Topology, Metrics and Data: Computational Methods and Applications.

PhD Theses.The eld of topological data analysis (TDA) combines computational geometry and algebraic topology notions for analyzing data. This thesis presents methods and e cient algorithms that extend the TDA toolset. After introducing the needed background information about Euler characteristic curves and persistent homology, the former objects are extended to bi-dimensional ltrations. The result are Euler characteristic surfaces, which capture insights about data over a pair of parameters. Moreover, algorithms to compute these objects are described for both image and point data. Persistent homology in `1 metric is also studied. It is proven that in this setting Alpha and Cech ltration are not equivalent in general. On the other hand, two new ltrations | Alpha ag and Minibox | are de ned and proven equivalent to Cech ltrations in homological dimensions zero and one. Algorithms for nding Minibox edges are described, and Minibox ltrations are empirically shown to speed up the computation of Cech persistence diagrams with computational experiments. Then a new family of summary functions of persistence diagrams is de ned, which is related to persistence landscapes. These are called cumulative landscapes and are used to vectorize the information contained in persistence diagrams. In particular, discretizations of these functions and their Fourier coe cients are used to obtain feature vectors that can be applied in supervised classi cation problems. The e ectiveness of these feature vectors for the classi cation of data is compared against vectors obtained using persistence landscapes on two open-source datasets. Finally, a novel method is described for the analysis of high-dimensional genomics data. Optimized metrics are de ned on genomic vectors making use of a loss function. These are used in combination with a distance-based classi cation method, showing good performance compared to standard machine learning algorithms. Moreover, the structure of the given optimized metrics helps identify coordinates of the genomic vectors, which are most important for the classi cation task under study

Real-Time Simulation of Indoor Air Flow using the Lattice Boltzmann Method on Graphics Processing Unit

This thesis investigates the usability of the lattice Boltzmann method (LBM) for the simulation of indoor air flows in real-time. It describes the work undertaken during the three years of a Ph.D. study in the School of Mechanical Engineering at the University of Leeds, England. Real-time fluid simulation, i.e. the ability to simulate a virtual system as fast as the real system would evolve, can benefit to many engineering application such as the optimisation of the ventilation system design in data centres or the simulation of pollutant transport in hospitals. And although real-time fluid simulation is an active field of research in computer graphics, these are generally focused on creating visually appealing animation rather than aiming for physical accuracy. The approach taken for this thesis is different as it starts from a physics based model, the lattice Boltzmann method, and takes advantage of the computational power of a graphics processing unit (GPU) to achieve real-time compute capability while maintaining good physical accuracy. The lattice Boltzmann method is reviewed and detailed references are given a variety of models. Particular attention is given to turbulence modelling using the Smagorinsky model in LBM for the simulation of high Reynolds number flow and the coupling of two LBM simulations to simulate thermal flows under the Boussinesq approximation. A detailed analysis of the implementation of the LBM on GPU is conducted. A special attention is given to the optimisation of the algorithm, and the program kernel is shown to achieve a performance of up to 1.5 billion lattice node updates per second, which is found to be sufficient for coarse real-time simulations. Additionally, a review of the real-time visualisation integrated within the program is presented and some of the techniques for automated code generation are introduced. The resulting software is validated against benchmark flows, using their analytical solutions whenever possible, or against other simulation results obtained using accepted method from classical computational fluid dynamics (CFD) either as published in the literature or simulated in-house. The LBM is shown to resolve the flow with similar accuracy and in less time