a dynamically computed
linear rational interpolant, along with a scaling procedure for robustness.
The paper also discusses the computation of rational divided differences via matrix functions. (Download a MATLAB implementation of NLEIGS.)

SIAM J. Sci. Comput., 36(6):A2842--A2864, 2014.
@article{GV14,
title = {{NLEIGS}: {A} class of fully rational {K}rylov methods for nonlinear eigenvalue problems},
author = {G{\"u}ttel, Stefan and Van Beeumen, Roel and Meerbergen, Karl and Michiels, Wim},
journal = {SIAM J. Sci. Comput.},
volume = {36},
number = {6},
pages = {A2842--A2864},
year = {2014}
}

- Efficient high-order rational integration and deferred correction with equispaced data

*with G. Klein.*
We analyze the convergence of integrals of barycentric rational interpolants to those of analytic functions as well as functions with a finite number of continuous derivatives.
A new deferred correction scheme based on this quadrature approach is presented.

Electron. Trans. Numer. Anal., 41:443--464, 2014.
@article{GK14,
title = {Efficient high-order rational integration and deferred correction with equispaced data},
author = {G{\"u}ttel, Stefan and Klein, Georges},
journal = {Electron. Trans. Numer. Anal.},
volume = {41},
pages = {443--464},
year = {2014}
}

- Efficient and stable Arnoldi restarts for matrix functions based on quadrature

*with A. Frommer and M. Schweitzer.*
We utilize an integral representation for the error of Arnoldi approximants
to develop an efficient quadrature-based restarting method suitable for a
large class of functions, including the so-called Stieltjes functions and the exponential function.
Our method is applicable for functions of Hermitian and non-Hermitian matrices, requires no a-priori
spectral information, and runs with essentially constant computational work per restart cycle. (Download FUNM_QUAD and documentation.)

SIAM J. Matrix Anal. Appl., 35(2):661--683, 2014.
@article{FGS14a,
title = {Efficient and stable {A}rnoldi restarts for matrix functions based on quadrature},
author = {Frommer, Andreas and G{\"u}ttel, Stefan and Schweitzer, Marcel},
journal = {SIAM J. Matrix Anal. Appl.},
volume = {35},
number = {2},
pages = {661--683},
year = {2014}
}

- A spatially adaptive iterative method for a class of nonlinear operator eigenproblems

*with E. Jarlebring.*
We present a new algorithm for the iterative solution of nonlinear operator eigenvalue problems arising from partial differential equations.
This algorithm combines automatic spatial resolution of linear operators with the infinite Arnoldi method for nonlinear matrix eigenproblems.

Electron. Trans. Numer. Anal., 41:21--41, 2014.
@article{JG14,
title = {A spatially adaptive iterative method for a class of nonlinear operator eigenproblems},
author = {Jarlebring, Elias and G{\"u}ttel, Stefan},
journal = {Electron. Trans. Numer. Anal.},
volume = {41},
pages = {21--41},
year = {2014}
}

- Some observations on weighted GMRES

*with J. Pestana.*
We investigate the convergence of the weighted GMRES method for solving linear systems.

Numer. Algorithms, 67(4):733--752, 2014.
@article{GP14,
title = {Some observations on weighted {GMRES}},
author = {G{\"u}ttel, Stefan and Pestana, Jennifer},
journal = {Numer. Algorithms},
volume = {67},
number = {4},
pages = {733--752},
year = {2014}
}

- Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection

We review various rational Krylov methods for the computation of large-scale matrix functions.

GAMM-Mitt., 36(1):8--31, 2013.
@article{Gue13b,
title = {Rational {K}rylov approximation of matrix functions: {N}umerical methods
and optimal pole selection},
author = {G{\"u}ttel, Stefan},
journal = {GAMM-Mitt.},
volume = {36},
number = {1},
pages = {8--31},
year = {2013}
}

- A black-box rational Arnoldi variant for Cauchy-Stieltjes matrix functions

*with L. Knizhnerman.*
We present and investigate a novel strategy for the automated parameter
selection when the function to be approximated is of Cauchy-Stieltjes (or Markov) type, such as the
matrix square root or the logarithm.
(Download MARKOVFUN.)

BIT Numer. Math., 53(3):595--616, 2013.
@article{GK13,
title = {A black-box rational {A}rnoldi variant for {C}auchy--{S}tieltjes matrix functions},
author = {G{\"u}ttel, Stefan and Knizhnerman, Leonid},
journal = {BIT Numer. Math.},
volume = {53},
number = {3},
pages = {595--616},
year = {2013}
}

- Robust Padé approximation via SVD

*with P. Gonnet and L. N. Trefethen.*
Padé approximation is considered from the point of view of robust methods of
numerical linear algebra, in particular the singular value decomposition. This leads to an algorithm
for practical computation that bypasses most problems of solution of nearly-singular systems and
spurious pole-zero pairs caused by rounding errors.

