Luca Dieci

Shui-Nee Chow, Luca Dieci, Wuchen Li et al.

We propose a new semi-discretization scheme to approximate nonlinear Fokker-Planck equations, by exploiting the gradient flow structures with respect to the 2-Wasserstein metric. We discretize the underlying state by a finite graph and define a discrete 2-Wasserstein metric. Based on such metric, we introduce a dynamical system, which is a gradient flow of the discrete free energy. We prove that the new scheme maintains dissipativity of the free energy and converges to a discrete Gibbs measure at exponential (dissipation) rate. We exhibit these properties on several numerical examples.

Luca Dieci, Manuela Manetta, Haomin Zhou

In this work, we present a method to explore the landscape of a smooth potential in the search of global minimizers,combining a double-descent technique and a basin-escaping technique based on intermittent colored diffusion. Numerical results illustrate the performance of the method.

Qujiangxue Chen, Jianbo Cui, Luca Dieci et al.

In this paper, we study dynamical optimal transport on a connected graph from the perspective of the Benamou-Brenier formulation, where densities are assigned to vertices and velocities to edges. However, directly using Newton's method on the resulting nonlinear systems encounters two potential difficulties: (i) if the graph contains cycles, edge variables are not unique, and (ii) there is no guarantee that the density variables remain positive. To address these challenges, we introduce a finite-difference-type Newton method that eliminates cycle-induced redundancies through a spanning-tree gauge, resulting in a reduced set of independent variables and a well-posed, sparse linear system. For the lattice graph arising from the continuous optimal transport problem, density positivity can also be guaranteed by using an upwind discretization subject to a CFL-type condition. We further demonstrate the versatility of the proposed scheme by applying it to a range of problems, including optimal transport on lattices and random graphs, inverse optimal transport problems, and social network analysis.

Luca Dieci, Daniyar Omarov

In this work, we solve a discrete optimal transport problem in a nonuniform environment. To solve the optimal transport problem, we build the cost matrix and then use classical solvers for discrete optimal transport. The challenge is to form the cost matrix, which requires finding the optimal path between two points, and for this task we formulate and solve the associated Euler-Lagrange equations. A main contribution of ours is to provide verifiable sufficient conditions of optimality of the solution of the Euler-Lagrange equation and to propose new algorithms to to check optimality a-posteriori, thus validating the (exact) computation of the cost matrix. We illustrate our results and performance of the algorithms on several numerical examples in 2 and 3 dimensions.

Luca Dieci, J. D. Walsh

We introduce a new technique, which we call the boundary method, for solving semi-discrete optimal transport problems with a wide range of cost functions. The boundary method reduces the effective dimension of the problem, thus improving complexity. For cost functions equal to a p-norm with p in (1,infinity), we provide mathematical justification, convergence analysis, and algorithmic development. Our testing supports the boundary method with these p-norms, as well as other, more general cost functions.