5 research outputs found
Conditional gradients for total variation regularization with PDE constraints: a graph cuts approach
Total variation regularization has proven to be a valuable tool in the
context of optimal control of differential equations. This is particularly
attributed to the observation that TV-penalties often favor piecewise constant
minimizers with well-behaved jumpsets. On the downside, their intricate
properties significantly complicate every aspect of their analysis, from the
derivation of first-order optimality conditions to their discrete approximation
and the choice of a suitable solution algorithm. In this paper, we investigate
a general class of minimization problems with TV-regularization, comprising
both continuous and discretized control spaces, from a convex geometry
perspective. This leads to a variety of novel theoretical insights on
minimization problems with total variation regularization as well as tools for
their practical realization. First, by studying the extremal points of the
respective total variation unit balls, we enable their efficient solution by
geometry exploiting algorithms, e.g. fully-corrective generalized conditional
gradient methods. We give a detailed account on the practical realization of
such a method for piecewise constant finite element approximations of the
control on triangulations of the spatial domain. Second, in the same setting
and for suitable sequences of uniformly refined meshes, it is shown that
minimizers to discretized PDE-constrained optimal control problems approximate
solutions to a continuous limit problem involving an anisotropic total
variation reflecting the fine-scale geometry of the mesh.Comment: 47 pages, 12 figure
PDE-betinga optimering : prekondisjonerarar og metodar for diffuse domene
This thesis is mainly concerned with the efficient numerical solution of optimization problems subject to linear PDE-constraints, with particular focus on robust preconditioners and diffuse domain methods. Associated with such constrained optimization problems are the famous first-order KarushKuhn-Tucker (KKT) conditions. For certain minimization problems, the functions satisfying the KKT conditions are also optimal solutions of the original optimization problem, implying that we can solve the KKT system to obtain the optimum; the so-called “all-at-once” approach. We propose and analyze preconditioners for the different KKT systems we derive in this thesis.Denne avhandlinga ser i hovudsak på effektive numeriske løysingar av PDE-betinga optimeringsproblem, med eit særskilt fokus på robuste prekondisjonerar og “diffuse domain”-metodar. Assosiert med slike optimeringsproblem er dei velkjende Karush-Kuhn-Tucker (KKT)-føresetnadane. For mange betinga optimeringsproblem, vil funksjonar som tilfredstillar KKT-vilkåra samstundes vere ei optimal løysing på det opprinnelege optimeringsproblemet. Dette impliserar at vi kan løyse KKT-likningane for å finne optimum. Vi konstruerar og analyserar prekondisjonerar for dei forskjellige KKT-systema vi utleiar i denne avhandlinga
Fast Acquisition and Reconstruction Techniques in MRI
The aim of this thesis was to develop fast reconstruction and acquisition techniques for MRI that can support clinical applications where time is a limiting factor. In general, fast acquisition techniques were realized by undersampling k-space, while fast reconstruction techniques were achieved by using efficient numerical algorithms. In particular, undersampled acquisitions were processed in a CS and MRF framework. Preconditioning techniques were used to accelerate CS reconstructions, and a number of challenges encountered in MRF were addressed using appropriate post-processing techniques. European Research Council (ERC) Advanced Grant (670629 NOMA MRI)LUMC / Geneeskund