Résumé
- since 2011 Postdoc at University of Oxford (supported by a DFG grant)
- from 2010 to 2011 Postdoc at University of Geneva
- in 2010 obtained doctorate degree (Dr. rer. nat.) from TU Bergakademie Freiberg
- in 2008 visited the National Institute of Informatics in Japan (supported by a JSPS grant)
- in 2007 awarded the Georgius-Agricola medal of the university
- from 2006 to 2010 research associate and PhD candidate at TU Bergakademie Freiberg
- in 2006 received Diploma degree (Dipl.-Math.)
- from 2005 to 2006 student at University of Cyprus (supported by a DAAD grant)
- from 2001 to 2006 studied Applied Mathematics at TU Bergakademie Freiberg
- born 1981 in Dresden, Germany
Publications
- 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.
Submitted to GAMM-Mitteilungen, 2012.
- 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.
Submitted to SIAM Journal on Scientific Computing, 2012.
- 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.
Submitted 2012.
- 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.
Submitted to SIAM Journal on Scientific Computing, 2011.
- 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.
Accepted for publication in SIAM Review, 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., 121:205--236, 2012.
- Automated parameter selection for 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.
Accepted for publication in PAMM, 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. This parallel method exploits the
fact that homogeneous initial-value problems can be integrated much faster
than inhomogeneous problems using a fast Arnoldi approximation for the
matrix exponential function.
Accepted for publication in DD20 proceedings, 2011.
- 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 Thesis, 2010.
Published online (citable): http://nbn-resolving.de/urn:nbn:de:bsz:105-qucosa-27645
- Deflated restarting for matrix functions
with M. Eiermann and O. G. Ernst. We investigate an acceleration technique for restarted Krylov
subspace methods for computing the action of a function of a large sparse matrix on a vector. Its effect is
to ultimately deflate a specific invariant subspace of the matrix which most impedes the convergence of the restarted
approximation process. An approximation to the subspace to be deflated is successively refined in
the course of the underlying restarted Arnoldi process by extracting Ritz vectors and using those
closest to the spectral region of interest as exact shifts. The approximation is constructed with the
help of a generalization of Krylov decompositions to linearly dependent vectors. A description of
the restarted process as a successive interpolation scheme at Ritz values is given in which the exact
shifts are replaced with improved approximations of eigenvalues in each restart cycle. Numerical
experiments demonstrate the efficacy of the approach.
SIAM J. Matrix Anal. Appl., 32:621--641, 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. Under
fairly general assumptions 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. In particular, we quantify how rational Ritz values are attracted by poles. Our results
generalize recent findings on superlinear convergence of Krylov subspace methods. In particular, we
will also consider a constrained extremal problem in logarithmic potential theory where an additional
external field of a special form is required for taking into account the poles. Our analytic results are
illustrated by several examples.
SIAM J. Matrix Anal. Appl., 31:1740--1774, 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.
Electronic Transactions on Numerical Analysis. 28:206--222, 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.
Linear Algebra and its Applications. 429:2293--2314, 2008.
Talks and Posters
- Parallel integration of linear ODEs using rational Chebyshev
It is demonstrated how the solution of linear ODEs can be parallelized in time
with almost perfect speedup using an overlapping time-decomposition. You may
download a Matlab file of the rational Chebyshev method.
International Linear Algebra Society Conference, Pisa (Italy), 24.06.2010
- Time-parallel integration of linear ODEs
In this talk we report on joint work with M. Gander. It is demonstrated how the solution of linear ODEs can be parallelized in time
with almost perfect speedup.
AIMS Conference on Dynamical Systems, Differential Equations and Applications, Dresden (Germany), 26.05.2010
- Further talks on rational Krylov methods:
Swiss Numerics Day, ETH Zürich (Switzerland), 16.04.2010
DWCAA09 Conference, Canazei (Italy), 08.09.2009
Seminar TU Dresden, Dresden (Germany), 14.07.2009
- Rational Krylov methods and approximation of f(A)b
On this poster we report on joint work with B. Beckermann, M. Eiermann,
O. G. Ernst and R. Vandebril, and on some results from my dissertation. In particular, we show how to obtain a practical estimator for
the error arrising from
inexact solves of the shifted linear systems in the rational Krylov method, making use of a corrected Rayleigh quotient.
Additionally, an error estimator based on an extension of an idea of Saad (1992, "corrected schemes") is presented.
Conference on Scientific Computing, Geneva (Switzerland), 17.06.2009
- Various aspects of rational Krylov methods for matrix functions
In this talk we report on joint work with B. Beckermann, M. Eiermann,
O. G. Ernst and R. Vandebril. We characterize different approximations to f(A)b from rational Krylov spaces as approximation and interpolation problems. The latter characterization opens the question which nodes underlie the
interpolation process. This question is answered in an asymptotic sense with the help of weighted potential theory.
Academy of Sciences of the Czech Republic (Prague), 15.05.2009
- On Restarted Krylov Methods for the Approximation of Matrix Functions
We consider an interesting conjecture made by Forsythe and Motzkin in the early 1950s in connection with the directions of the optimum s-gradient method. We show that the asymptotic forward-backward
behavior of these directions can be explained by a local strong version of the conjecture, which is provable. We briefly describe why this conjecture is useful to understand the convergence of the restarted Lanczos method for the approximation of matrix functions.
Rolling Waves in Leuven (Leuven, Belgium), 15.12.2008
- Rational Krylov Methods for the Approximation of Matrix Functions
Numerik-Seminar (Freiberg), 04.12.2008
- Implementation of Restarted Krylov Method for the Approximation of Matrix Functions
Computational Linear Algebra with Applications (Harrachov, Czech Republic), 23.08.2007
- Restarted Krylov Subspace Approximations of Matrix Functions
ICIAM 2007 (Zürich, Switzerland), 19.07.2007
- On the Steepest Descent Method for the Evaluation of Matrix Functions
Südostdeutsches Kolloqium zur Numerischen Mathematik (Freiberg, Germany), 27.04.2007
- A Comparison of Krylov Subspace Methods and Time-Stepping for TEM Simulation
with M. Afanasjew, R.-U. Börner, and K. Spitzer. On this poster we compare the performance of Krylov Subspace Methods and
time-stepping methods for Transient Electromagnet simulations, March 2007
Miscancellous
Projects
- Restart code for the numerical evaluation of matrix functions
This Matlab code computes restarted Krylov approximations matrix functions.
An error bound (indicator for non-Hermitian matrices) is provided.
It is also possible to perform thick restarts or harmonic restarts.
- PARTYMAT.DE
PARTYMAT is a company providing a web-based event management system.
It was founded by Daniel Kroemer, Daniel Nimptsch and myself
in 2008 and has served more than 200,000 unique visitors since then.
- moondrops
Before moving to UK recently, I have been actively involved in various bands as lead guitarist and singer.
I also like composing songs and recording my own music.
- xetrader
Some time ago I have tried to code an automated virtual stock trading system in Matlab.
Xetrader is based on the heuristic assumption that market values of major companies tend to increase after
having been decreasing (relative to outperformers) over a long period.
Testing this assumption on a large enough data set,
the depot showed remarkable performance.
