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Luca Dieci

Publications and source records attributed to Luca Dieci.

At least 19 recordsLinked to original sources
arXiv2026

On the Optimal Laplacian Jordan Structure for Synchronizability

In this work, for a network of differential equations with a prescribed graph structure, our goal is to show how to select the Laplacian of the network in order to obtain the most favorable outcome insofar as synchronizability. That is, we will want to: (i) guarantee asymptotic stability of a synchronous solution (as measured by a negative value of the master stability function), (ii) minimize the normalized spread of the Laplacian eigenvalues, and (iii) have a transient as short as possible. Within the class of tridiagonal Laplacians, we give both necessary and sufficient conditions for satisfying our three criteria above, and give extension to banded Laplacians as well. Finally, we give extensive numerical results to elucidate our theoretical results and to compare to existing works.

math.OC↗
arXiv2026

Newton's method for optimal transport problem on graphs

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.

math.NA↗
arXiv2025

Computing Optimal Trajectories for Optimal Transport in Nonuniform Environments

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.

math.OC↗
arXiv2025

Generic Cuspidal Points and Their Localization

In this work we consider generic coalescing of eigenvalues of smooth complex valued matrix functions depending on 2 parameters. We call generic cuspidal points the parameter values where eigenvalues coalesce and we discuss the relation between cuspidal points and the closely related exceptional points studied in the literature. By considering loops in parameter space enclosing the cuspidal points, we rigorously prove when there is a phase accumulation for the eigenvectors and further detail how, by looking at the periodicity of the eigenvalues along the loop, and/or by looking at the aforementioned phase accumulation, one may be able to localize generic cuspidal points.

math.RA↗
arXiv2024

On an inverse tridiagonal eigenvalue problem and its application to synchronization of network motion

In this work, motivated by the study of stability of the synchronous orbit of a network with tridiagonal Laplacian matrix, we first solve an inverse eigenvalue problem which builds a tridiagonal Laplacian matrix with eigenvalues $\lambda_1=0<\lambda_2<\cdots <\lambda_N$ and null-vector $\boldsymbol{e} = \begin{bmatrix} 1 \\ \vdots \\ 1 \end{bmatrix}$. Then, we show how this result can be used to guarantee -- if possible -- that a synchronous orbit of a connected tridiagonal network associated to the matrix $L$ above is asymptotically stable, in the sense of having an associated negative Master Stability Function (MSF). We further show that there are limitations when we also impose symmetry for $L$.

math.DS↗
arXiv2024

SVD, joint-MVD, Berry phase, and generic loss of rank for a matrix valued function of 2 parameters

In this work we consider generic losses of rank for complex valued matrix functions depending on two parameters. We give theoretical results that characterize parameter regions where these losses of rank occur. Our main results consist in showing how following an appropriate smooth SVD along a closed loop it is possible to monitor the Berry phases accrued by the singular vectors to decide if -- inside the loop -- there are parameter values where a loss of rank takes place. It will be needed to use a new construction of a smooth SVD, which we call the "joint-MVD" (minimum variation decomposition).

math.RA↗
arXiv2024

Nonnegative moment coordinates on finite element geometries

In this paper, we introduce new generalized barycentric coordinates (coined as {\em moment coordinates}) on nonconvex quadrilaterals and convex hexahedra with planar faces. This work draws on recent advances in constructing interpolants to describe the motion of the Filippov sliding vector field in nonsmooth dynamical systems, in which nonnegative solutions of signed matrices based on (partial) distances are studied. For a finite element with $n$ vertices (nodes) in $\mathbb{R}^2$, the constant and linear reproducing conditions are supplemented with additional linear moment equations to set up a linear system of equations of full rank $n$, whose solution results in the nonnegative shape functions. On a simple (convex or nonconvex) quadrilateral, moment coordinates using signed distances are identical to mean value coordinates. For signed weights that are based on the product of distances to edges that are incident to a vertex and their edge lengths, we recover Wachspress coordinates on a convex quadrilateral. Moment coordinates are also constructed on a convex hexahedra with planar faces. We present proofs in support of the construction and plots of the shape functions that affirm its properties.

math.NA↗
arXiv2023

Forming a symmetric, unreduced, tridiagonal matrix with a given spectrum

Given a set of $n$ distinct real numbers, our goal is to form a symmetric, unreduced, tridiagonal, matrix with those numbers as eigenvalues. We give an algorithm which is a stable implementation of a naive algorithm forming the characteristic polynomial and then using a technique of Schmeisser.

math.NA↗
arXiv2023

Solving Semi-Discrete Optimal Transport Problems: star shapedeness and Newton's method

In this work, we propose a novel implementation of Newton's method for solving semi-discrete optimal transport (OT) problems for cost functions which are a positive combination of $p$-norms, $1<p<\infty$. It is well understood that the solution of a semi-discrete OT problem is equivalent to finding a partition of a bounded region in Laguerre cells, and we prove that the Laguerre cells are star-shaped with respect to the target points. By exploiting the geometry of the Laguerre cells, we obtain an efficient and reliable implementation of Newton's method to find the sought network structure. We provide implementation details and extensive results in support of our technique in 2-d problems, as well as comparison with other approaches used in the literature.

