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Andrea Natale

Publications and source records attributed to Andrea Natale.

At least 19 recordsLinked to original sources
arXiv2026

Semi-discrete convex order and Laguerre tessellation fitting

Laguerre tessellations offer an efficient way to parameterize a large class of convex partitions of Euclidean space using only a set of points and scalar weights. For this reason, they have become popular in computational geometry, imaging and numerical analysis, both as a modeling and a discretization tool. In this paper we study the problem of reconstructing a Laguerre tessellation with prescribed cell volumes from the barycenters of its cells. We establish a geometric interpretation of this problem in terms of the set of discrete measures dominated in convex order by an absolutely continuous measure. In particular, we show that the reconstruction problem can be solved approximately by computing a Wasserstein projection onto this set. More generally, our method can also be applied to fit a Laguerre tessellation to an arbitrary set of barycenters. We give a concrete application of this in materials science, of fitting a Laguerre tessellation to an electron backscatter diffraction (EBSD) image of a steel.

math.OC↗
arXiv2025

Infinitely many saturated travelling waves for a degenerate Fisher-KPP equation not in divergence form

We consider an epidemic model with distributed-contacts. When the contact kernel concentrates, one formally reaches a very degenerate Fisher-KPP equation with a diffusion term that is not in divergence form. We make an exhaustive study of its travelling waves. For every admissible speed, there exist not only a unique non-saturated (smooth) wave but also infinitely many saturated (sharp) ones. Furthermore their tails may differ from what is usually expected. These results are thus in sharp contrast with their counterparts on related models.

math.AP↗
arXiv2024

Discretizing the Fokker-Planck equation with second-order accuracy: a dissipation driven approach

We propose a fully discrete finite volume scheme for the standard Fokker-Planck equation. The space discretization relies on the well-known square-root approximation, which falls into the framework of two-point flux approximations. Our time discretization is novel and relies on a tailored nonlinear mid-point rule, designed to accurately capture the dissipative structure of the model. We establish well-posedness for the scheme, positivity of the solutions, as well as a fully discrete energy-dissipation inequality mimicking the continuous one. We then prove the rigorous convergence of the scheme under mildly restrictive conditions on the unstructured grids, which can be easily satisfied in practice. Numerical simulations show that our scheme is second order accurate both in time and space, and that one can solve the discrete nonlinear systems arising at each time step using Newton's method with low computational cost.

math.AP↗
arXiv2024

Metric extrapolation in the Wasserstein space

In this article we study a variational problem providing a way to extend for all times minimizing geodesics connecting two given probability measures, in the Wasserstein space. This is simply obtained by allowing for negative coefficients in the classical variational characterization of Wasserstein barycenters. We show that this problem admits two equivalent convex formulations: the first can be seen as a particular instance of Toland duality and the second is a barycentric optimal transport problem. We propose an efficient numerical scheme to solve the latter formulation based on entropic regularization and a variant of Sinkhorn algorithm.

math.OC↗
arXiv2024

Efficient preconditioners for solving dynamical optimal transport via interior point methods

In this paper we address the numerical solution of the quadratic optimal transport problem in its dynamical form, the so-called Benamou-Brenier formulation. When solved using interior point methods, the main computational bottleneck is the solution of large saddle point linear systems arising from the associated Newton-Raphson scheme. The main purpose of this paper is to design efficient preconditioners to solve these linear systems via iterative methods. Among the proposed preconditioners, we introduce one based on the partial commutation of the operators that compose the dual Schur complement of these saddle point linear systems, which we refer as $\boldsymbol{B}\boldsymbol{B}$-preconditioner. A series of numerical tests show that the $\boldsymbol{B}\boldsymbol{B}$-preconditioner is the most efficient among those presented, despite a performance deterioration in the last steps of the interior point method. It is in fact the only one having a CPU-time that scales only slightly worse than linearly with respect to the number of unknowns used to discretize the problem.

math.NA↗
arXiv2023

From geodesic extrapolation to a variational BDF2 scheme for Wasserstein gradient flows

