3 research outputs found
A small remark on Bernstein’s theorem
We investigate splitting-type variational problems with some
linear growth conditions. For balanced solutions of the associated Euler–
Lagrange equation, we receive a result analogous to Bernstein’s theorem
on non-parametric minimal surfaces. Without assumptions of this type,
Bernstein’s theorem cannot be carried over to the splitting case, which
follows from an elementary counterexample. We also include some modifications of our main theorem
Modeling and controlling nanoscale patterns formed by bombardment with a broad ion beam
2017 Summer.Includes bibliographical references.For over half a century it has been known that bombarding a solid surface with a broad ion beam can produce periodic nanoscale structures. Given the virtually limitless promise of nanotechnology, the potential of ion bombardment to produce nanopatterned surfaces over large areas in a simple and economical way has attracted substantial interest. In the decades since its discovery, there has been a wealth of experimental and theoretical work examining the phenomenon in detail, with the eventual goal of using ion beam sputtering (IBS) to produce useful nanostructures. Despite the body of work, there are many open questions and unsurmounted challenges remain- ing. In this thesis, I present work that I have conducted in collaboration with my advisor, Mark Bradley, with whom I addressed some of these challenges. I show how we developed a formalism which connects information about single ion impacts to the evolution of a surface which sustains > 1016 such impacts per square centimeter. We have also produced theoretical results for the case of a binary material being bombarded while rotated azimuthally, with some unexpected findings. I also discuss some very exciting theoretical predictions for the case in which an elemental target is bombarded while the polar angle of ion incidence periodically changes. In this case we find the temporal driving can induce a surface pattern which is nearly perfectly periodic in the long time limit. I also discuss our work on using templated surfaces in conjunction with IBS to produce ii high quality blazed gratings and multilayer blazed gratings. This work is the subject of a current collaboration with Carmen Menoni and her students
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New PDE models for imaging problems and applications
Variational methods and Partial Differential Equations (PDEs) have been extensively employed for the mathematical formulation of a myriad of problems describing physical phenomena such as heat propagation, thermodynamic transformations and many more. In imaging, PDEs following variational principles are often considered. In their general form these models combine a regularisation and a data fitting term, balancing one against the other appropriately. Total variation (TV) regularisation is often used due to its edgepreserving and smoothing properties. In this thesis, we focus on the design of TV-based models for several different applications. We start considering PDE models encoding higher-order derivatives to overcome wellknown TV reconstruction drawbacks. Due to their high differential order and nonlinear nature, the computation of the numerical solution of these equations is often challenging. In this thesis, we propose directional splitting techniques and use Newton-type methods that despite these numerical hurdles render reliable and efficient computational schemes. Next, we discuss the problem of choosing the appropriate data fitting term in the case when multiple noise statistics in the data are present due, for instance, to different acquisition and transmission problems. We propose a novel variational model which encodes appropriately and consistently the different noise distributions in this case. Balancing the effect of the regularisation against the data fitting is also crucial. For this sake, we consider a learning approach which estimates the optimal ratio between the two by using training sets of examples via bilevel optimisation. Numerically, we use a combination of SemiSmooth (SSN) and quasi-Newton methods to solve the problem efficiently. Finally, we consider TV-based models in the framework of graphs for image segmentation problems. Here, spectral properties combined with matrix completion techniques are needed to overcome the computational limitations due to the large amount of image data. Further, a semi-supervised technique for the measurement of the segmented region by means of the Hough transform is proposed