1,516 research outputs found
Variational approximation of functionals defined on 1-dimensional connected sets: the planar case
In this paper we consider variational problems involving 1-dimensional
connected sets in the Euclidean plane, such as the classical Steiner tree
problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal
partition problems and provide a variational approximation through
Modica-Mortola type energies proving a -convergence result. We also
introduce a suitable convex relaxation and develop the corresponding numerical
implementations. The proposed methods are quite general and the results we
obtain can be extended to -dimensional Euclidean space or to more general
manifold ambients, as shown in the companion paper [11].Comment: 30 pages, 5 figure
Indirect Image Registration with Large Diffeomorphic Deformations
The paper adapts the large deformation diffeomorphic metric mapping framework
for image registration to the indirect setting where a template is registered
against a target that is given through indirect noisy observations. The
registration uses diffeomorphisms that transform the template through a (group)
action. These diffeomorphisms are generated by solving a flow equation that is
defined by a velocity field with certain regularity. The theoretical analysis
includes a proof that indirect image registration has solutions (existence)
that are stable and that converge as the data error tends so zero, so it
becomes a well-defined regularization method. The paper concludes with examples
of indirect image registration in 2D tomography with very sparse and/or highly
noisy data.Comment: 43 pages, 4 figures, 1 table; revise
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
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