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Hyperspectral image analysis works with image data where each pixel contains hundreds of wavelengths acquired from spectral measurements. It is a growing field of research in the sciences and industries because it can distinguish visually similar objects. While many machine-learning methods work well for analysing regular images, little is known about how they perform on hyperspectral data. Standard methods for quantifying and classifying hyperspectral data include the chemometric methods PLS, PLS-DA and SIMCA. They provide rapid computations along with intuitive modelling and diagnostic tools, but cannot capture more complex data. I benchmarked the chemometric methods against machine learning methods from Microsoft's ML.NET library on six classification and two quantification problems. The ML.NET methods proved to be good complements to the chemometric methods. In particular, the decision tree methods provided accurate classification and quantification while the maximum entropy classification methods balanced between accuracy and computational time the best. While the remaining ML.NET methods performed equally well or better than the chemometric methods, finding their use requires testing on data sets with a wider range of properties. The best ML.NET methods are suitable for analysing more complex hyperspectral images by capturing nonlinearities disregarded by standard image analysis.

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Motivated by the study of the hypoxia problem in cancerous tissues, we propose a system of coupled partial differential equations defined on a heterogeneous, periodically perforated  domain describing the flux of oxygen from blood vessels towards the tissue and the corresponding oxygen diffusion within the tissue. Using heuristics based on dimensional analysis, we rephrase the initially parabolic problem as a semi-linear elliptic transmission problem. Focusing on the elliptic case, we are able to define a microscopic $\varepsilon$-dependent problem that is the starting point of our mathematical analysis; here $\varepsilon$ is linked to the scale of heterogeneity.   We study the well-posedness of the microscopic problem as well as the passage to the periodic homogenization limit. Additionally, we derive the strong formulation of the two-scale macroscopic limit problem. Finally, we prove a corrector estimate. This specific ingredient allows us to estimate, in an {\em a priori} way,  the discrepancy between solutions to the microscopic and, respectively, macroscopic problem. Our working techniques include energy-type estimates, fixed-point type iterations, monotonicity arguments, as well as the two-scale convergence tool.

In early drug discovery and the study of the effects of new chemical compounds on cancer cells, the change in cell shape over time provides vital information about cell health. Live-cell image analysis systems can be used to extract cell-shape describing parameters of individual cells during exposure to new drugs. Multivariate statistical analysis is then applied to understand cell morphology and the correlation between various shape descriptors. Principal component analysis integrated with histogram distribution analysis is a method to compress and summarize important cellular data features without loss of information about the individual cell shapes. A workflow for this kind of analysis is being developed at Sartorius and aims to aid in the biological interpretation of different experimental results. However, methods for exploring the time dimension in the experiments are not yet fully explored, and a temporal view of the data would increase understanding of the change in cell morphology metrics over time. In this study, we implement the workflow to a data set generated from the microscope IncuCyte and investigate a possible continuation of time-series analysis on the data. The results demonstrate how we can use principal component analysis in two steps together with histogram distributions of different experimental conditions to study cell shapes over time. Scores and loadings from the analysis are used as new observations representing the original data, and the evolution of score-value can be backtracked to cell morphology metrics changing in time. The results show a comprehensive way of studying how cells from all experimental conditions relate to each other during the course of an experiment.

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The present licentiate thesis is devoted to the rigorous derivation of the equations governing thin-film flow of incompressible Newtonian and non-Newtonian fluids. More precisely, we consider flow in a generalized Hele-Shaw cell, which is a thin three-dimensional domain confined between two surfaces connected by cylindrical obstacles of various shapes. Thin-film flows arise naturally in several applications. For instance, it is commonly used when the domain itself has different characteristic lengths in different directions, i.e. when the domain is a thin layer or a slender tube. Mathematically, the flow is described by a set of partial differential equations in a thin domain which depends on a small parameterε, e.g. the ratio of two characteristic lengths. By letting ε tend to zero, one can obtain a better understanding of the properties of solutions of such equations. In this limit process, all variables involved (e.g. velocity and pressure) depend on ε and the resulting limit problem yields a simplified model of the flow. There exist several mathematical techniques that have been developed to deal with such problems, e.g. asymptotic expansions, two-scale convergence for thin domains, etc. The scientific results in this thesis are presented in two papers (I and II) and a complimentary appendix. The results are discussed in a more general context in an introduction which also gives an overview of the subject. In both papers, we assume that the flow is governed by the Stokes system posed in a generalized Hele-Shaw cell satisfying a mixed boundary condition. The so-called no-slip and no-penetration conditions require that the velocity vanishes on the solid surfaces of the cell. This condition is complemented by the normal stress condition on the lateral boundary which is defined by an external pressure. Physically this means that the motion of the fluid is caused by the external pressure gradient, which acts in a direction parallel to the surfaces. One of the main objectives of this thesis is to develop a rigorous mathematical description of pressure-driven flow in thin domains. In paper I, we consider Hele-Shaw flow of an incompressible Newtonian fluid. The results are based on the formal asymptotic expansion method, i.e. by introducing a small parameter ε representing the thickness of the domain, rescaling the problem to a fixed domain, and considering solutions in the form of power series of ε. It is shown that the leading term of the velocity satisfies the so-called Poiseuille-law, i.e. the velocity is a linear function of the pressure gradient, whereas the leading pressure term satisfies the generalized Hele-Shaw equation. The main result is the construction of an approximate solution, which is justified by estimating the L2-norm of the error, i.e. the difference between the exact solution and the approximation. In paper II, the situation is similar to that of paper I, but the fluid obeys a more general constitutive relationship between the stress and the shear rate. More precisely, the functional relationship between the viscosity and the symmetrical part of the velocity gradient is given by a power-law. We develop techniques of functional analysis and calculus of variations in order to justify theorems concerning the existence and uniqueness of weak solutions of the corresponding Stokes system. The nonlinear Poiseuille-law, i.e. the limit velocity and the limit pressure gradient follow a power-law, is derived by using a two-scale convergence procedure and monotonicity arguments. Finally, uniqueness and regularity results for the solution of the limit problem are proved.

This thesis consists of an introduction, and one research paper with results related to potential theory both in the classical Euclidean setting, as well as in quite general metric spaces. The introduction contains a theoretical and historical background of some basic concepts, and their more modern generalisations to metric spaces developed in the last 30 years. By using upper gradients it is possible to define such notions as first order Sobolev spaces, p-harmonic functions and capacity on metric spaces. When generalising classical results to metric spaces, one often needs to impose some structure on the space by making additional assumptions, such as a doubling condition and a Poincaré inequality. In the included research paper, we study a certain type of metric spaces called bow-ties, which consist of two metric spaces glued together at a single designated point. For a doubling measure μ, we characterise when μ supports a Poincar´e inequality on the bow-tie, in terms of Poincaré inequalities on the separate parts together with a variational p-capacity condition and a quasiconvexity-type condition. The variational p-capacity condition is then characterised by a sharp measure decay condition at the designated point. We also study the special case when the bow-tie consists of the positive and negative hyperquadrants in Rn, equipped with a radial doubling measure. In this setting, we characterise the validity of the p-Poincaré inequality in various ways, and then provide a formula for the variational p-capacity of annuli centred at the origin.i