Doctoral School Mathematics and Scientific Computing

Abstract: Mapper graphs are a tool to summarize the connectivity of a topological space with respect to a choice of a lens function and a choice of a cover. To capture more topological information we decorate the mapper graph with local homology information of the base space. Such a decorated mapper graph corresponds to a discretization of the so-called Leray cosheaf with respect to the lens function. We show that the decorated mapper graph converges to the continuous Leray cosheaf if the resolution of the cover goes to zero.[hide abstract]

Abstract: We analyze semilinear reaction–diffusion systems that are mass controlled, and have nonlinearities that satisfy critical growth rates. The systems under consideration are only assumed to satisfy natural assumptions, namely the preservation of non-negativity and a control of the total mass. It is proved in dimension one that if nonlinearities have (slightly super-) cubic growth rates then the system has a unique global classical solutions. Moreover, in the case of mass dissipation, the solution is bounded uniformly in time in sup-norm.

Abstract: State estimation is crucial for data assimilation but can be computationally demanding. Conventional approaches employ variational formulations, at high computational cost. To address this issue, reduced-order models have been introduced, replacing the full-order model. The state estimation can be also cast as an optimal recovery approach. In this context, the Parametrized Background Data-Weak method offers a promising strategy. It enables state estimation without parameter estimation, circumventing the complexity of nonlinear inverse problems. We explore applications to the reconstruction time-dependent problems, incorporating artificial sensor measurements and sensor selection algorithms to enhance performance.

Abstract: Given a data set X, the standard pipeline of topological data analysis involves the construction of a commutative diagram of simplicial complexes F(X) as an intermediate step. A simplicial bifiltration is a diagram F(X) indexed by a product of two totally ordered sets. A further classification is then made depending on the number of generators of each simplex, classifying F(X) to be either k-critical for or ∞-critical. We will construct examples of bifiltrations and sketch a way to reduce k-critical bifiltrations to the 1-critical case.

Abstract: The Delaunay filtration D(X) of a point cloud X is a central tool of computational topology. Its use is justified by the topological equivalence of D(X) and the offset (i.e., union-of-balls) filtration of X. Given a function γ:X→R, we introduce a Delaunay bifiltration DC(γ) that satisfies an analogous topological equivalence and is of small size, ensuring that DC(γ) topologically encodes the offset filtrations of all sublevel sets of γ, as well as the topological relations between them