The computation of stationary states of rotating Bose-Einstein condensates.
This project is devoted to solving nonlinear Schrödinger equations with an angular momentum rotation. Such equations occur in the context of rotating superfluids. One example are rotating Bose-Einstein condensates. Such condensates are formed when a dilute gas of Bosons is trapped in a magnetic potential and cooled down to ultra-low temperatures close to absolute zero. It is characterized by the property that the particles can no longer be separated from each other. They lose their identity and behave like one single super-atom, which in particular has a superfluid character. Superfluidity is expressed through a lattice of density singularities (vortices). In this project we aim at simulating realistic scenarios and to study the number of vortices and the patterns that they form. This requires to solve nonlinear eigenvalue problems.Handledare: Patrick Henning
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Approximation of the mean curvature vector
Accurate computation of the mean curvature vector on a discrete surface plays an important role in computer graphics as well as in ce rtain surface evolution problems. An accurate approximation of the mean curvature vector is for example essential for the approximati on of the surface tension force in simulations of multiphase flow problems. In this work we study a stabilized finite element method for approximation of the mean curvature vector.Handledare: Sara Zahedi
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Numerical methods for pattern recognition
Description: In modern society, huge amounts of data are collected and stored in various databases. The data is often unstructured an d information cannot be analyzed directly by a human being. The computerized extraction of qualitative information from enormous data sets is usually called ``data mining''. A typical example in data mining is to reliably identify hand-written characters in an autom atic way, for instance to determine the hand-written postal code on letters. In this project we study a specific approach to automati cally recognize hand-written digits. The approach is based on the numerical method called the singular value decomposition.Handledare: Elias Jarlebring
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Rear-wheel steering
One of the most challenging tasks for a beginner in car driving is the problem of backwards manoevring in space-limited parking places. It is so much easier to drive forwards! Why? Your task consists of constructing a model for the steering of a car, investigating some special cases analytically with respect to stability, and finally numerically investigate the general case.Handledare: Michael Hanke
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Multiscale methods for highly oscillatory ordinary differential equations
In this project we consider ODEs with multiple time scales, i.e. ODEs whose solution consists of fast oscillations with period ε<<1, superimposed on a slowly varying function. With standard numerical ODE methods the time step Δt must be taken smaller than ε to get an accurate result. The computational cost is therefore O(1/ε) which is very high when ε is small. Instead, we will study what is known as multiscale methods which are able to solve the problem at a computational cost that is almost independent of ε. The ODE problem is an example of a multiscale problem. They are common in many areas, from physics and chemistry to biology. They are characterized by the need to take very fine scale information into account even when only the coarse scales of the solution is of interest. This typically makes them very difficult to treat with numerical methods.Handledare: Olof Runborg
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Woodpecker
Mechanical systems systems of rigid bodies are often modelled by differential-algebraic systems. The woodpecker is a curious toy which features contact discontinuities. In practice one can observe that this toy leads to periodic oscillations. With this project, we try to model this behavior. Your task will be to develop a mathematical model and the corresponding numerical methods for simulating the woodpecker. In particular, you will show that stable oscillations exist and find the oscillation frequency.Handledare: Michael Hanke
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Trafik och folkträngsel med varierande framkomlighet
Avsikten med detta projekt är att modellera och numeriskt lösa flöde av gång, cykel eller biltrafik.Handledare: Anders Szepessy
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Numerical quadrature for singular integrals
Integrals with singular integrands occur for example when solving integral equations. In this project we will consider the numerical evaluation of integrals where the integrand is singular at some point within the integration interval, but where the integral is well defined. This requires special numerical quadrature schemes, and we will investigate some strategies to design such schemes.Handledare: Anna-Karin Tornberg
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Numerical issues of the elastic wave equation in highly heterogenous media
We consider the multiscale elastic wave equations. Elastic wave equations have a significant relevance for determining subsurface formations in the earth’s crust. Practically, an image of the underground is determined by generating an energy impulse (or vibrating source) that sends a seismic wave into the ground. A fraction of the wave is reflected, where the reflection pattern depends on the multiscale structures in the subsurface. When the reflected wave is measured by microphones, so called seismometers, it is possible to reconstruct the subsurface structures from the measured data. In this project we shall study so-called multiscale methods for solving the elastic wave equation and we discuss the numerical issues that one might face.Handledare: Patrick Henning
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Solving PDEs in “free space” by the use of FFTs
In many applications, one wants to numerically solve a partial differential equation (PDE) without imposing boundary conditions on an enclosing boundary and rather mimic an “infinite domain”. This is the case for example when simulating electromagnetic or acoustic waves that hit an object and scatters outwards. However, discretizing the PDE with traditional grid based techniques, a finite size domain must be used, and quite advanced techniques have been developed to avoid the artificial reflection that occurs at boundaries of the computational domain.Handledare: Anna-Karin Tornberg
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Optimal konsumtion och sparande
Avsikten med detta projekt är att formulera ett optimeringsproblem för sparande och konsumtion, beskriva problemets teoretiska bakgrund och numeriskt lösa problemet. Optimal konsumtion kan formuleras och analyseras med optimal styrteori, som är en generalisering av variationskalkyl. Ett optimalt styrproblem är ett optimeringsproblem med en differentialekvation som bivillkor.Handledare: Anders Szepessy
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