We introduce a time discretization for Wasserstein gradient flows based on the classical Backward Differentiation Formula of order two. The main building block of the scheme is the notion of geodesic extrapolation in the Wasserstein space, which in general is not uniquely defined. We propose several possible definitions for such an operation, and we prove convergence of the resulting scheme to the limit PDE, in the case of the Fokker-Planck equation. For a specific choice of extrapolation we also prove a more general result, that is convergence towards EVI flows. Finally, we propose a variational finite volume discretization of the scheme which numerically achieves second order accuracy in both space and time.

math.AP↗
arXiv2023

Gradient flows of interacting Laguerre cells as discrete porous media flows

We study a class of discrete models in which a collection of particles evolves in time following the gradient flow of an energy depending on the cell areas of an associated Laguerre (i.e. a weighted Voronoi) tessellation. We consider the high number of cell limit of such systems and, using a modulated energy argument, we prove convergence towards smooth solutions of nonlinear diffusion PDEs of porous medium type.

math.NA↗
arXiv2022

Atrial Fibrillation Recurrence Risk Prediction from 12-lead ECG Recorded Pre- and Post-Ablation Procedure

Introduction: 12-lead electrocardiogram (ECG) is recorded during atrial fibrillation (AF) catheter ablation procedure (CAP). It is not easy to determine if CAP was successful without a long follow-up assessing for AF recurrence (AFR). Therefore, an AFR risk prediction algorithm could enable a better management of CAP patients. In this research, we extracted features from 12-lead ECG recorded before and after CAP and train an AFR risk prediction machine learning model. Methods: Pre- and post-CAP segments were extracted from 112 patients. The analysis included a signal quality criterion, heart rate variability and morphological biomarkers engineered from the 12-lead ECG (804 features overall). 43 out of the 112 patients (n) had AFR clinical endpoint available. These were utilized to assess the feasibility of AFR risk prediction, using either pre or post CAP features. A random forest classifier was trained within a nested cross validation framework. Results: 36 features were found statistically significant for distinguishing between the pre and post surgery states (n=112). For the classification, an area under the receiver operating characteristic (AUROC) curve was reported with AUROC_pre=0.64 and AUROC_post=0.74 (n=43). Discussion and conclusions: This preliminary analysis showed the feasibility of AFR risk prediction. Such a model could be used to improve CAP management.

cs.LG↗
arXiv2022

A mixed finite element discretization of dynamical optimal transport

In this paper we introduce a new class of finite element discretizations of the quadratic optimal transport problem based on its dynamical formulation. These generalize to the finite element setting the finite difference scheme proposed by Papadakis et al. [SIAM J Imaging Sci, 7(1):212--238,2014]. We solve the discrete problem using a proximal splitting approach and we show how to modify this in the presence of regularization terms which are relevant for physical data interpolation.

math.NA↗
arXiv2021

Convergence of a Lagrangian discretization for barotropic fluids and porous media flow

When expressed in Lagrangian variables, the equations of motion for compressible (barotropic) fluids have the structure of a classical Hamiltonian system in which the potential energy is given by the internal energy of the fluid. The dissipative counterpart of such a system coincides with the porous medium equation, which can be cast in the form of a gradient flow for the same internal energy. Motivated by these related variational structures, we propose a particle method for both problems in which the internal energy is replaced by its Moreau-Yosida regularization in the L2 sense, which can be efficiently computed as a semi-discrete optimal transport problem. Using a modulated energy argument which exploits the convexity of the problem in Eulerian variables, we prove quantitative convergence estimates towards smooth solutions. We verify such estimates by means of several numerical tests.

math.AP↗
arXiv2021

Computation of Optimal Transport with Finite Volumes

We construct Two-Point Flux Approximation (TPFA) finite volume schemes to solve the quadratic optimal transport problem in its dynamic form, namely the problem originally introduced by Benamou and Brenier. We show numerically that these type of discretizations are prone to form instabilities in their more natural implementation, and we propose a variation based on nested meshes in order to overcome these issues. Despite the lack of strict convexity of the problem, we also derive quantitative estimates on the convergence of the method, at least for the discrete potential and the discrete cost. Finally, we introduce a strategy based on the barrier method to solve the discrete optimization problem.

