424 research outputs found
The Monge-Ampere equation: various forms and numerical methods
We present three novel forms of the Monge-Ampere equation, which is used,
e.g., in image processing and in reconstruction of mass transportation in the
primordial Universe. The central role in this paper is played by our Fourier
integral form, for which we establish positivity and sharp bound properties of
the kernels. This is the basis for the development of a new method for solving
numerically the space-periodic Monge-Ampere problem in an odd-dimensional
space. Convergence is illustrated for a test problem of cosmological type, in
which a Gaussian distribution of matter is assumed in each localised object,
and the right-hand side of the Monge-Ampere equation is a sum of such
distributions.Comment: 24 pages, 2 tables, 5 figures, 32 references. Submitted to J.
Computational Physics. Times of runs added, multiple improvements of the
manuscript implemented
A global method for deterministic and stochastic homogenisation in BV
In this paper we study the deterministic and stochastic homogenisation of free-discontinuity functionals under linear growth and coercivity conditions. The main novelty of our deterministic result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Combining this result with the pointwise Subadditive Ergodic Theorem by Akcoglu and Krengel, we prove a stochastic homogenisation result, in the case of stationary random integrands. In particular, we characterise the limit integrands in terms of asymptotic cell formulas, as in the classical case of periodic homogenisation
Low-rank and sparse reconstruction in dynamic magnetic resonance imaging via proximal splitting methods
Dynamic magnetic resonance imaging (MRI) consists of collecting multiple MR images in time, resulting in a spatio-temporal signal. However, MRI intrinsically suffers from long acquisition times due to various constraints. This limits the full potential of dynamic MR imaging, such as obtaining high spatial and temporal resolutions which are crucial to observe dynamic phenomena. This dissertation addresses the problem of the reconstruction of dynamic MR images from a limited amount of samples arising from a nuclear magnetic resonance experiment. The term limited can be explained by the approach taken in this thesis to speed up scan time, which is based on violating the Nyquist criterion by skipping measurements that would be normally acquired in a standard MRI procedure. The resulting problem can be classified in the general framework of linear ill-posed inverse problems. This thesis shows how low-dimensional signal models, specifically lowrank and sparsity, can help in the reconstruction of dynamic images from partial measurements. The use of these models are justified by significant developments in signal recovery techniques from partial data that have emerged in recent years in signal processing. The major contributions of this thesis are the development and characterisation of fast and efficient computational tools using convex low-rank and sparse constraints via proximal gradient methods, the development and characterisation of a novel joint reconstruction–separation method via the low-rank plus sparse matrix decomposition technique, and the development and characterisation of low-rank based recovery methods in the context of dynamic parallel MRI. Finally, an additional contribution of this thesis is to formulate the various MR image reconstruction problems in the context of convex optimisation to develop algorithms based on proximal splitting methods
Interpolating orientation fields : an axiomatic approach
International audienc
A novel approach to shape optimisation with Lipschitz domains
This article introduces a novel method for the implementation of shape
optimisation with Lipschitz domains. We propose to use the shape derivative to
determine deformation fields which represent steepest descent directions of the
shape functional in the topology. The idea of our approach is
demonstrated for shape optimisation of -dimensional star-shaped domains,
which we represent as functions defined on the unit -sphere. In this
setting we provide the specific form of the shape derivative and prove the
existence of solutions to the underlying shape optimisation problem. Moreover,
we show the existence of a direction of steepest descent in the
topology. We also note that shape optimisation in this context is closely
related to the Laplacian, and to optimal transport, where we highlight
the latter in the numerics section. We present several numerical experiments in
two dimensions illustrating that our approach seems to be superior over a
widely used Hilbert space method in the considered examples, in particular in
developing optimised shapes with corners
IST Austria Thesis
This PhD thesis is primarily focused on the study of discrete transport problems, introduced for the first time in the seminal works of Maas [Maa11] and Mielke [Mie11] on finite state Markov chains and reaction-diffusion equations, respectively. More in detail, my research focuses on the study of transport costs on graphs, in particular the convergence and the stability of such problems in the discrete-to-continuum limit. This thesis also includes some results concerning
non-commutative optimal transport. The first chapter of this thesis consists of a general introduction to the optimal transport problems, both in the discrete, the continuous, and the non-commutative setting. Chapters 2 and 3 present the content of two works, obtained in collaboration with Peter Gladbach, Eva Kopfer, and Jan Maas, where we have been able to show the convergence of discrete transport costs on periodic graphs to suitable continuous ones, which can be described by means of a homogenisation result. We first focus on the particular case of quadratic costs on the real line and then extending the result to more general costs in arbitrary dimension. Our results are the first complete characterisation of limits of transport costs on periodic graphs in arbitrary dimension which do not rely on any additional symmetry. In Chapter 4 we turn our attention to one of the intriguing connection between evolution equations and optimal transport, represented by the theory of gradient flows. We show that discrete gradient flow structures associated to a finite volume approximation of a certain class of diffusive equations (Fokker–Planck) is stable in the limit of vanishing meshes, reproving the convergence of the scheme via the method of evolutionary Γ-convergence and exploiting a more variational point of view on the problem. This is based on a collaboration with Dominik Forkert and Jan Maas. Chapter 5 represents a change of perspective, moving away from the discrete world and reaching the non-commutative one. As in the discrete case, we discuss how classical tools coming from the commutative optimal transport can be translated into the setting of density matrices. In particular, in this final chapter we present a non-commutative version of the Schrödinger problem (or entropic regularised optimal transport problem) and discuss existence and characterisation of minimisers, a duality result, and present a non-commutative version of the well-known Sinkhorn algorithm to compute the above mentioned optimisers. This is based on a joint work with Dario Feliciangeli and Augusto Gerolin. Finally, Appendix A and B contain some additional material and discussions, with particular attention to Harnack inequalities and the regularity of flows on discrete spaces
- …