Geometry Helps to Compare Persistence Diagrams

Exploiting geometric structure to improve the asymptotic complexity of discrete assignment problems is a well-studied subject. In contrast, the practical advantages of using geometry for such problems have not been explored. We implement geometric variants of the Hopcroft--Karp algorithm for bottleneck matching (based on previous work by Efrat el al.) and of the auction algorithm by Bertsekas for Wasserstein distance computation. Both implementations use k-d trees to replace a linear scan with a geometric proximity query. Our interest in this problem stems from the desire to compute distances between persistence diagrams, a problem that comes up frequently in topological data analysis. We show that our geometric matching algorithms lead to a substantial performance gain, both in running time and in memory consumption, over their purely combinatorial counterparts. Moreover, our implementation significantly outperforms the only other implementation available for comparing persistence diagrams

Calculation of Structures Lying On an Anisotropic Basis

The definition of the nucleus and the influence function of a transversally isotropic half-space is considered.  Expressions of deflection and internal forces in an infinite base plate are obtained, taking into account their deepening into the rock mass, as well as the effect of the anisotropy of the base on the distribution of deflections and internal forces. The results obtained serve as a reference for the reconciliation of the results obtained by numerical and computer methods. This calculation algorithm allows us to estimate the bearing capacity of an anisotropic soil base