Along with experiment and theory, computation is nowadays widely
recognized as a crucial component for modern research in science and engineering.
Instead of carrying out experiments, which can be expensive
and time-consuming,
a prediction
using an advanced simulation
on a computer can be much cheaper and time-efficient.
At some point in such a research process
we often face a formulation or characterization
using the mathematical concepts
of
matrices, vectors and theory from linear algebra.
My research interest consists of all
aspects related to scientific computing and numerical analysis,
and my main research contributions have
been on the study of numerical
analysis for linear algebra operations,
also known as numerical linear algebra
or matrix computations.
Due
to the fact
that numerical linear algebra
appears in a wide variety of fields,
it has an unusual role as
a very interdisciplinary subject
and
my research has involved topics
in a number of related fields, such as
systems and control, quantum mechanics, optimization and
some applications in mechanical engineering.
Selection of current projects
Iterative methods for nonlinear eigenvalue problems
In many situations, a discretization of a partial
differential equation and a linearization lead to the problem of
computing eigenvalues of a large matrix. In contrast to this,
in this project derive and study algorithms for new types of
eigenvalue problems which are nonlinear, thereby providing the
possibility to study new recent types of physical models. The project
involves algorithms, theory, applications and software. The new
algorithms are analyzed and developed with tools from numerical
linear algebra and complex analysis, in particular Krylov methods and
the theory for analytic functions. The algorithms are used to
study specific problems in acoustics, in particular models of
waveguides, and also problems in quantum mechanics and quantum
chemistry.
Example of description of convergence as a function of a parameter for two choices of the Rayleigh-quotient vectors (E. Jarlebring and W. Michiels, Analyzing the convergence factor of residual inverse iteration, BIT numerical mathematics, 51(4):937-957, 2011). The iterative method
is convergent for all values inside the thick contour:
Example of a solution to the Gross-Pitaevskii equation
computed with the iterative method in E. Jarlebring, S. Kvaal and W. Michiels,
An inverse iteration method for eigenvalue problems with eigenvector nonlinearities, SIAM J. Sci. Comput., 2014:
Related publications:
R. Van Beeumen, E. Jarlebring and W. Michiels A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems Numer. Linear Algebra Appl., 23(4):607-628, 2016
E. Jarlebring, S. Kvaal and W. Michiels An inverse iteration method for eigenvalue problems with eigenvector nonlinearities SIAM J. Sci. Comput., 36-4:A1978-A2001, 2014
E. Jarlebring, K. Meerbergen and W. Michiels Computing a partial Schur factorization of nonlinear
eigenvalue problems using the infinite Arnoldi method SIAM J. Matrix Anal. Appl., 35(2):411-436, 2014
E. Jarlebring, W. Michiels and K. Meerbergen A linear eigenvalue algorithm for the nonlinear eigenvalue problem Numerische Mathematik, 122(1):169-195, 2012
E. Jarlebring and W. Michiels Analyzing the convergence factor of residual inverse iteration BIT numerical mathematics, 51(4):937-957, 2011
E. Jarlebring, K. Meerbergen and W. Michiels A Krylov method for the delay eigenvalue problem SIAM J. Sci. Comput., 32(6):3278-3300, 2010
E. Jarlebring Convergence factors of Newton methods for nonlinear eigenvalue problems Linear Algebra Appl., 436:3943-3953, 2012
E. Jarlebring and H. Voss Rational Krylov for Nonlinear Eigenproblems, an Iterative Projection Method Applications of Mathematics, 50(6):543-554, 2005
Computational solutions for time-delay systems
When external forces act on a
real-world system, say
a mechanical object, a chemical process,
a control device in a factory or
the gravitation from the sun acting on earth,
the reaction often
does not take place instantaneously. This
situation appears for many phenomena in, e.g., engineering, biology and chemistry.
In these situations,
the system and the
non-instantaneous action can be naturally
modeled using a differential equation
with a time delay
also known as a delay-differential equation.
In a simplified case, this
is mathematically described by
\[
\dot{x}(t)=A_0x(t)+A_1x(t-\tau_1).
