Information for master students
Contents
Graduating in the numerical analysis group (chair Prof.dr.ir. C.
Vuik).
In the investigation of physical, biological and economical phenomena
numerical mathematics and computer simulations play an important role.
As an
example, we show a detail of the blood flow near a heart valve
(video 1
and
video 2).
In order to simulate such phenomena, a mathematical model of the
reality is
set up. Mathematical research is necessary to validate such a model.
For this, results, principles and techniques from mathematical analysis
are
often used. The model should, of course, be an adequate representation
of the
reality. The necessary knowledge to judge this may be obtained from
mathematical physics (biology, economy).
Numerical mathematics is then the basis to solve the mathematical model
efficiently and accurately. The solution is typically obtained by a
computer
simulation.
During a graduation at the chair for numerical analysis, the topics
analysis,
mathematical physics, linear algebra are treated. Most important is,
however,
the numerical analysis. The Master's thesis research can take place in
a
variety of topics in numerical mathematics, for example, in reducing
numerical errors (of a problem's discretization, for example), or to
improve
the efficiency of a solution process, to analyse the convergence
behaviour of
an iterative solution method, or in parallel computing.
The numerical questions always arise from practical applications.
If you would like to graduate in the numerical analysis group, the
typical procedure is as follows:
- First, you inform Prof. Vuik (or one of the co-workers) and
discuss
the courses that will be included in your
graduation package.
- Some months before starting the Master's thesis you contact Prof.
Vuik
(or one of the co-workers), so that your wishes concerning the topic of
the
thesis can be included while searching a Master's project.
- After that we will look for a suitable place to carry out the
research.
Examples of research institutions include: Philips NatLab, Delft
Hydraulics,
TNO-TPD, Shell, academic hospitals etc.
It is, of course, also possible to stay at TUD and have a regular
contact
with a member of the numerical analysis group.
-
Initially, you perform a literature study to get a good overview of
the
topic you will work on. This study (duration: 3 months) is finalized
in a
written report and with a short presentation.
-
After this, the research starts, which ends in a Master's thesis and
with an
oral presentation. This final stage of the graduation takes about 6
months.
Possibilities for Master projects
At this moment there are a number of Master projects available. It is
possible to formulate new projects where we take wishes of a masters
student into account.
-
Accelerating Computational Quantum Chemistry software package ADF using GPU's
-
Developing a fast (fast-time) solver for large sparse matrices for MARIN
-
European Modelling of the Ocean,
Example grid
-
Numerical method for airbag simulations
-
Grid generation for the Airbus 380
-
Numerical Modeling of Electromagnetic Fields
with Applications to Eddy Current Imaging
-
Eddy Current Imaging of Electrically Conducting Media
-
Multiscale Techniques for the Optimal Control of PDEs
with Application to the Design of Catalysts
-
Modeling Contactless Energy Transfer using a System of Spiral Antennas
-
Wave impacts: experimental and numerical research
-
Reducing the sensitivity of a preconditioner
-
Computation of thermo-acoustic modes in combustors (CERFACS)
-
Improving the performance of the MODFLOW solver for the
simulation of groundwater flow (Deltares)
-
Development of serious games (in collaboration with VSTEP)
-
Solving partial differential equations related to option pricing with
numerical methods
For further information about these project and graduation at the chair
Numerical Analysis we refer to:
Prof.dr.ir. C. Vuik 015 2785530
Prof.dr.ir. C.W. Oosterlee 015 2788283
Ir. A. Segal 015 2785535
Dr.ir. F. Vermolen 015 2787298
Dr.ir. M. van Gijzen 015 2782519
Dr.ir. D.R. van der Heul 015 2782632
Dr. D.J.P. Lahaye 015 2787257
Dr. J.K. Ryan 015 2789755
- Developing a fast (fast-time) solver for large sparse matrices for
MARIN
(Nederlands)
MARIN (Maritime Research Institute Netherlands) provides ship
manoeuvring simulators that offer a variety of maritime operations for
virtually every type of ship and of propulsion. The current
computation model for the wave field is based on cosine wave spectra,
that are converted to time signals through Fourier transformation.
This has the benefit of being deterministic in time and place and
therefore is easy to implement on our distributed simulation systems.
However, this model is not interactive, that is diffraction,
reflection, refraction and depth dependency are not taken into
account. From a visualization point of view, this model is limited
too. Better visualization models (in e.g. Waterworld, Titanic, Perfect
Storm) lack physical realism.
MARIN wishes to use the so-called Variational Boussinesq Model to
compute and visualize the wave field. This physically realistic model
does provide interaction with objects, diffraction, reflection etc. At
this moment, the model can be used for computing a wave field of 500m
by 500m (i.e., 10000 points) in real-time. To be useful in our
manoeuvring simulator, a much larger wave field of at least 5km by 5km
(1000000 points) must be computed in real time.
Part of this Variational Boussinesq Model is a CG sparse matrix
solver. The purpose of the work will be to speed up the current solver
or develop another (super)fast one for this kind of large matrices.
Possibilities include a parallell iterative CG method, a Box
Multi-grid method or implementation of a solver on the GPU by means of
CUDA.
CUDA.
Location
The work will be conducted at MARIN in Wageningen. MARIN has been an
independent and innovative service provider for the maritime sector
worldwide for more that 75 years now. The research is carried out by
model tests in large basins and by lots of computer simulations.
Above: the progression of a small wave coming from the left in a small
area, with a hollow (bottom left), a small harbour (bottom right) and
a small beach (top right).
-
Grid generation for the Airbus 380
(Nederlands)
An opportunity for an MSc project exists at Airbus Bremen on advanced grid
optimization, for further details contact Prof. Vuik.
