Senior Lecturer in Numerical Analysis

The University of Manchester

School of Mathematics

Alan Turing Building, Room 2.114

M13 9PL, Manchester, UK

+44 161 275 5849 (office)

+44 7449 348 273 (mobile)

stefan@guettel.com

Résumé

My primary research interests are numerical analysis, in particular, the design
and analysis of numerical algorithms for the solution of partial differential equations and related high-dimensional
linear algebra problems, rational approximation, scientific computing, and parallel algorithms.

My current research is devoted to rational Krylov methods for the efficient solution of Maxwell's equations and nonlinear eigenvalue problems, optimized transparent boundary conditions, and deferred correction methods.

My current research is devoted to rational Krylov methods for the efficient solution of Maxwell's equations and nonlinear eigenvalue problems, optimized transparent boundary conditions, and deferred correction methods.

- since 2016 Senior Lecturer in Numerical Analysis
- from 2012 to 2016 Lecturer in Numerical Analysis at The University of Manchester
- from 2011 to 2012 Postdoc at University of Oxford (funded by DFG)
- 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 (funded by JSPS)
- 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 DAAD)
- from 2001 to 2006 studied Applied Mathematics at TU Bergakademie Freiberg

Publications

- 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. - 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.

Submitted, 2016. - A rational deferred correction approach to PDE-constrained optimization

*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.

Submitted, 2016. - 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.

Submitted, 2015. - 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. - 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. - 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. - 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. - 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. - 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. - 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. - 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. - 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. - 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. - 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. - 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. - 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., 35:661--683, 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. - 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. - 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 Mitteilungen, 36(1):8--31, 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. - 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. - 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. - 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., 50:2560--2580, 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.

Proc. Appl. Math. Mech., 11:15--18, 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. - 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.

Electron. Trans. Numer. Anal., 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. (Download MATLAB implementation of FUNM_KRYL.)

Linear Algebra Appl., 429:2293--2314, 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.

Teaching

- MATH36001 Matrix Analysis (semester 1, see Blackboard for course materials)
- MATH65740 Introduction to Matlab (semester 1 in the academic year 2015/16, week 1)

Software

- A Rational Krylov Toolbox for MATLAB

The Rational Krylov Toolbox contains MATLAB implementations of Ruhe's rational Krylov sequence method, algorithms for the implicit and explicit relocation of the poles of a rational Krylov space, and an implementation of RKFIT, an algorithm for rational least squares fitting. - FUNM_KRYL: Restart code for matrix function evaluation

The FUNM_KRYL code computes restarted Krylov approximations matrix functions. An error indicator is provided. It is also possible to perform thick restarts or harmonic restarts.**Update:**An improved version based on numerical quadrature can be found here: FUNM_QUAD.

An nice overview of restarted Krylov methods is given in the dissertation of Marcel Schweitzer. - NLEIGS: Solver for nonlinear eigenproblems

NLEIGS is a rational Krylov method for the solution of nonlinear eigenvalue problems. It is based on a companion-type linearization obtained from a dynamically computed linear rational interpolant, along with a scaling procedure for robustness. - MARKOVFUNMV: A black-box rational Arnoldi method
for f(A)b

This is a Matlab implementation of a black-box rational Arnoldi method for Markov functions of matrices times a vector which is based on an automated pole selection heuristics. - PARTYMAT.DE

PARTYMAT.DE is a commercial company providing a web-based event management system. It was founded by Daniel Kroemer, Daniel Nimptsch and myself in 2008 and currently serves about 1.3 million unique visitors/year. - Partiso.co.uk

Partiso is the UK version of PARTYMAT.DE, developed from scratch and online since 2013. - xetrader

Some time ago I attempted to code an automated virtual stock trading system in Matlab. Xetrader was 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. This assumption was tested on a large enough data set and the depot showed a remarkable performance.