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A numerical inverse method called FlowPaths is presented to solve for the hydraulic conductivity field of an isotropic heterogeneous porous medium from a known specific discharge field (and constant-head boundary conditions). This method makes possible a new approach to reactive transport experiments, aimed at understanding the dynamic spatial and temporal evolution of hydraulic conductivity, which simultaneously record the evolving reaction and the evolving flow geometry. This inverse method assumes steady, two-dimensional flow through a square matrix of grid blocks. A graph-theoretical approach is used to find a set of flow paths through the porous medium using the known components of the specific discharge, where every vertex is traversed by at least one path from the upstream high-head boundary to the downstream low-head boundary. Darcy’s law is used to create an equation for the unknown head drop across each edge. Summation of these edge equations along each path through the network generates a set of linearly independent head-drop equations that is solved directly for the hydraulic conductivity field. FlowPaths is verified by generating 12,740 hydraulic conductivity fields of varying size and heterogeneity, calculating the corresponding specific discharge field for each, and then using that specific discharge field to estimate the underlying hydraulic conductivity field. When estimates from FlowPaths are compared to the simulated hydraulic conductivity fields, the inverse method is demonstrated to be accurate and numerically stable. Accordingly, within certain limitations, FlowPaths can be used in field or laboratory applications to find hydraulic conductivity from a known velocity field. 

The separability of clusters is one of the most desired properties in clustering. There is a wide range of settings in which different clusterings of the same data set appear. We are interested in applications for which there is a need for an explicit, gradual transition of one separable clustering into another one. This transition should be a sequence of simple, natural steps that upholds separability of the clusters throughout. We design an algorithm for such a transition. We exploit the intimate connection of separability and linear programming over bounded-shape partition and transportation polytopes: separable clusterings lie on the boundary of partition polytopes and form a subset of the vertices of the corresponding transportation polytopes, and circuits of both polytopes are readily interpreted as sequential or cyclical exchanges of items between clusters. This allows for a natural approach to achieve the desired transition through a combination of two walks: an edge walk between two so-called radial clusterings in a transportation polytope, computed through an adaptation of classical tools of sensitivity analysis and parametric programming, and a walk from a separable clustering to a corresponding radial clustering, computed through a tailored, iterative routine updating cluster sizes and reoptimizing the cluster assignment of items. Funding: Borgwardt gratefully acknowledges support of this work through National Science Foundation [Grant 2006183] Circuit Walks in Optimization, Algorithmic Foundations, Division of Computing and Communication Foundations; through Air Force Office of Scientific Research [Grant FA9550-21-1-0233] The Hirsch Conjecture for Totally-Unimodular Polyhedra; and through Simons Collaboration [Grant 524210] Polyhedral Theory in Data Analytics. Happach has been supported by the Alexander von Humboldt Foundation with funds from the German Federal Ministry of Education and Research.