We consider an overdetermined problem of Serrin-type with respect to an operator in divergence form with piecewise constant coefficients subject to both Dirichlet and Neumann conditions on the free boundary. These  problems appear in solid mechanics where the goal is to find the best shape of a cylindrical bar made of two different isotropic materials. A recent result by Cavallina and Yachimura asserts that if a fixed core is sufficiently close to a ball, then the two-phase Serrin-type problem admits a solution with a domain sufficiently close to a ball. In this work, we develop a level set method for approximating the solution to a two-phase Serrin-type overdetermined boundary problem based on the Osher-Sethian formulation and the boundary variation algorithm for shape optimization employing gradient descent of a Kohn-Vogelius-type functional. The implicit nature of the proposed method naturally handles topological changes and eliminates the need for numerical surgeries to handle mesh-related issues. Moreover, we present our computational examples that are consistent with those of Cavallina and Yachimura.

Admixture analysis is a method to infer an individual’s origins based on genetic ancestry. It aims to identify two or more previously isolated populations that interbred in the individual’s ancestors.  The recent availability of genome-wide high-density single nucleotide polymorphisms (SNPs) data has facilitated the study of admixture in humans. Admixture algorithms use statistical methods and are computationally intensive. In this study, we use persistent homology to analyze admixture in a population of captive marmosets. Persistent homology (PH) is an applied mathematics tool that is attractive to mathematicians and computer scientists due to its foundations in algebraic topology, linear algebra and computer science.  It can be used to analyze complex multi-dimensional data. It does so by identifying fundamental structures in the data through persistence diagrams and barcodes -- objects that summarizes the topological structure of data. We use persistent homology to analyze phased SNP data obtained from whole genome sequencing of 123 marmosets.

Using a computational pipeline based on persistent homology, we show that admixed datasets generally admit higher values related to their barcodes and persistence diagrams (PDs), particularly in birth and death times and number of cycles in their persistence diagrams. Furthermore, we employ bottleneck and Wasserstein distances, which are metrics on persistence diagrams, to differentiate PDs of admixed and non-admixed datasets. Results show that chromosome 10 has the most differentiation in its PDs. Additionally, we compared distances of PDs of the same class (admixed/non-admixed), which were then plotted to heatmaps for visualization. Heatmaps showed clearer dissimilarity of PDs using the Wasserstein metric.