- Computation of thermo-acoustic modes in combustors
(Nederlands)
in collaboration with Cerfacs
see, also the work by
Jan-willem van Leeuwen
Combustion oscillations are frequently encountered during the
development of
many combustion chambers for gas turbines. Testing burners in
simplified
combustions chambers is a common method to verify their stability but
this is
also an ambiguous approach because a given burner can be unstable in
one
chamber and not in the other. Predicting methods are therefore
requested.
A proper framework to analyse the combustion stability is the wave
equation in
a reacting flow. The thermo-acoustic modes can then be computed from
the
Helmholtz equation, the frequency domain version of the wave equation,
by
solving a large nonlinear eigenproblem.
The goal of this master's research is to improve the existing method
for
solving the nonlinear eigenproblem. The solution of this eigenproblem
is one
of the most time consuming parts of the analysis. The currently used
solution
method is a fixed point method in which in each iteration a large
quadratic
eigenproblem is solved. In the graduation research another approach,
based on
the Jacobi-Davidson method, will be investigated for solving the
nonlinear
eigenproblem. This techniques combines a Newton-iteration with a
subspace
acceleration. The Jacobi-Davidson method will be evaluated and
compared with
the fixed point iteration for a number of test problems, ranging from
academic to realistic.
- Improving the performance of the MODFLOW solver for the
simulation of groundwater flow
(Nederlands)
in collaboration with Deltares
In densely populated areas, land use and planning are closely related to
demands on water management of, for example, natural, agricultural and
recreational areas. It is therefore important to base management of both
groundwater and surface water systems on these demands. For this
purpose, more and more detailed large scale models are developed at
Deltares in which both saturated and unsaturated groundwater flow can be
coupled to the surface water flow.
Examples of developed high resolution models (25m horizontal) are the
MIPWA model for the Northern part of The Netherlands, consisting of
about 271 million active grid nodes, and the Limburg model, consisting
of over 383 million active grid nodes (see Figure).
Computed mean highest groundwater levels for a part of Limburg.
In these models the finite-difference code MODFLOW is used to solve the
groundwater flow equations that are based on Darcy?s law for flow
through porous media. MODFLOW computes approximate solutions of these
nonlinear flow equations by means of so-called Picard iterations (the
outer-loop). Inside each Picard iteration a linear system is solved with
the iterative preconditioned Conjugate Gradient method (the inner-loop).
Since the models used can be extremely large good performance of the
solver is important. So far, not much effort has been taken to optimize
the solver and usually the default solver settings are chosen.
Goal of the Master's project is to improve the solver performance.
Possibilities for research are:
- Optimization of the termination criterion of the inner iteration to the
convergence process of the outer iteration (like inexact Newton
methods);
- Improvement of the linearization (and its parameters) applied in the
Picard iterations (the outer loop);
- Improvement of the preconditioner (and its parameters) applied in the
inner iteration loop;
- Improvement of boundary conditions
Suitable methods are to be implemented in the MODFLOW code. The Master's
project will be carried out both at location Delft (formerly WL | Delft
Hydraulics) and in Utrecht (formerly TNO groundwater group).
-
Development of serious games
(Nederlands)
in collaboration with VSTEP
Daily supervisor: Kees Vuik
In the first generation of games the appearance and the contents of the
games were very simple. In for instance Pacman: a hungry symbol moves
along a grid and the challenge was to move
your cursor away from this symbol. In the current games, advanced 3D
visualitions are used and the plot resembles reality. For the next
generation of games the rate of reality should be increased further.
However, the game industry again encounters the bounds of the computer
power. To move beyond these bounds they challenge the scientific
computing community.
In this game the ship movements should be very realistic, with respect
to waves, currents and various propulsion mechanisms. Scientific
computing has a lot of experience in the simulation of a real ship with
high accuracy, which is necessary in order to predict the drag
accurately. For such a simulation it is necessary that the computer time
is less than one hour. To include a ship simulation algorithm
in a game, the computer
time should be less than one milli-second, however the accuracy can be
much less. So the challenge is to simplify the models and accelerate the
methods in such a way that the small computer time is attained.
In the world of games, a number of Dutch companies are active, such as
VSTEP, which are very successfull.
Since the TU Delft has a tradition in simulating ships, aircraft and
water flows, the game companies like to have TU Delft students join
their teams in order to develop the next generation of games.
It appears that these programs are also used for serious simulations
for the training of ship personnel, navy and firemans.
For more information klick
here.
-
Solving partial differential equations related to option pricing with
numerical methods
(Nederlands)
In these Master's thesis projects you solve partial differential
equations (pdes) or partial integro-differential equations in
financial mathematics with numerical methods. It is possible, under
certain assumptions on the development of stock prices, to compute the
value of an option with the help of pdes. The basic equation is the
well-known Black-Scholes equation. A standard option is a contract to
buy or sell stocks at a certain time in the future for a prescribed
amount of money. Nowadays, there are many different option variants on
the market. Some of them lead to interesting numerical aspects in the
discretization and numerical solution of the pde, like discontinuous
final conditions or higher dimensionality. An additional integral term
is appearring in the differential equation if jumps are included in
the model for the stock prices. This brings other interesting
numerical issues.
In these Master's thesis projects you will get acquainted with the
world of stocks and options by a literature studies.Furthermore, you
will solve a pde accurately and efficiently. We are interested in
options on more than one stock, but also in other exotic options.
Contactperson:
Prof.dr.ir. Kees Oosterlee
Previous Master projects
Below is a list of previous Master projects
- 2010
- 2009
- 2008
- 2007
- 2006
- 2005
- 2004
- 2003
- 2002
- 2001
How to deal with computer problems?
Contact information:
Kees
Vuik
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