SIAM Rev., 55(1):101--117, 2013.
@article{GGT13,
title = {Robust {P}ad\'{e} approximation via {SVD}},
author = {Gonnet, Pedro and G{\"u}ttel, Stefan and Trefethen, Lloyd Nick},
journal = {SIAM Rev.},
volume = {55},
number = {1},
pages = {101--117},
year = {2013}
}

- PARAEXP: A parallel integrator for linear initial-value problems

*with M. J. Gander.*
A novel parallel algorithm for the integration of linear initial-value problems is proposed.
This algorithm is based on the observation that homogeneous problems can be integrated
faster than inhomogeneous problems if the inhomogeneity is suffciently difficult to integrate.

SIAM J. Sci. Comput., 35(2):C123--C142, 2013.
@article{GG13,
title = {{PARAEXP}: {A} parallel integrator for linear initial-value problems},
author = {Gander, Martin J. and G{\"u}ttel, Stefan},
journal = {SIAM J. Sci. Comput.},
volume = {35},
number = {2},
pages = {C123--C142},
year = {2013}
}

- Convergence of linear barycentric rational interpolation for analytic functions

*with G. Klein.*
With the help of logarithmic potential theory we derive
asymptotic convergence results for a class of linear barycentric rational interpolants proposed
by Floater and Hormann in 2007. We present suggestions on how to choose the involved
blending parameter in order to observe fast and stable convergence even with equispaced nodes.

SIAM J. Numer. Anal., 50(5):2560--2580, 2012.
@article{GK12,
title = {Convergence of linear barycentric rational interpolation for analytic functions},
author = {G{\"u}ttel, Stefan and Klein, Georges},
journal = {SIAM J. Numer. Anal.},
volume = {50},
number = {5},
pages = {2560--2580},
year = {2012}
}

- Superlinear convergence of the rational Arnoldi method

*with B. Beckermann.* We analyze the superlinear convergence behavior of the
rational Arnoldi method when being applied for the approximation of
Markov functions of matrices.

Numer. Math., 121(2):205--236, 2012.
@article{BG12,
title = {Superlinear convergence of the rational {A}rnoldi method},
author = {Beckermann, Bernhard and G{\"u}ttel, Stefan},
journal = {Numer. Math.},
volume = {121},
number = {2},
pages = {205--236},
year = {2012}
}

- Automated parameter selection for rational Arnoldi approximation of Markov functions

*with L. Knizhnerman.* Rational Arnoldi is a powerful method for approximating functions of large sparse matrices times a vector. The selection of
asymptotically optimal parameters for this method is crucial for its fast convergence. We present a heuristic for the automated
pole selection when the function to be approximated is of Markov type, such as the matrix square root. The performance of
this approach is demonstrated at several numerical examples.

Proc. Appl. Math. Mech., 11:15--18, 2011.
@article{GK11,
author = {G{\"u}ttel, Stefan and Knizhnerman, Leonid},
title = {Automated parameter selection for rational {A}rnoldi approximation of {M}arkov functions},
journal = {PAMM},
volume = {11},
number = {1},
publisher = {Wiley-VCH Verlag},
issn = {1617-7061},
url = {http://dx.doi.org/10.1002/pamm.201110005},
doi = {10.1002/pamm.201110005},
pages = {15--18},
year = {2011}
}

- A parallel overlapping time-domain decomposition method for ODEs

We introduce an overlapping time-domain decomposition method for linear
initial-value problems which gives rise to an efficient parallel solution method
without resorting to the frequency domain.

In R. Bank et al. (eds.), Domain Decomposition Methods in Science and Engineering XX, Lecture Notes in Computational Science and Engineering 91, pages 483--490. Springer-Verlag, Berlin, 2013.
@inproceedings{Gue13a,
author = {G{\"u}ttel, Stefan},
title = {A parallel overlapping time-domain decomposition method for {ODE}s},
year = {2013},
volume = {91},
editor = {Bank, R.E. and Holst, M. and Widlund, O.B. and Xu, J.},
booksubtitle = {Lecture Notes in Computational Science and Engineering},
booktitle = {Domain Decomposition Methods in Science and Engineering XX},
publisher = {Springer-Verlag, Berlin},
pages = {483--490}
}

- Rational Krylov Methods for Operator Functions

We present a unified and self-contained treatment of rational Krylov
methods for approximating the product of a function of a linear operator
with a vector. With the help of general rational Krylov decompositions
we reveal the connections between seemingly different approximation
methods, such as the Rayleigh--Ritz or shift-and-invert method, and derive
new methods, for example a restarted rational Krylov method and
a related method based on rational interpolation in prescribed nodes...

