Sparsity-based algorithms for inverse problems

Inverse problems are problems where we want to estimate the values of certain parameters of a system given observations of the system. Such problems occur in several areas of science and engineering. Inverse problems are often ill-posed, which means that the observations of the system do not uniquely define the parameters we seek to estimate, or that the solution is highly sensitive to small changes in the observation. In order to solve such problems, therefore, we need to make use of additional knowledge about the system at hand. One such prior information is given by the notion of sparsity. Sparsity refers to the knowledge that the solution to the inverse problem can be expressed as a combination of a few terms. The sparsity of a solution can be controlled explicitly or implicitly. An explicit way to induce sparsity is to minimize the number of non-zero terms in the solution. Implicit use of sparsity can be made, for e.g., by making adjustments to the algorithm used to arrive at the solution.In this thesis we studied various inverse problems that arise in different application areas, such as tomographic imaging and equation learning for biology, and showed how ideas of sparsity can be used in each case to design effective algorithms to solve such problems.Financial support was provided by the European Union's Horizon 2020 Research and Innovation programme under the Marie Sklodowska-Curie grant agreement no.~765604Number theory, Algebra and Geometr

Atomic-scale and three-dimensional transmission electron microscopy of nanoparticle morphology

The burgeoning field of nanotechnology motivates comprehensive elucidation of nanoscale materials. This thesis addresses transmission electron microscope characterisation of nanoparticle morphology, concerning specifically the crystal- lographic status of novel intermetallic GaPd2 nanocatalysts and advancement of electron tomographic methods for high-fidelity three-dimensional analysis. Going beyond preceding analyses, high-resolution annular dark-field imaging is used to verify successful nano-sizing of the intermetallic compound GaPd2. It also reveals catalytically significant and crystallographically intriguing deviations from the bulk crystal structure. So-called ‘non-crystallographic’ five-fold twinned nanoparticles are observed, adding a new perspective in the long standing debate over how such morphologies may be achieved. The morphological complexity of the GaPd2 nanocatalysts, and many cognate nanoparticle systems, demands fully three-dimensional analysis. It is illustrated how image processing techniques applied to electron tomography reconstructions can facilitate more facile and objective quantitative analysis (‘nano-metrology’). However, the fidelity of the analysis is limited ultimately by artefacts in the tomographic reconstruction. Compressed sensing, a new sampling theory, asserts that many signals can be recovered from far fewer measurements than traditional theories dictate are necessary. Compressed sensing is applied here to electron tomographic reconstruction, and is shown to yield far higher fidelity reconstructions than conventional algorithms. Reconstruction from extremely limited data, more robust quantitative analysis and novel three-dimensional imaging are demon- strated, including the first three-dimensional imaging of localised surface plasmon resonances. Many aspects of transmission electron microscopy characterisation may be enhanced using a compressed sensing approach

Single atom imaging with time-resolved electron microscopy

Developments in scanning transmission electron microscopy (STEM) have opened up new possibilities for time-resolved imaging at the atomic scale. However, rapid imaging of single atom dynamics brings with it a new set of challenges, particularly regarding noise and the interaction between the electron beam and the specimen. This thesis develops a set of analytical tools for capturing atomic motion and analyzing the dynamic behaviour of materials at the atomic scale. Machine learning is increasingly playing an important role in the analysis of electron microscopy data. In this light, new unsupervised learning tools are developed here for noise removal under low-dose imaging conditions and for identifying the motion of surface atoms. The scope for real-time processing and analysis is also explored, which is of rising importance as electron microscopy datasets grow in size and complexity. These advances in image processing and analysis are combined with computational modelling to uncover new chemical and physical insights into the motion of atoms adsorbed onto surfaces. Of particular interest are systems for heterogeneous catalysis, where the catalytic activity can depend intimately on the atomic environment. The study of Cu atoms on a graphene oxide support reveals that the atoms undergo anomalous diffusion as a result of spatial and energetic disorder present in the substrate. The investigation is extended to examine the structure and stability of small Cu clusters on graphene oxide, with atomistic modelling used to understand the significant role played by the substrate. Finally, the analytical methods are used to study the surface reconstruction of silicon alongside the electron beam-induced motion of adatoms on the surface. Taken together, these studies demonstrate the materials insights that can be obtained with time-resolved STEM imaging, and highlight the importance of combining state-ofthe- art imaging with computational analysis and atomistic modelling to quantitatively characterize the behaviour of materials with atomic resolution.The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement 291522–3DIMAGE, as well as from the European Union Seventh Framework Programme under Grant Agreement 312483-ESTEEM2 (Integrated Infrastructure Initiative -I3)

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Magnetic Hybrid-Materials

Externally tunable properties allow for new applications of suspensions of micro- and nanoparticles in sensors and actuators in technical and medical applications. By means of easy to generate and control magnetic fields, fluids inside of matrices are studied. This monnograph delivers the latest insigths into multi-scale modelling, manufacturing and application of those magnetic hybrid materials