1 August 2022
Full-time (part-time possible)
The goal of this thesis is to design a new data structure for simplicial complexes (in arbitrary dimensions) that is completely compatible with distributed computing analysis pipelines. Simplicial complexes are becoming more and more crucial in the analysis pipeline since they intrinsically provide topological information of the data. The raw (point cloud) data of interest is mainly produced by e.g. numerical simulations exploiting HPC resources or high-resolution sensor systems mounted on satellites. On such raw data a simplicial complex is constructed (either using a Delaunay meshing strategy, in low dimensions, or by generating a Vietoris-Rips complex in higher dimensions). Handling such a representation, it has been historically troublesome, and only in recent years, a novel model for efficient topological data structures over a broad range of simplicial and cell complexes has been defined, the Stellar decomposition.
A Stellar decomposition of a complex is a collection of regions indexing the complex’s vertices and cells such that each region has sufficient information to locally reconstruct the star of its vertices, i.e., the cells incident in the region’s vertices. Stellar decompositions are general in that they can compactly represent and efficiently traverse arbitrary complexes with a manifold or non-manifold domain. As a concrete realization of this model for spatially embedded complexes, the Stellar tree has been defined. A Stellar tree combines a nested spatial tree with a simple tuning parameter to control the number of vertices in a region. Stellar trees exploit the complex’s spatial locality by reordering vertex and cell indices according to the spatial decomposition and by compressing sequential ranges of indices. The major limitation of Stellar trees is how it is encoded the simplicial complex. This is encoded in a global data structure, that does not really support distributed computation in HPC facilities.
In this thesis we want to design another concrete realization of the Stellar decomposition model that entirely support distributed computations while at the same time providing a much more efficient strategy for encoding the simplicial complex. Some expected key features, this representation should have, are: (i) de-centralized encoding of the simplicial complex, such that the mesh is encoded independently by multiple MPI workers (for example) and still be able to provide a comprehensive and coherent representation of the simplicial complex, (ii) generic representation that can provide a compact way of encoding a simplicial complex also in single workstation configuration environments (compared to a Stellar tree), and (iii) to have comparable performance to Stellar trees at performing Topology Data Analysis operation on the indexed data.
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Dr. Riccardo Fellegara
Institute for Software Technology
Phone: +49 53 12953-429