Wavelet and Multiscale Methods

Various scientific models demand finer and finer resolutions of relevant features. Paradoxically, increasing computational power serves to even heighten this demand. Namely, the wealth of available data itself becomes a major obstruction. Extracting essential information from complex structures and developing rigorous models to quantify the quality of information leads to tasks that are not tractable by standard numerical techniques. The last decade has seen the emergence of several new computational methodologies to address this situation. Their common features are the nonlinearity of the solution methods as well as the ability of separating solution characteristics living on different length scales. Perhaps the most prominent examples lie in multigrid methods and adaptive grid solvers for partial differential equations. These have substantially advanced the frontiers of computability for certain problem classes in numerical analysis. Other highly visible examples are: regression techniques in nonparametric statistical estimation, the design of universal estimators in the context of mathematical learning theory and machine learning; the investigation of greedy algorithms in complexity theory, compression techniques and encoding in signal and image processing; the solution of global operator equations through the compression of fully populated matrices arising from boundary integral equations with the aid of multipole expansions and hierarchical matrices; attacking problems in high spatial dimensions by sparse grid or hyperbolic wavelet concepts. This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computation and to promote the exchange of ideas emerging in various disciplines

Methods for detecting and characterising clusters of galaxies

The main theme of this PhD-thesis is the observation of clusters of galaxies at submillimetric wavelengths. The Sunyaev-Zel'dovich (SZ) effect due to interaction of cosmic microwave background (CMB) photons with electrons of the hot intra-cluster medium causes a distinct modulation in the spectrum of the CMB and is a very promising tool for detecting clusters out to very large distances. Especially the European PLANCK-mission, a satellite dedicated to the mapping of CMB anisotropies, will be the first experiment to routinely detect clusters of galaxies by their SZ-signature. This thesis presents an extensive simulation of PLANCK's SZ-capabilities, that combines all-sky maps of the SZ-effect with a realisation of the fluctuating CMB and submillimetric emission components of the Milky Way and of the Solar system, and takes instrumental issues such as the satellite's point-spread function, the frequency response, scan paths and detector noise of the receivers into account. For isolating the weak SZ-signal in the presence of overwhelming spurious components with complicated correlation properties across PLANCK's channels, multifrequency filters based on matched and scale-adaptive filtering have been extended to spherical topologies and applied to simulated data. These filters were shown to efficiently amplify and extract the SZ-signal by combining spatial band-filtering and linear combination of observations at different frequencies, where the filter shapes and the linear combination coefficients follow from the cross- and autocorrelation properties of the sky maps, the anticipated profile of SZ clusters and the known SZ spectral dependence. The characterisation of the resulting SZ-sample yielded a total number of 6000 detections above a statistical significance of 3 sigma and the distribution of detected clusters in mass, redshift, and position on the sky. In a related project, a method of constructing morphological distance estimators for resolved SZ cluster images is proposed. This method measures a cluster's SZ-morphology by wavelet decomposition. It was shown that the spectrum of wavelet moments can be modeled by elementary functions and has characteristic properties that are non-degenerate and indicative of cluster distance. Distance accuracies following from a maximum likelihood approach yielded values as good as 5% for the relative deviation, and deteriorate only slightly when noise components such as instrumental noise or CMB fluctuations were added. Other complications like cool cores of clusters and finite instrumental resolution were shown not to affect the wavelet distance estimation method significantly. Another line of research is the Rees-Sciama (RS) effect, which is due to gravitational interaction of CMB photons with non-stationary potential wells. This effect was shown to be a second order gravitational lensing effect arising in the post-Newtonian expansion of general relativity and measures the divergence of gravitomagnetic potentials integrated along the line-of-sight. The spatial autocorrelation function of the Rees-Sciama effect was derived in perturbation theory and projected to yield the angular autocorrelation function while taking care of the differing time evolution of the various terms emerging in the perturbation expansion. The RS-effect was shown to be detectable by PLANCK as a correction to the primordial CMB power spectrum at low multipoles. Within the same perturbative formalism, the gravitomagnetic corrections to the autocorrelation function of weak gravitational lensing observables such as cosmic shear could be determined. It was shown that those corrections are most important on the largest scales beyond 1~Gpc, which are difficult to access observationally. For contemporary weak lensing surveys, gravitomagnetic corrections were confirmed not play a significant role. A byproduct of the simulation of CMB fluctuations on the basis of Gaussian random fields was a new way of generating coded mask patterns for X-ray and gamma-ray imaging. Coded mask cameras observe a source by recording the shadow cast by a mask onto a position-sensitive detector. The distribution of sources can be reconstructed from this shadowgram by correlation techniques. By using Gaussian random fields, coded mask patterns can be specifically tailored for a predefined point-spread function which yields significant advantages with respect to sensitivity in the observation of extended sources while providing a moderate performance compared to traditional mask generation schemes in the observation of point sources. Coded mask patterns encoding Gaussian point-spread functions have been subjected to extensive ray-tracing studies where their performance has been evaluated