math.NA↗
arXiv2020

TPFA Finite Volume Approximation of Wasserstein Gradient Flows

Numerous infinite dimensional dynamical systems arising in different fields have been shown to exhibit a gradient flow structure in the Wasserstein space. We construct Two Point Flux Approximation Finite Volume schemes discretizing such problems which preserve the variational structure and have second order accuracy in space. We propose an interior point method to solve the discrete variational problem, providing an efficient and robust algorithm. We present two applications to test the scheme and show its order of convergence.

math.NA↗
arXiv2019

Generalized compressible flows and solutions of the H(div) geodesic problem

We study the geodesic problem on the group of diffeomorphism of a domain M$\subset$Rd, equipped with the H(div) metric. The geodesic equations coincide with the Camassa-Holm equation when d=1, and represent one of its possible multi-dimensional generalizations when d>1. We propose a relaxation {à} la Brenier of this problem, in which solutions are represented as probability measures on the space of continuous paths on the cone over M. We use this relaxation to prove that smooth H(div) geodesics are globally length minimizing for short times. We also prove that there exists a unique pressure field associated to solutions of our relaxation. Finally, we propose a numerical scheme to construct generalized solutions on the cone and present some numerical results illustrating the relation between the generalized Camassa-Holm and incompressible Euler solutions.

math.AP↗
arXiv2019

Metric completion of $Diff([0,1])$ with the $H1$ right-invariant metric

We consider the group of smooth increasing diffeomorphisms Diff on the unit interval endowed with the right-invariant $H^1$ metric. We compute the metric completion of this space which appears to be the space of increasing maps of the unit interval with boundary conditions at $0$ and $1$. We compute the lower-semicontinuous envelope associated with the length minimizing geodesic variational problem. We discuss the Eulerian and Lagrangian formulation of this relaxation and we show that smooth solutions of the EPDiff equation are length minimizing for short times.

math.OC↗
arXiv2018

Embedding Camassa-Holm equations in incompressible Euler

In this article, we show how to embed the so-called CH2 equations into the geodesic flow of the Hdiv metric in 2D, which, itself, can be embedded in the incompressible Euler equation of a non compact Riemannian manifold. The method consists in embedding the incompressible Euler equation with a potential term coming from classical mechanics into incompressible Euler of a manifold and seeing the CH2 equation as a particular case of such fluid dynamic equation.

math.AP↗
arXiv2016

Scale-selective dissipation in energy-conserving finite element schemes for two-dimensional turbulence

We analyse the multiscale properties of energy-conserving upwind-stabilised finite element discretisations of the two-dimensional incompressible Euler equations. We focus our attention on two particular methods: the Lie derivative discretisation introduced in Natale and Cotter (2016a) and the Streamline Upwind/Petrov-Galerkin (SUPG) discretisation of the vorticity advection equation. Such discretisations provide control on enstrophy by modelling different types of scale interactions. We quantify the performance of the schemes in reproducing the non-local energy backscatter that characterises two-dimensional turbulent flows.

math.NA↗
arXiv2016

A variational H(div) finite element discretisation approach for perfect incompressible fluids

We propose a finite element discretisation approach for the incompressible Euler equations which mimics their geometric structure and their variational derivation. In particular, we derive a finite element method that arises from a nonholonomic variational principle and an appropriately defined Lagrangian, where finite element H(div) vector fields are identified with advection operators; this is the first successful extension of the structure-preserving discretisation of Pavlov et al. (2009) to the finite element setting. The resulting algorithm coincides with the energy-conserving scheme presented in Guzm\'an et al. (2016). Through the variational derivation, we discover that it also satisfies a discrete analogous of Kelvin's circulation theorem. Further, we propose an upwind-stabilised version of the scheme which dissipates enstrophy whilst preserving energy conservation and the discrete Kelvin's theorem. We prove error estimates for this version of the scheme, and we study its behaviour through numerical tests.

math.NA↗