Spectral preconditioners for the efficient numerical solution of a continuous branched transport model

We consider the efficient solution of sequences of linear systems arising in the numerical solution of a branched transport model whose long time solution for specific parameter settings is equivalent to the solution of the Monge\u2013Kantorovich equations of optimal transport. Galerkin Finite Element discretization combined with explicit Euler time stepping yield a linear system to be solved at each time step, characterized by a large sparse very ill conditioned symmetric positive definite (SPD) matrix . Extreme cases even prevent the convergence of Preconditioned Conjugate Gradient (PCG) with standard preconditioners such as an Incomplete Cholesky (IC) factorization of , which cannot always be computed. We investigate several preconditioning strategies that incorporate partial approximated spectral information. We present numerical evidence that the proposed techniques are efficient in reducing the condition number of the preconditioned systems, thus decreasing the number of PCG iterations and the overall CPU time

Numerical solution of the three dimensional Optimal Transport Problem

In this thesis we analyze a model introduced in [14, 17] and its extensions [14, 15]. These works conjectured a new formulation of Optimal Transport, an expanding area of mathematics whose aim is the identification of the most efficient strategy to reallocate resources from one place to another. The numerical approximation of the equations of the model represents a simple yet effective numerical approach to solve Optimal Transport problems. However, the numerical scheme was analyzed only in the two dimensional case. The aim of this thesis is to exploit the model to solve three dimensional Optimal Transport problems, where few examples of numerical solution are known from the literature. We present all the non trivial challenges required by the three dimensional extension, together with an ample series of numerical experiments, that confirms the conjectured equivalence with the Optimal Transport problem. The results show that the numerical scheme is robust and efficient, with ample space for improvement from the computational point of view

Fast Iterative Solution of the Optimal Transport Problem on Graphs

In this paper, we address the numerical solution of the Optimal Transport Problem on undirected weighted graphs, taking the shortest path distance as transport cost. The optimal solution is obtained from the long-time limit of the gradient descent dynamics. Among different time stepping procedures for the discretization of this dynamics, a backward Euler time stepping scheme combined with the inexact Newton-Raphson method results in a robust and accurate approach for the solution of the Optimal Transport Problem on graphs. It is found experimentally that the algorithm requires solving between $\mathcal{O}(1)$ and $\mathcal{O}(M^{0.36})$ linear systems involving weighted Laplacian matrices, where $M$ is the number of edges. These linear systems are solved via algebraic multigrid methods, resulting in an efficient solver for the Optimal Transport Problem on graphs

Emerging Developments in Interfaces and Free Boundaries

The field of the mathematical and numerical analysis of systems of nonlinear partial differential equations involving interfaces and free boundaries is a well established and flourishing area of research. This workshop focused on recent developments and emerging new themes. By bringing together experts in these fields we achieved progress in open questions and developed novel research directions in mathematics related to interfaces and free boundaries. This interdisciplinary workshop brought together researchers from distinct mathematical fields such as analysis, computation, optimisation and modelling to discuss emerging challenges

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells