Optimal transport methods for estimation and control in critical infrastructure

Abstract

This thesis studies computational optimal transport methods for modelling, estimation and control problems in networked systems. The focus is on scenarios where the available information is incomplete, aggregated, or constrained by the physical structure of the network. Such settings arise naturally in critical infrastructure systems, including water distribution networks, where contaminant flows must be inferred from sparse measurements, and transportation networks, where vehicles or agents must be steered under physical constraints. The thesis formulates these problems using multi-marginal entropy-regularized optimal transport and Schrödinger bridge methods, taking into consideration both theoretical and computational aspects, developing algorithms based on Sinkhorn-type iterations and entropic proximal schemes.

The first paper considers the problem of estimating the spread of contaminants in water distribution networks from sparse sensor measurements. The water flow is modelled as a time-varying Markov chain, and the pollutant evolution is recovered as a Schrödinger bridge problem with partial marginal observations. A dual formulation and a Sinkhorn-type algorithm are derived, and the method is illustrated on simulated water-network data.

The second paper extends this formulation to the case where the first marginal is also only partially observed, corresponding to an unknown contamination source. This leads to an incomplete-information problem with unknown total mass and possible non-uniqueness. The paper characterizes the optimal solution set in terms of observability of an associated time-varying linear system, and proposes an algorithm that combines an entropic proximal scheme with Sinkhorn-type iterations. The method is validated on experimental data collected at a water distribution laboratory.

The third paper studies the steering of electric fleets over networks with origin-destination, battery-charge, and capacity constraints. By augmenting the physical network with discrete charge states, the routing problem is formulated as a structured multi-marginal optimal transport problem. A dual coordinate ascent algorithm is developed, exploiting the structure and sparsity of the problem, and the method is demonstrated with numerical simulations on a grid network.

The fourth paper addresses the estimation of Markov transition matrices from aggregate observations of indistinguishable particles. The problem is formulated as a convex inverse optimal transport problem, where transport plans and the transition matrix are estimated jointly. The paper provides existence, uniqueness and duality results, proposing an entropic proximal algorithm for computing the solution. Numerical experiments show that the method can recover the underlying dynamics when the observations sufficiently excite the state space.

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