## Subjects for master dissertation (Mathematics, Engineering and Architecture, Ghent University)

Several topics are possible, related to the courses
• Introduction to Numerical Mathematics (E002910),
• Partial Differential Equations (C000802),
• Applied Functional Analysis (C001307),
• Approximation Methods for Boundary Value Problems (C001497),
• Mathematical Techniques for Engineers: Advanced Topics (E002683).

These topics typically consider ordinary and partial differential equations (e.g. their numerical analysis, their numerical implementation), (applied) functional analysis and inverse problems.

The master dissertation of some of my previous students can give you a better idea about the possibilities:

• Bjarne Dewilde (topic: image processing with fractional partial differential equations)
• Arnaud Devos (topic: image processing with Perona-Malik equation)
• Fien Gistelinck (topic: inverse problem - convolution kernel reconstruction)

Interested? Contact me! Some examples are (but not limited):

#### Fractional calculus and image processing.

• It has been demonstrated that many systems in science and engineering can be modeled more accurately by fractional-order than integer-order derivatives, and many methods are developed to solve problems including fractional derivatives. Fractional-order based methods have been used also in the image processing field (for instance in astronomy, see e.g. here and here). In this field, PDEs are used to restore pictures of poor quality. By using specific mathematical models, an image of much higher quality can be obtained. The goal of this thesis is (a) to give an introduction on image processing and fractional calculus, (b) to study different methods (e.g. Perona-Malik, fractional diffusion) and to compare the results with each other. Matlab can be used to implement the different models. An image processing toolbox is already available. For some interesting papers related to this subject, we refer to here, here and here.
• Also other subjects in the area of image processing are possible. Subject related to the courses C000802, C001307 and C001497 can focus more on the mathematical analysis of the different models.

#### The implementation of flow through porous media.

• There are numerous applications in which it is of interest to study the flow of multiple fluids through porous media. Some examples of these include the transport of dissolved nutrients through biological tissue and the extraction of petroleum from underground deposits. In many practical situations, the physics of such flows are characterised by coupled governing equations that are nonlinear and transient, and the problem is further complicated by material heterogeneity and irregular geometries. The goal of this thesis is to study (e.g. the derivation) and to implement existing numerical algorithms for such systems of equations (Darcy's law, porous medium equation, Navier-Stokes equation). The code will be implemented using the FEniCS Project (Python), which is based on the variational framework and the finite element method. There are different demos available for solving this type of equations. A computer can be provided for performing the simulations.

#### A heat transfer problem and a wave propagation problem with Robin type boundary condition in 1D: a case study.

• A heat and wave problem on a bounded domain in 1D accompanied by Robin type boundary conditions with real coefficients at the endpoints is considered. The goal of this thesis is to solve these problems by the method of separation of variables. The exact solution to these problems depend on the coefficients appearing in the Robin boundary conditions. The objective is to catch all possible situations in a phase diagram. Another goal is to develop a Python code such that numerical simulations for each situation can be established. If time is left, also an so-called inverse problem related to this type of equations can be studied. For instance, in the case of the heat equation, the reconstruction of the intial temperature can be studied based on a measurement of the temperature at some fixed time.

#### Inverse problems in Science and Engineering.

• In most of the course notes (partial) differential equations were only depicted as a tool for modelling physical phenomena. If some universal law is translated into mathematical language then this model can be used to predict the behavior of the quantity of interest. This is often called forward or direct problem. Inverse problems (IPs) -as the name suggests- do the opposite. They induce the reason which led to the result from the observed data.
• An example of an inverse problem is to determine the cause of a disease based on the results of a medical examination. It is easy to make a mistake when solving inverse problems. For example, symptoms that are associated with an HIV infection look like symptoms of other illnesses. It is thus impossible to tell, exclusively on the basis of symptoms, whether the problem is related to HIV or another medical condition. Therefore, the problem of determining the cause of a disease is called ill-posed, i.e. there is no unique cause (or solution). Additional medical investigations (measurements) are required to determine the correct cause. Similar issues are encountered when studying inverse problems for partial differential equations.
• Inverse problems arise in many areas of mathematical physics and applications are rapidly expanding to geophysics, chemistry, medicine and engineering. A typical example is the computed axial tomography (CAT or CT scan). CT provides clinically relevant anatomic and functional information, is relatively noninvasive, and has very low short- and long-term risks.
• The topics of the master dissertation range from the mathematical modelling and the theoretical analysis of inverse problems for partial differential equations where some parameters (right-hand side (heat/load source), kernel, diffusion coefficient, etc.), unknown boundary condition(s) or portion of the boundary are to be found, to the development of efficient numerical schemes and their practical application in sciences, engineering (eg. diffusion equation, beam equation, Maxwell's equations) and finance.
• The exact topic of the master dissertation is worked out in consultation with the thesis advisors. For each problem under consideration, the most important questions are:
• Which additional measurement is required for the unique reconstruction of the solution?
• How can the solution be reconstructed?

#### Inverse problems and regularization.

• Inverse problems often lead to mathematical problems that are not well-posed in the sense of Hadamard, i.e. to ill-posed problems. This means especially that their solution is unstable under data perturbations. Numerical methods that can cope with this problem are so-called regularization methods. This thesis is devoted to the study of inverse problems and/or regularization methods. Several topics are possible, see for instance here for a more detailed description.

#### The finite element method for Maxwell's equations.

• Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. The goal of this thesis is to learn how numerical solutions to (practical) problems related to these fundamental equations can be obtained. The goal of the dissertation is
• To introduce the physical laws that describe the phenomena of electromagnetism. This leads to the Maxwell’s equations in their classical form;
• To study Sobolev spaces for vector functions (definition/trace theorem/Green's theorem);
• To study finite elements on tetrahedra;
• To study and implement (using the finite element library DOLFIN of the FEniCS project) a particular problem derived from Maxwell equations (curl-curl formulation in terms of magnetic or electric field, scattering problem,...).
The exact topic of the master dissertation is worked out in consultation with the thesis advisors.

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#### Computational methods for flow problems in porous media.

• Het onderzoek naar stroming en transport doorheen poreuze media heeft duidelijke ecologische imperatieven, bijvoorbeeld inzake de indringing van zout water in de bodem, de remediëring van grondwatervervuiling (o.m. rond afvalstorten), de mogelijke opslag van radioactief afval in zoutmijnen of kleiputten, enzomeer. In een 1ste deel van de studie wordt het wiskundig vraagstuk van gekoppelde stroming en transport in poreuze media bestudeerd, zowel wat het model zelf betreft als inzake de numerieke methoden (algoritmen en hun analyse). Thema's die kunnen aan bod komen zijn o.m.: verzadigde versus onverzadigde stroming, één-fasige versus meer-fasige stroming, adsorptie en reacties van polluenten, eindige elementen en/of eindige differentiemethoden, methode der karakteristieken. In een 2e deel komen meer specifieke, zeer actuele onderwerpen aan bod: (1) densiteits-gebonden stroming (density driven flow) wat o.m. bij vraagstukken van zeewaterintrusie optreedt; (2) doorblazing van met vluchtige organische componenten verontreinigde grondlagen (soil venting). De graad van wiskundigheid kan in overleg worden bepaald, maar interesse voor wiskundige analyse, numerieke analyse en modellering wordt ondersteld. Dit onderwerp is meer geschikt voor een PhD-thesis.