math.NA↗
arXiv2021

Takagi factorization of matrices depending on parameters and locating degeneracies of singular values

In this work we consider the Takagi factorization of a matrix valued function depending on parameters. We give smoothness and genericity results and pay particular attention to the concerns caused by having either a singular value equal to $0$ or multiple singular values. For these phenomena, we give theoretical results showing that their co-dimension is $2$, and we further develop and test numerical methods to locate in parameter space values where these occurrences take place. Numerical study of the density of these occurrences is performed.

math.NA↗
arXiv2021

A continuation multiple shooting method for Wasserstein geodesic equation

In this paper, we propose a numerical method to solve the classic $L^2$-optimal transport problem. Our algorithm is based on use of multiple shooting, in combination with a continuation procedure, to solve the boundary value problem associated to the transport problem. We exploit the viewpoint of Wasserstein Hamiltonian flow with initial and target densities, and our method is designed to retain the underlying Hamiltonian structure. Several numerical examples are presented to illustrate the performance of the method.

math.NA↗
arXiv2021

Decompositions and coalescing eigenvalues of symmetric definite pencils depending on parameters

In this work, we consider symmetric positive definite pencils depending on two parameters. That is, we are concerned with the generalized eigenvalue problem $A(x)-λB(x)$, where $A$ and $B$ are symmetric matrix valued functions in ${\mathbb R}^{n\times n}$, smoothly depending on parameters $x\in Ω\subset {\mathbb R}^2$; further, $B$ is also positive definite. In general, the eigenvalues of this multiparameter problem will not be smooth, the lack of smoothness resulting from eigenvalues being equal at some parameter values (conical intersections). We first give general theoretical results on the smoothness of eigenvalues and eigenvectors for the present generalized eigenvalue problem, and hence for the corresponding projections, and then perform a numerical study of the statistical properties of coalescing eigenvalues for pencils where $A$ and $B$ are either full or banded, for several bandwidths. Our numerical study will be performed with respect to a random matrix ensemble which respects the underlying engineering problems motivating our study.

math.NA↗
arXiv2020

Time Discretizations of Wasserstein-Hamiltonian Flows

We study discretizations of Hamiltonian systems on the probability density manifold equipped with the $L^2$-Wasserstein metric. Based on discrete optimal transport theory, several Hamiltonian systems on graph (lattice) with different weights are derived, which can be viewed as spatial discretizations to the original Hamiltonian systems. We prove the consistency and provide the approximate orders for those discretizations. By regularizing the system using Fisher information, we deduce an explicit lower bound for the density function, which guarantees that symplectic schemes can be used to discretize in time. Moreover, we show desirable long time behavior of these schemes, and demonstrate their performance on several numerical examples.

math.NA↗
arXiv2019

General auction method for real-valued optimal transport

Optimal transportation theory is an area of mathematics with real-world applications in fields ranging from economics to optimal control to machine learning. We propose a new algorithm for solving discrete transport (network flow) problems, based on classical auction methods. Auction methods were originally developed as an alternative to the Hungarian method for the assignment problem, so the classic auction-based algorithms solve integer-valued optimal transport by converting such problems into assignment problems. The general transport auction method we propose works directly on real-valued transport problems. Our results prove termination, bound the transport error, and relate our algorithm to the classic algorithms of Bertsekas and Castanon.

math.OC↗
arXiv2018

A study of deformation localization in nonlinear elastic lattices

The paper investigates localized deformation patterns resulting from the onset of instabilities in lattice structures. The study is motivated by previous observations on discrete hexagonal lattices, where the onset of non-uniform, quasi-static deformation patterns was associated with the loss of convexity of the interaction potential, and where a variety of localized deformations were found depending on loading configuration, lattice parameters and boundary conditions. These observations are here conducted on other lattice structures, with the goal of identifying models of reduced complexity that are able to provide insight into the key parameters that govern the onset of instability-induced localization. To this end, we first consider a two-dimensional square lattice consisting of point masses connected by in-plane axial springs and vertical ground springs. Results illustrate that depending on the choice of spring constants and their relative values, the lattice exhibits in-plane or out-of plane instabilities leading to folding and unfolding. This model is further simplified by considering the one-dimensional case of a spring-mass chain sitting on an elastic foundation. A bifurcation analysis of this lattice identifies the stable and unstable branches and illustrates its hysteretic and loading path-dependent behaviors. Finally, the lattice is further reduced to a minimal four mass model which undergoes a folding/unfolding process qualitatively similar to the same process in the central part of a longer chain, helping our understanding of localization in more complex systems. In contrast to the widespread assumption that localization is induced by defects or imperfections in a structure, this work illustrates that such phenomena can arise in perfect lattices as a consequence of the mode-shapes at the bifurcation points.

cond-mat.soft↗
arXiv2017

The boundary method for semi-discrete optimal transport partitions and Wasserstein distance computation

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.

math.NA↗