Chunk Reduction for Multi-Parameter Persistent Homology

The extension of persistent homology to multi-parameter setups is an algorithmic challenge. Since most computation tasks scale badly with the size of the input complex, an important pre-processing step consists of simplifying the input while maintaining the homological information. We present an algorithm that drastically reduces the size of an input. Our approach is an extension of the chunk algorithm for persistent homology (Bauer et al., Topological Methods in Data Analysis and Visualization III, 2014). We show that our construction produces the smallest multi-filtered chain complex among all the complexes quasi-isomorphic to the input, improving on the guarantees of previous work in the context of discrete Morse theory. Our algorithm also offers an immediate parallelization scheme in shared memory. Already its sequential version compares favorably with existing simplification schemes, as we show by experimental evaluation

Chunk Reduction for Multi-Parameter Persistent Homology

The extension of persistent homology to multi-parameter setups is an algorithmic challenge. Since most computation tasks scale badly with the size of the input complex, an important pre-processing step consists of simplifying the input while maintaining the homological information. We present an algorithm that drastically reduces the size of an input. Our approach is an extension of the chunk algorithm for persistent homology (Bauer et al., Topological Methods in Data Analysis and Visualization III, 2014). We show that our construction produces the smallest multi-filtered chain complex among all the complexes quasi-isomorphic to the input, improving on the guarantees of previous work in the context of discrete Morse theory. Our algorithm also offers an immediate parallelization scheme in shared memory. Already its sequential version compares favorably with existing simplification schemes, as we show by experimental evaluation

Do Finite-Size Lyapunov Exponents Detect Coherent Structures?

Ridges of the Finite-Size Lyapunov Exponent (FSLE) field have been used as indicators of hyperbolic Lagrangian Coherent Structures (LCSs). A rigorous mathematical link between the FSLE and LCSs, however, has been missing. Here we prove that an FSLE ridge satisfying certain conditions does signal a nearby ridge of some Finite-Time Lyapunov Exponent (FTLE) field, which in turn indicates a hyperbolic LCS under further conditions. Other FSLE ridges violating our conditions, however, are seen to be false positives for LCSs. We also find further limitations of the FSLE in Lagrangian coherence detection, including ill-posedness, artificial jump-discontinuities, and sensitivity with respect to the computational time step.Comment: 22 pages, 7 figures, v3: corrects the z-axis labels of Fig. 2 (left) that appears in the version published in Chao

Current Trends for 4D Space-Time Topology for Semantic Flow Segmentation

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Solving the incompressible surface Navier-Stokes equation by surface finite elements

We consider a numerical approach for the incompressible surface Navier-Stokes equation on surfaces with arbitrary genus $g(\mathcal{S})$. The approach is based on a reformulation of the equation in Cartesian coordinates of the embedding $\mathbb{R}^3$, penalization of the normal component, a Chorin projection method and discretization in space by surface finite elements for each component. The approach thus requires only standard ingredients which most finite element implementations can offer. We compare computational results with discrete exterior calculus (DEC) simulations on a torus and demonstrate the interplay of the flow field with the topology by showing realizations of the Poincar\'e-Hopf theorem on $n$-tori