  The multiscale simplicial flat norm (MSFN) of a d-cycle is a family of
optimal homology problems indexed by a scale parameter {\lambda} >= 0. Each
instance (mSFN) optimizes the total weight of a homologous d-cycle and a
bounded (d + 1)-chain, with one of the components being scaled by {\lambda}.We
propose a min-cost flow formulation for solving instances of mSFN at a given
scale {\lambda} in polynomial time in the case of (d + 1)-dimensional
simplicial complexes embedded in {R^(d + 1)} and homology over Z. Furthermore,
we establish the weak and strong dualities for mSFN, as well as the
complementary slackness conditions. Additionally, we prove optimality
conditions for directed flow formulations with cohomology over Z+.
  Next, we propose an approach based on the multiscale flat norm, a notion of
distance between objects defined in the field of geometric measure theory, to
compute the distance between a pair of planar geometric networks. Using a
triangulation of the domain containing the input networks, the flat norm
distance between two networks at a given scale can be computed by solving a
linear program. In addition, this computation automatically identifies the 2D
regions (patches) that capture where the two networks are different. We
demonstrate through 2D examples that the flat norm distance can capture the
variations of inputs more accurately than the commonly used Hausdorff distance.
As a notion of stability, we also derive upper bounds on the flat norm distance
between a simple 1D curve and its perturbed version as a function of the radius
of perturbation for a restricted class of perturbations. We demonstrate our
approach on a set of actual power networks from a county in the USA. Our
approach can be extended to validate synthetic networks created for multiple
infrastructures such as transportation, communication, water, and gas networks.
