In this paper a compact Riemannian manifold with strictly convex boundary is reconstructed from its partial travel time data. This data assumes that an open measurement region on the boundary is given, and that for every point in the manifold, the respective distance function to the points on the measurement region is known. This geometric inverse problem has many connections to seismology, in particular to microseismicity. The reconstruction is based on embedding the manifold in a function space. This requires the differentiation of the distance functions. Therefore this paper also studies some global regularity properties of the distance function on a compact Riemannian manifold with strictly convex boundary.

}, journal={INVERSE PROBLEMS AND IMAGING}, author={Pavlechko, Ella and Saksala, Teemu}, year={2022}, month={May} } @article{anaya_anipchenko-ulaj_ashfaq_chiu_kaiser_ohsawa_owen_pavlechko_st john_suleria_et al._2020, title={Properties for the Frechet mean in Billera-Holmes-Vogtmann treespace}, volume={120}, ISSN={["1090-2074"]}, DOI={10.1016/j.aam.2020.102072}, abstractNote={Abstract The Billera-Holmes-Vogtmann (BHV) space of weighted trees can be embedded in Euclidean space, but the extrinsic Euclidean mean often lies outside of treespace. Sturm showed that the intrinsic Frechet mean exists and is unique in treespace. This Frechet mean can be approximated with an iterative algorithm, but bounds on the convergence of the algorithm are not known, and there is no other known polynomial algorithm for computing the Frechet mean nor even the edges present in the mean. We give the first necessary and sufficient conditions for an edge to be in the Frechet mean. The conditions are in the form of inequalities on the weights of the edges. These conditions provide a pre-processing step for finding the treespace orthant containing the Frechet mean. This work generalizes to orthant spaces.}, journal={ADVANCES IN APPLIED MATHEMATICS}, author={Anaya, Maria and Anipchenko-Ulaj, Olga and Ashfaq, Aisha and Chiu, Joyce and Kaiser, Mahedi and Ohsawa, Max Shoji and Owen, Megan and Pavlechko, Ella and St John, Katherine and Suleria, Shivam and et al.}, year={2020}, month={Sep} }