The p-Poisson Equation: Regularity Analysis and Adaptive Wavelet Frame Approximation

This thesis is concerned with an important class of quasilinear elliptic equations: the p-Poisson equations -div(|\nabla u|^{p-2} \nabla u) = f in Ω, where 1 =2. Equations of this type appear, inter alia, in various problems in continuum mechanics, for instance in the mathematical modelling of non-Newtonian fluids. Furthermore, the p-Poisson equations possess a certain model character for more general quasilinear elliptic problems. The central aspect of this thesis is the regularity analysis of solutions u to the p-Poisson equation in the so-called adaptivity scale B^σ_τ(L_τ(Ω)), 1/τ = σ/d + 1/p, σ > 0, of Besov spaces. It is well-known that the smoothness parameter σ determines the approximation rate of the best n-term wavelet approximation, and hence provides information on the maximal convergence rate of certain adaptive numerical wavelet methods. To derive Besov regularity estimates for solutions to the p-Poisson equation, two approaches are pursued in this work. The first approach makes use of the fact that under appropriate conditions the solutions to the p-Poisson equation admit certain higher regularity in the interior of the domain, in the sense that they are locally Hölder continuous. In general, the Hölder semi-norms may explode as one approaches the boundary of the domain, but this singular behavior can be controlled by some power of the distance to the boundary. It turns out that the combination of global Sobolev regularity and locally weighted Hölder regularity can be used to derive Besov smoothness in the adaptivity scale for solutions to the p-Poisson equation. The results of the first approach are stated in two steps. At first, a general embedding theorem is proved, which says that the intersection of a classical Sobolev space with a Hölder space having the above mentioned properties can be embedded into certain Besov spaces in the adaptivity scale. The proof of this result is based on extension arguments in connection with the characterization of Besov spaces by wavelet expansion coefficients. Subsequently, it is verified that in many cases the solutions u to the p-Poisson equation indeed satisfy the conditions of the embedding theorem, so that its application yields the desired regularity result. As it is shown, in many cases the Besov smoothness σ of the solution is significantly higher than its Sobolev smoothness, so that the development of adaptive schemes for the p-Poisson problem is completely justified. It is worthwhile noting that this universal approach is applicable for the general class of Lipschitz domains. The aim of the second approach is to make a first step in improving some of the derived Besov regularity results for solutions on polygonal domains. To this end the regularity is examined in a neighborhood of the corners of the domain, since generally the critical singularities of the solutions occur there. As it is shown, this approach leads to regularity assertions which are – in a local sense – indeed stronger in some cases than those derived with the first approach. The proofs are based on known results on the singular expansion of the solution in a neighborhood of a conical boundary point, as well as on embeddings of the intersection of Babuska-Kondratiev spaces K^l_{p,a}(Ω) with certain Besov spaces into the adaptivity scale of Besov spaces. As it is shown, in some cases the solutions to the p-Poisson equation admit arbitrary high weighted Sobolev regularity l in a neighborhood of the corners, and hence arbitrary high Besov regularity σ. Because of this fact the borderline case of this embedding for l equal infinity is analyzed in addition. It is shown that the resulting Fréchet spaces are continuously embedded into the corresponding F-spaces. It is worth mentioning that by these embeddings – independent of the p-Poisson setting – universal functional analytical tools are provided. The second central issue of this thesis is the numerical solution of the p-Poisson equation for 1 < p < 2. In this context, the focus is put on the implementation and numerical testing of a relaxed Kačanov-type iteration scheme for the approximate solution of the p-Poisson equation with homogeneous Dirichlet boundary conditions. For the numerical solution of the occurring linear elliptic subproblems an adaptive wavelet frame method is used. The resulting algorithm is studied in a series of numerical tests. Here, it turns out that in practice the implemented algorithm shows a stable convergence behavior

Combining Pati-Salam and Flavour Symmetries

We construct an extension of the Standard Model (SM) which is based on grand unification with Pati-Salam symmetry. The setup is supplemented with the idea of spontaneous flavour symmetry breaking which is mediated through flavon fields with renormalizable couplings to new heavy fermions. While we argue that the new gauge bosons in this approach can be sufficiently heavy to be irrelevant at low energies, the fermionic partners of the SM quarks, in particular those for the third generation, can be relatively light and provide new sources of flavour violation. The size of the effects is constrained by the observed values of the SM Yukawa matrices, but in a way that is different from the standard minimal-flavour violation approach. We determine characteristic deviations from the SM that could eventually be observed in future precision measurements.Comment: 26 pages, 5 figure

