# Mathematical Sciences

## Mathematics LTH

• Modelling and simulation of flotation
• Flotation is a process commonly used in mineral processing and wastewater treatment to separate some material from a suspension by the use of bubbles and the degree of hydrophobicity of different materials. The volume fraction of bubbles in a flotation column can be modelled by a nonlinear partial differential equations and simulated by a suitable numerical method. Prerequisites: good grades in courses in mathematics (up to Kontinuerliga system) and numerical analysis
• Modelling, simulation and flux identification of traffic flow
• Traffic flow can be modelled with a nonlinear partial differential equation (PDE), whose solution gives the vehicle density along the road. The project means some theory, implementation of a numerical method, simulation of traffic-flow situations and identification of the traffic-flow flux function from traffic movies. Prerequisites: good grades in courses in mathematics (up to Kontinuerliga system) and numerical analysis
• Genetic selection under recurrent mutation and finite population size
• The simplest description of evolution in genetic terms is that mutations form new beneficial alleles (=gene variants) that enter a population and then increase in frequency until they have replaced the previous variants. However, if the population size is limited, random fluctuation will occur with respect to the frequency of the beneficial allele and it may be lost in spite of it being better than all other existing alleles. If the same, or equivalent, mutations are allowed to occur then the beneficial allele will eventually reach fixation (reach the frequency 100%). This project will investigate the distribution of times to fixation in this scenario. The underlying mathematics is based on diffusion processes.
• Multimodal 3D image analysis of oat and barley seeds – segmentation and machine learning
• Analysis of 3D images using X-rays and neutrons allow for complementary non-destructive analysis of structure and chemical composition of biological materials such as oat and barley seeds using machine learning techniques. Contrast differences between the modes arise due to the differences in interaction with seed matter. Due to the high sensitivity to hydrogen, neutrons are useful for separating liquid water or hydrogenous phases from the underlying structure while X-rays resolve the solid structure.
• Adherent raindrop modeling and detection
• Develop and test existing algorithms for detecting raindrops on the frontglas/dome on Axis cameras. The algorithm should be accurate, efficient and can not be CPU intensive since it should smoothly run on a 1 GHz CPU camera