\]
In the field of systems and control, one
usually also consider an input function u
and an output function y,
\begin{align}\dot{x}(t)&=A_0x(t)+A_1x(t-\tau_1)+Bu(t)\\
y(t)&=Cx(t).
\end{align}
I have contributed to the following problems and fields:
Compute eigenvalues which are
solutions to a nonlinear eigenvalue problem
Computing and characterize stability by computing
conditions such that there is an imaginary eigenvalue
Compute the ℋ₂ norm of a time-delay systems
Stability of a machine-tool milling model
Model reduction techniques for
time-delay systems based on
balanced truncation, dominant pole selection
and moment matching
Example of description of stability with manually computed eigenvalues and the expansion from theory with an application from machine tool milling:
Related publications:
E. Jarlebring, T. Damm and W. Michiels Model reduction of time-delay systems using position balancing and delay Lyapunov equations Mathematics of Control, Signals, and Systems, 25(2):147-166, 2013
M. Saadvandi, K. Meerbergen and E. Jarlebring On dominant poles and model reduction of second order time-delay systems Appl. Numer. Math., 62(1):21-34, 2012
J. Vanbiervliet, W. Michiels and E. Jarlebring Using spectral discretization for the optimal ℋ₂ design of time-delay systems Int. J. Control, 84(2):228-241, 2011
E. Jarlebring, J. Vanbiervliet and W. Michiels Characterizing and computing the ℋ₂ norm of time-delay systems by solving the delay Lyapunov equation IEEE Trans. Autom. Control, full paper, 56(4):814-825, 2011
W. Michiels, E. Jarlebring and K. Meerbergen Krylov-based model order reduction of time-delay systems SIAM J. Matrix Anal. Appl., 32(4):1399-1421, 2011
E. Jarlebring, W. Michiels, K. Meerbergen The infinite Arnoldi method and an application to time-delay systems with distributed delays chapter in edited volume, in In R. Sipahi, T. Vyhlidal, P. Pepe, S.-I. Niculescu, Eds., 'Time Delay Systems - Methods, Applications and New Trends'
E. Jarlebring, K. Meerbergen and W. Michiels A Krylov method for the delay eigenvalue problem SIAM J. Sci. Comput., 32(6):3278-3300, 2010
E. Jarlebring and W. Michiels Invariance properties in the root sensitivity of time-delay systems with double imaginary roots Automatica, 46:1112-1115, 2010
E. Jarlebring Critical delays and Polynomial Eigenvalue Problems J. Comput. Appl. Math., 224(1):296-306, 2009
E. Jarlebring and T. Damm The Lambert W function and the spectrum of some multidimensional time-delay systems Automatica, 43(12):2124-2128, 2007
E. Jarlebring The spectrum of delay-differential equations: numerical methods, stability and perturbation.
PhD thesis, Inst. Comp. Math, TU Braunschweig, 2008
Computational methods
and applications for multiparameter and double eigenvalue problems
This project concerns the problem of
determining choices of the scalar parameter μ such that
the matrix A+μB has a double eigenvalue.
The problem occurs in several fields in science, and
particular attention is given to problems in
quantum chemistry.
In particular, we construct a method to accurately find all
such solutions accurately. The method
is based on characterizing
the solution as an approximation
of a two-parameter eigenvalue
problem which can be solved accurately with
methods available in the literature. Other properties
of double eigenvalues and two-parameter eigenvalue
problems are also studied.
Related publications:
E. Jarlebring, S. Kvaal and W. Michiels Computing all pairs (λ,μ) such that λ is a double eigenvalue of A+μB SIAM J. Matrix Anal. Appl., 32(3):902-927, 2011
S. Kvaal, E. Jarlebring and W. Michiels Computing singularities of perturbation series Phys. Rev. A, 83:032505, 2011
E. Jarlebring and W. Michiels Invariance properties in the root sensitivity of time-delay systems with double imaginary roots Automatica, 46:1112-1115, 2010
E. Jarlebring and M.E. Hochstenbach Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations Linear Algebra Appl., 431(3-4):369-380, 2009