Doctor of Philosophy

dissertationWhile boundary representations, such as nonuniform rational B-spline (NURBS) surfaces, have traditionally well served the needs of the modeling community, they have not seen widespread adoption among the wider engineering discipline. There is a common perception that NURBS are slow to evaluate and complex to implement. Whereas computer-aided design commonly deals with surfaces, the engineering community must deal with materials that have thickness. Traditional visualization techniques have avoided NURBS, and there has been little cross-talk between the rich spline approximation community and the larger engineering field. Recently there has been a strong desire to marry the modeling and analysis phases of the iterative design cycle, be it in car design, turbulent flow simulation around an airfoil, or lighting design. Research has demonstrated that employing a single representation throughout the cycle has key advantages. Furthermore, novel manufacturing techniques employing heterogeneous materials require the introduction of volumetric modeling representations. There is little question that fields such as scientific visualization and mechanical engineering could benefit from the powerful approximation properties of splines. In this dissertation, we remove several hurdles to the application of NURBS to problems in engineering and demonstrate how their unique properties can be leveraged to solve problems of interest

Numerical simulation of droplet dispersion and deposition in pipes

Multiphase flows are commonly encountered in industrial processes but remain challenging to predict. The role of droplets in the setting of various flow patterns seen in pipes is capital. Being able to simulate accurately the motion, the dispersion, the deposition and the entrainment of droplets from a liquid film or pool would allow refining the various numerical models and would provide a useful insight to people involved with such flows. The PhD work summarised in this thesis aims at answering that ambitious goal, i.e. to reproduce the whole "life" of a cloud of droplets, with application to pipes and industrial systems. To the author’s knowledge, such study has never been realized with any open source computational fluid dynamics code such as OpenFOAM and in such details. An original surface-tracking motion has also been developed to solve wavy-stratified flows and droplets entrainment by extending OpenFOAM’s capabilities. The Lagrangian framework has been selected for this study as the relationship with various forces could be expressed directly and statistical information, including any Eulerian field if needed, could be retrieved

Magnetoencephalography

This is a practical book on MEG that covers a wide range of topics. The book begins with a series of reviews on the use of MEG for clinical applications, the study of cognitive functions in various diseases, and one chapter focusing specifically on studies of memory with MEG. There are sections with chapters that describe source localization issues, the use of beamformers and dipole source methods, as well as phase-based analyses, and a step-by-step guide to using dipoles for epilepsy spike analyses. The book ends with a section describing new innovations in MEG systems, namely an on-line real-time MEG data acquisition system, novel applications for MEG research, and a proposal for a helium re-circulation system. With such breadth of topics, there will be a chapter that is of interest to every MEG researcher or clinician

Simulation of the evolution of large scale structure elements with adaptive multigrid method

http://www.ester.ee/record=b1053241~S1*es

The influence of environment on galaxy formation

The dynamical evolution of the matter content of the universe is modelled throughout this study as that of self and mutually gravitating Lagrangian fluids in the so called ΛCDM-Concordance cosmological framework which leads to the Hierarchical Clustering paradigm for the formation of cosmic structures. As a numerical tool for investigating galaxy formation scenarios in this context, we employed GADGET2 (see Springel 2005) and the more recent GADGET3 (see Springel et al. 2008): we describe the numerical solvers implemented in the code and test their behaviour in both gravitational and hydrodynamical setups of relevance for cosmological calculations (Tasker et al. 2008). Using the outputs of the MILLENNIUM simulation and the relative Semi Analytical galaxy catalogues produced by Croton et al. 2006, we developed an algorithm aimed at the identification of large spherical underdense regions in the simulated Large Scale Structure (LSS), at z = 0. Focusing on this peculiar environment, we found a confirmation in numerical simulations for the observations by Trujillo, Carretero & Patiri (2006). The Tidal Torque Theory can predict the spatial distribution of the orientation of both the angular momentum vector of Milky Way size galaxies located on the surface of large spherical voids, and of their host DM halos. We re–simulated the 5 GIMIC regions (Crain et al. 2009) following the gravitational evolution of the CDM component only. We then applied a Semi Analytical Model (SAM) of galaxy formation (De Lucia & Blaizot 2007) obtaining the galaxy catalogues and merger histories for the 5 different volumes simulated. It is not yet well understood if and how the LSS environment can influence the Star Formation (SF) histories of galaxies. Starting from the stellar mass content of semi–analytical galaxies at z = 0, we defined characteristic epochs for their build up and, as a preliminary study, investigated how these distribute as a function of different LSS environments