Besov regularity of solutions to the p-Poisson equation

In this paper, we study the regularity of solutions to the $p$-Poisson equation for all $1<p<\infty$. In particular, we are interested in smoothness estimates in the adaptivity scale $B^\sigma_{\tau}(L_{\tau}(\Omega))$, $1/\tau = \sigma/d+1/p$, of Besov spaces. The regularity in this scale determines the order of approximation that can be achieved by adaptive and other nonlinear approximation methods. It turns out that, especially for solutions to $p$-Poisson equations with homogeneous Dirichlet boundary conditions on bounded polygonal domains, the Besov regularity is significantly higher than the Sobolev regularity which justifies the use of adaptive algorithms. This type of results is obtained by combining local H\"older with global Sobolev estimates. In particular, we prove that intersections of locally weighted H\"older spaces and Sobolev spaces can be continuously embedded into the specific scale of Besov spaces we are interested in. The proof of this embedding result is based on wavelet characterizations of Besov spaces.Comment: 45 pages, 2 figure

Thermal Conductivity from Core and Well log Data

The relationships between thermal conductivity and other petrophysical properties have been analysed for a borehole drilled in a Tertiary Flysch sequence. We establish equations that permit us to predict rock thermal conductivity from logging data. A regression analysis of thermal conductivity, bulk density, and sonic velocity yields thermal conductivity with an average accuracy of better than 0.2 W/(m K). As a second step, logging data is used to compute a lithological depth profile, which in turn is used to calculate a thermal conductivity profile. From a comparison of the conductivity-depth profile and the laboratory data it can be concluded that thermal conductivity can be computed with an accuracy of less than 0.3 W/(m K)from conventional wireline data. The comparison of two different models shows that this approach can be practical even if old and incomplete logging data is used. The results can be used to infer thermal conductivity for boreholes without appropriate core data that are drilled in a similar geological setting.Comment: 18 pages, 9 figure, 3 table

Evaluation of the develoPPP.de programme

In the past two decades, development cooperation actors have launched wide-reaching approaches to strengthen cooperation with the private sector as an active partner in financing and implementing development projects. Development partnerships with the private sector are intended to pool public and private resources, making it possible to use business know-how and capital for economic and social development in partner countries. DEval has evaluated the develoPPP.de programme, the largest programme set up by the German Federal Ministry for Economic Cooperation and Development (BMZ) to promote such partnerships. The evaluation comprised document and literature analyses, a portfolio review of all develoPPP.de projects since 2009, expert interviews and company surveys as well as 12 comprehensive case studies in four countries. The data provide key findings with regard to the way in which the programme was steered and implemented, and its results and sustainability. The findings will be used to further develop the programme. They will also be used at policy and implementation level, and enable BMZ to comply with its accountability obligations

Modeling of flexible side chains for protein-ligand docking

This work comprises new approaches that are developed to support structure-based drug design in cases where side-chain conformations are uncertain, be it through exibility or the devised modeling procedure. A knowledge-based scoring function ROTA is derived that can successfully identify correct rotamers and near-native ligand placements. ROTA is also able to reliably estimate the binding anity of a protein-ligand complex, even if the conformations of one or both binding partners contain small errors. The side-chain prediction algorithm IRECS is developed for generating protein models that contain ensembles of rotamers for flexible side chains. IRECS is guided by ROTA and can accurately predict single and multiple side-chain conformations that represent the exibility and conformational space of the respective side chains. IRECS is also able to include knowledge of side-chain conformations from a homologous protein used as a template directly in its optimization procedure. A modeling and docking pipeline is constructed that comprises IRECS, ROTA and the docking program FlexE. This pipeline is tested on 40 targets of the screening database DUD, where it is shown that the application of ROTA and IRECS can signicantly increase the performance of screening experiments in cases in which side chains are exible or were modeled.Diese Arbeit stellt neue Methoden vor, die die strukturbasierte Suche nach Wirkstoffen in solchen Fällen unterstützen soll, in denen Seitenkettenkonformationen durch Flexibilität der Seitenketten oder durch die verwendete Modellierungstechnik nicht sicher bestimmt werden können. Die Bewertungsfunktion ROTA wurde abgeleitet um richtige Rotamere und Ligandplazierungen zu erkennen. ROTA ist außerdem in der Lage die Bindungsaffinität eines Protein-Ligand-Komplexes zuverlässig zu bestimmen, auch wenn die Konformationen der Bindungspartner geringe Fehler aufweisen. Das Programm IRECS wurde entwickelt um Proteinmodelle zu erzeugen, die Ensembles von Rotameren für flexible Seitenketten enthalten. IRECS verwendet ROTA zur Bewertung von Proteinkonformationen und kann zuverlässig Ensembles von Rotameren bestimmen, die die Flexibilität und den konformellen Raum der jeweiligen Seitenketten repräsentieren. IRECS ist auch in der Lage zusätzliche Informationen über Seitenketten eines homologen Proteins, das der Modellierung als Vorlage diente, während seiner Optimierungsprozedur zu nutzen. IRECS, ROTA und das Dockingprogramm FlexE wurden zu einer Modellierungs- und Dockingpipeline vereinigt und auf den 40 Proteinen der Screening-Datenbank DUD getestet. Es konnte gezeigt werden, dass in Fällen mit flexiblen oder modellierten Seitenketten die Anwendung von ROTA und IRECS die Leistung von Screening-Experimenten deutlich steigern kann