## Mathematical Statistics

• Lika vård i hela Sverige?
• I Sverige eftersträvas jämstäld vård i hela landet. En viktig fråga vid utvärderingen av vård är om skillnader i vård kvalitet mellan landsting är slumpmässiga eller om de tyder på strukturella skillnader mellan landstingen. Ett sätt att modellera vårdkvalitet som försöker att identifiera faktiska skillnader mellan landstingen är att använda cox-regression med ett lämpligt LASSO straff som grupperar liknande effekter. De resulterande modellerna kan anpassas med hjälp av convexa optimeringsmetoder (t.ex. ADMM eller proximal gradients). En stor del av examensarbetet kommer utgöras av implementering av dessa optimerings metoder (d.v.s. att generalisera tidigare resultat för cox-regression med enklare LASSO straff). Metoderna kommer sen att testas på ett dataset bestående av över 20 000 patienter med en kronisk sjukdom. Arbetet är ett fortsättning på ett tidigare kandidatarbete https://lup.lub.lu.se/student-papers/search/publication/8925320
• Cross-spectral analysis of HRV data connected to work related stress
• Cross-spectral analysis between the Heart Rate Variability (HRV) and the breathing signals, can be seen as a novel and refined technique with higher sensitivity than the usual HRV power, a standardized measure in medical care related to cardiac diseases. This project aims for classification of groups of patients with stress related diagnosis using cross-spectral techniques. The work includes analysis and evaluation the Welch method and multitaper based methods, on a novel set of metronome guided HRV measurement data. Prerequisites: FMSF10/MASC04, (FMSN35/MASM26)
• Classification of bird song syllables using low-rank approximations of time-frequency images
• A bird's song is used as an identification tool, serving as a recognition signal to indicate the individual, the kinship and the species. The studies so far are however impaired by the lack of methods which would automatically and objectively analyse the song structure. In this project, the singular vectors when decomposing the multitaper spectrogram image are proposed to be used as feature vectors in classification of bird song syllables. The approach is expected to be especially suitable for signals consisting stochastic components with variance in amplitudes as well as time- and frequency locations. Prerequisites: FMSF10/MASC04, FMSN35/MASM26
• Feature extraction for dialect classification
• Speech signals provide both linguistic information, as well as information about the specific speaker. The aim of this project is to go beyond the traditional speech and speaker analysis and extract suitable features for swedish dialect classification based on the word ‘hallå’, spoken in a number of specifically defined situations, e.g. answering the phone or meeting a friend. Available data are recordings of 36 subjects with different dialects. Additionally, to remove the speaker specific influence of the analysis, unique recordings have been made by the same professional imitator using different dialects. Prerequisites: FMSF10/MASC04, (FMS051/MASM17), FMSN35/MASM26
• Suppression of interference in time-frequency representations using penalty functions
• In the sense of energy concentration the Wigner-Ville distribution the most optimal time-frequency representation of a signal. The problem with the Wigner-Ville distribution is that it suffers heavily from interference in the form of cross-terms. The distribution can be filtered by a kernel to supress the cross-terms, however this often also smooths the auto-terms, resulting in loss of resolution. In this project the connection between the Wigner-Ville distribution and the multitaper spectrogram is used to design penalty functions that suppress cross-terms. The aim is to keep the resolution of the Wigner-Ville distribution, but still suppressing interference in an effective way. This approach can be applied to many different non-stationary signals, depending on how the penalty kernels are designed. Prerequisites: FMSF10/MASC04, FMSN35/MASM26
• Wavelet denoising of non-stationary signals before calculation of the reassigned spectrogram
• For non-stationary signals the readability of the spectrogram can be improved by reassigning mass to the centre of gravity, thus increasing the concentration and improving the time-frequency localisation of signal components. The reassignment of noisy signals can however result in erroneous peaks and biased reassignment. This project investigates if reducing noise in signals using the discrete wavelet transform before calculating the reassigned spectrogram can yield better time-frequency representations. The discrete wavelet transform is a popular method for denoising signals, but the effects of doing so before reassignment has not been evaluated. In the project different wavelets and thresholding techniques can investigated. Prerequisites: FMSF10/MASC04, FMSN35/MASM26
• Spatiotemporal modelling of sound envelope for improved auditory attention identification
• Not a lot is known about the remarkable ability of humans to separate a single sound source from a dense mixture of sound sources in a crowded background, the cocktail-party scenario. More knowledge could lead to a breakthrough for the next-generation hearing aids to have the ability to be cognitively controlled. However, a key that helped the field to progress is that human cortical activity follows the sound envelope. The aim of this project is to investigate if robust time-frequency estimation can be used to accurately ‘reconstruct’ speech from the recorded brain responses and identify the sound source of the listener’s interest. There are two different available datasets. The first dataset contains recordings of 30 subjects, where were instructed to attend to a specific sound source, on either the left or right side during the entire experiment. The second dataset contains recordings of 30 subjects, where the subjects were instructed to switch their attention from one sound source to another during the experiment. The project is performed in close collaboration with Eriksholm Research Centre, Oticon A/S, Denmark. Prerequisites: FMSF10/MASC04, (FMS051/MASM17), FMSN35/MASM26
