Overview Research @ NALAG
An overview of current and past research
Current research
The research of the Numerical Approximation & Linear Algebra Group embraces several topics which are mainly categorized as Numerical Approximation or as Linear Algebra.
Linear algebra and rational approximation
There is the tight interplay between
structured matrices like
Toeplitz, Hankel, Loewner, etc., and corresponding problems of
rational interpolation and more generally
rational approximation.
Eigenvalue problems, acoustics and vibration
Large scale matrices with parameters arise in PDE constraint optimization, uncertainty modeling, structured distance problems, and graph applications. In order to reduce the computational cost to work with such matrices and related tensors, suitable model reduction techniques are required. Related is the solution of eigenvalue problems with specific structures, e.g. symmetric matrices, matrices consisting Kronecker sums, non-linear eigenvalue problems, etc. An important class of applications are structures and vibration.
Tensor decomposition
In signal processing, data mining, scientific computing and many other fields, quantities that are characterized by more than two indices are becoming increasingly important. Polynomials and functions in three or more variables are naturally represented in tensor form. Exponential signals, which play a fundamental role in signal processing, can be represented by rank-1 tensors. Semantic graphs, multilayer networks and hyperlink documents can also be represented by tensors. In scientific computing, the ubiquitous Kronecker product corresponds to the tensor outer product. Tensor techniques allow one to get around the "curse of dimensionality". This shift of paradigm concerns the most diverse aspects of mathematical engineering and goes together with the explosion of available information and the increase in computing power.
Systems, control, signal processing
Rational approximation and structured matrices play an important role in several aspects of linear systems, H-infinity control, and signal processing. Such applications are occasionally persued. See the DYSCO project.
Isogeometric analysis with Powell-Sabin splines
Numerical simulation and computer aided design are linked in the framework of isogeometric analysis. We investicate how classical tensor product NURBS can be replaced by Powell-Sabin splines in this context.
Generic Linear Algebra Software
Another software project GLAS is to produce C++ code for Generic Linear Algebra Software.
Exascale computing
The NAAM Unit is partner in the Flanders ExaScale Lab project, hosted by Intel. The goal is to prepare algorithms and hardware for the next generation supercomputers. Major challenges are the reduction of the energy cost of algorithms (rather than the classical flop count), the complex memory hierarchy that such systems will have, and dealing with hardware faults that may occur during a computation. The NALAG research group is involved in the development of Krylov methods and preconditioning for the solution of linear systems.
Rational approximation and orthogonal functions
Such linear algebra problems are often solved recursively, which links this up with recurrence relations, continued fractions, (bi)orthogonal polynomials, and (bi)orthogonal rational functions.
Fractional transforms
The fractional Fourier transform is a fractional power of the classical Fourier transform. It is another time-frequency technique that can be used for signal an image analysis.
Old research
Powell-Sabin splines and PDE
Multiresolution ideas in combination with Powell-Sabin splines give rise to subdivision techniques that allow for the design of efficient methods to compute preconditioners for the numerical solution of PDE's by FEM.
Powell-Sabin splines and CAGD
Multiresolution ideas in combination with Powell-Sabin splines give rise to subdivision techniques that allow for local modification of a cagd model.
Wavelets and multiresolution
Also
wavelets
became an important tool in signal and image processing.
This reseach is done in collaboration with
the research group Scientific Computing.
An alternative for the compression of horizon images is given
by normal offsets which is an adaptive
method to catch line disconinuities in images.
Splines and data fitting
Certainly for applications of data fitting the use of spline functions is essential. Spline functions and their use in curve and surface fitting is an important subject with a long tradition in the group. It resulted in a world-wide distributed software package FITPACK.
Didactics research
The members of the group have a heavy
teaching load, mainly in
undergraduate courses. A project was started about
the use of information technology in undergraduate mathematics
courses and the influence that this may have on the
didactics
of undergraduate math teaching.
There are projects visnue and
IPON
for stimulation of active learning in numerical analysis.
Besides the solution of linear systems, also
Subspace methods
iterative methods
for the solution of linear systems or
eigenvalue problems,
such as the
Lanczos
algorithm were done. Research on restarting eigenvalue
solvers has been a main topic.
This research is discontinued since 1998.
ROLLS
A highlight in the integration of linear algebra, linear systems, rational approximation and fast algorithms was certainly the project ROLLS covering Rational Approximation, Orthogonal Functions, Linear Algebra, Linear Systems, and Signal Processing.
Cryptography
With the recent developments in electronic commerce, elliptic curve
cryptosystems
have gained more attention. Because of their relative
short keys, these systems can be implemented in various settings, e.g.
chipcards. The research focusses on the mathematical background, in
cooperation with the department of mathematics.
This research is discontinued since 1999-2000.