Dissertation, TU Bergakademie Freiberg, 2010.
@PhDThesis{Gue10,
title = {Rational {K}rylov Methods for Operator Functions},
author = {G{\"u}ttel, Stefan},
school = {Technische Universit{\"a}t Bergakademie Freiberg},
address = {Germany},
year = {2010},
note = {Dissertation available as MIMS Eprint 2017.39.},
url = {http://eprints.ma.man.ac.uk/2586/}
}

- Deflated restarting for matrix functions

*with M. Eiermann and O. G. Ernst.* We investigate an acceleration technique for restarted Krylov
subspace methods for approximating the action of a function of a large sparse matrix on a vector. The approximation is constructed with the
help of a generalization of Krylov decompositions to linearly dependent vectors. The restarted process is characterized as a successive interpolation scheme at Ritz values in which the exact
shifts are replaced with improved eigenvalue approximations in each cycle.

SIAM J. Matrix Anal. Appl., 32(2):621--641, 2011.
@article{EEG11,
title = {Deflated restarting for matrix functions},
author = {Eiermann, Michael and Ernst, Oliver G. and G{\"u}ttel, Stefan},
journal = {SIAM J. Matrix Anal. Appl.},
volume = {32},
number = {2},
pages = {621--641},
year = {2011}
}

- On the convergence of rational Ritz values

*with B. Beckermann and R. Vandebril.* The rational Krylov method is a method for computing parts of the spectrum of
a large Hermitian matrix. It is well known that its convergence behavior depends not only on the
distribution of eigenvalues but also on the choice of the poles which are free parameters. We characterize the region of good convergence for the rational Arnoldi
process, and obtain various results on the rate of approximation of a given eigenvalue by a rational
Ritz value.

SIAM J. Matrix Anal. Appl., 31(4):1740--1774, 2010.
@article{BGV10,
title = {On the convergence of rational Ritz values},
author = {Beckermann, Bernhard and G{\"u}ttel, Stefan and Vandebril, Raf},
journal = {SIAM J. Matrix Anal. Appl.},
volume = {31},
number = {4},
pages = {1740--1774},
year = {2010}
}

- A generalization of the steepest descent method for matrix functions

*with M. Afanasjew, M. Eiermann, and O. G. Ernst.* We consider the special case of the restarted Arnoldi method for approximating the product of a
function of a Hermitian matrix with a vector which results when the restart length is set to one. When applied
to the solution of a linear system of equations, this approach coincides with the method of steepest descent. We
show that the method is equivalent to an interpolation process in which the node sequence has at most two points of
accumulation. This knowledge is used to quantify the asymptotic convergence rate.

Electron. Trans. Numer. Anal., 28:206--222, 2008.
@article{AEEG08b,
title = {A generalization of the steepest descent method for matrix functions},
author = {Afanasjew, Martin and Eiermann, Michael and Ernst, Oliver G. and G{\"u}ttel, Stefan},
journal = {Electron. Trans. Numer. Anal.},
volume = {28},
pages = {206--222},
year = {2008}
}

- Implementation of a restarted Krylov subspace method for the evaluation of matrix functions

*with M. Afanasjew, M. Eiermann, and O. G. Ernst.* A new implementation of restarted Krylov subspace methods for evaluating f(A)b for a function f, a
matrix A and a vector b is proposed. In contrast to an implementation proposed previously, it requires constant
work and constant storage space per restart cycle. The convergence behavior of this scheme is discussed
and a new stopping criterion based on an error indicator is given. The performance of the implementation is
illustrated for three parabolic initial value problems, requiring the evaluation of exp(A)b.
(Download MATLAB implementation of FUNM_KRYL.)

Linear Algebra Appl., 429(10):2293--2314, 2008.
@article{AEEG08a,
title = {Implementation of a restarted {K}rylov subspace method for the evaluation of
matrix functions},
author = {Afanasjew, Martin and Eiermann, Michael and Ernst, Oliver G. and G{\"u}ttel, Stefan},
journal = {Linear Algebra Appl.},
volume = {429},
number = {10},
pages = {2293--2314},
year = {2008}
}

- Convergence Estimates of Krylov Subspace Methods for the Approximation of Matrix Functions Using Tools from Potential Theory

This diploma thesis reviews various definitions of matrix functions and polynomial Krylov methods for their approximation. Relations to polynomial interpolation and best approximation problems are made. The convergence behavior of Ritz values associated with Hermitian matrices is investigated. A new algorithm for the solution of the constrained energy problem with a measure supported in the complex plane is developed. This algorithm is then used to study Ritz values associated with a normal non-Hermitian matrix.

Diploma Thesis, 2006.
@MastersThesis{Gue06,
title = {Convergence Estimates of {K}rylov Subspace Methods for the Approximation of Matrix
Functions Using Tools from Potential Theory},
author = {G{\"u}ttel, Stefan},
school = {Technische Universit{\"a}t Bergakademie Freiberg},
address = {Germany},
year = {2006},
note = {Diploma thesis available as MIMS Eprint 2015.34.},
url = {http://eprints.ma.man.ac.uk/2301/}
}