• Using Emulators to Assess the Climate Sensitivity of a Vegetation Model
• Vegetation models can be used to assess the impacts of climate change on future food production (i.e. farming). However, vegetation models are computationally expensive to run and to account for the uncertainty in future climate scenarios the models should, ideally, be run using as many different scenarios as possible. One way of reducing the computational cost of this procedure is to fit a statistical model to a suitable subset of vegetation model runs and then use the statistical model to cheaply obtain estimates of the vegetation model.
• Coherence and cross-spectral analysis for understanding the relations between sound and the ‘listening’ brain
• We are remarkably good at focusing on only one talker in a scene consisting of multiple, spatially separated talkers, also known as the cocktail-party scenario. However, our knowledge of the brain’s ability in these situations is very limited. More knowledge could lead to a breakthrough for the next-generation hearing aids to have the ability to be cognitively controlled. In this project, we touch on this challenge in terms of trying to determine where in the time- and frequency domain there is correlation between the sound and the brain response. Coherence and cross-spectral analysis of the sound and the brain responses should be investigated using different techniques, to find such relations. There are two different available datasets. The first dataset contains multichannel electroencephalogram (EEG) recordings of 30 subjects, instructed to attend to one sound source, on either the left or right side during the entire experiment. The second dataset contains similar recordings of 30 subjects, instructed to switch their attention from one sound source to another during the experiment. The project is performed in close collaboration with Eriksholm Research Centre, Oticon A/S, Denmark. Prerequisites: FMSF10/MASC04, (FMS051/MASM17), FMSN35/MASM26
• Using Emulators for Optimization: Fitting a Vegetation Model to Field Observations
• The basic idea of a Gaussian process emulator is to fit a statistical model to output from a (computationally expensive) model. The statistical model can then be used to cheaply obtain new values from the complex model. If the complex model contains parameters which have to be optimised the emulator can be used to aid the optimisation by suggesting new evalution points, and excluding unlikely regions from the optimisation. Here the emulator will be used to fit a vegetation model to observed data.
• Invasive Species: How Fast does it Spread?
• Based on aerial imaging during several years we wish to model the spread of an invasive plant species, Rosa rugosa, across a small island on the Baltic Sea Coast between Lubeck and Rostock. The aerial imaging gives us information regarding the location, size, and first ocurance of each plant and the thesis aim is to build a statistical model for growth rate of individual plants as well as the establishment of plants at new locations.
• Spatial Modelling of Insurance Claims and Cost
• An important question in insurance is the accurate pricing of risks. A common approach is to consider the socio-economic status (income, education, age, etc.) of both costumer and the area where said customer lives. However, recent work has shown that also accounting for spatial dependence between neighbouring regions leads to improved estimates of accident risks. The aim of this thesis is to expand on the previous work to provide joint models for both accident risk and cost of the resulting insurance claims.
• Equal Care in All of Sweden?
• Sweden strives for equal care across the country. An important question when evaluating and comparing the effects of care between different regions (landsting) is how much of the differences that are due to random effects and how much can be accounted for by structural differences between the regions. One option is to model the survival time of patients using a cox-regression with a suitable penalty term (lasso) that groups regions with similar effects together.
• Pricing of exotic derivatives in the presence of jumps and stochastic volatility
• In many cases we see that real assets cannot be modeled with reasonable accuracy by Geometric Brownian motions. So we need more complex models to get a good fit to real data. This project will deal with techniques to valuate derivatives using such models.

## Numerical Analysis

• Simulation of a wastewater treatment plant with positivity-preserving numerical methods
• Implementation of special numerical methods for simulation of a wastewater treatment plant consisting of biological reactors and a sedimentation tank, modelled with a system of coupled nonlinear ordinary and partial differential equations. Prerequisites: good grades in courses in mathematics (up to Kontinuerliga system) and numerical analysis
• Optimization of pedestrian evacuation modelled by PDE
• Pedestrian flow can be modelled with a nonlinear partial differential equation (PDE), whose solution gives the density along the path. The flux of people is a nonlinear function of the local density, project means some theory, implementation of a numerical method, simulation of traffic-flow situations and identification of the traffic-flow flux function from traffic movies. Prerequisites: good grades in courses in mathematics (up to Kontinuerliga system) and numerical analysis
• Time adaptivity in PDE constrained optimization
• The project is about using time adaptive methods within an optimization procedure where as a solution constraint, a time dependent PDE is given, for example the compressible Navier-Stokes equations for compressible gases like air. To obtain search directions in the optimization, so called adjoints are used, which require solving time dependent problem, where time adaptivity would greatly improve the time to solution. Course requirements: FMNN10, recommended are FMNN25, FMA051 and MMVN05
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