Quasi-Interpolation von Funktionen und Anwendungen auf Potentialprobleme

Ausgangspunkt der Dissertation ist ein von V. Maz'ya entwickeltes Verfahren, eine gegebene Funktion f : Rn ! R durch eine Linearkombination fh radialer glatter exponentiell fallender Basisfunktionen zu approximieren, die im Gegensatz zu den Splines lediglich eine näherungsweise Zerlegung der Eins bilden und somit ein für h ! 0 nicht konvergentes Verfahren definieren. Dieses Verfahren wurde unter dem Namen Approximate Approximations bekannt. Es zeigt sich jedoch, dass diese fehlende Konvergenz für die Praxis nicht relevant ist, da der Fehler zwischen f und der Approximation fh über gewisse Parameter unterhalb der Maschinengenauigkeit heutiger Rechner eingestellt werden kann. Darüber hinaus besitzt das Verfahren große Vorteile bei der numerischen Lösung von Cauchy-Problemen der Form Lu = f mit einem geeigneten linearen partiellen Differentialoperator L im Rn. Approximiert man die rechte Seite f durch fh, so lassen sich in vielen Fällen explizite Formeln für die entsprechenden approximativen Volumenpotentiale uh angeben, die nur noch eine eindimensionale Integration (z.B. die Errorfunktion) enthalten. Zur numerischen Lösung von Randwertproblemen ist das von Maz'ya entwickelte Verfahren bisher noch nicht genutzt worden, mit Ausnahme heuristischer bzw. experimenteller Betrachtungen zur sogenannten Randpunktmethode. Hier setzt die Dissertation ein. Auf der Grundlage radialer Basisfunktionen wird ein neues Approximationsverfahren entwickelt, welches die Vorzüge der von Maz'ya für Cauchy-Probleme entwickelten Methode auf die numerische Lösung von Randwertproblemen überträgt. Dabei werden stellvertretend das innere Dirichlet-Problem für die Laplace-Gleichung und für die Stokes-Gleichungen im R2 behandelt, wobei für jeden der einzelnen Approximationsschritte Konvergenzuntersuchungen durchgeführt und Fehlerabschätzungen angegeben werden.

Lagrangian approximations and weak solutions of the Navier-Stokes equations

The motion of a viscous incompressible fluid flow in bounded domains with a smooth boundary can be described by the nonlinear Navier-Stokes equations. This description corresponds to the so-called Eulerian approach. We develop a new approximation method for the Navier-Stokes equations in both the stationary and the non-stationary case by a suitable coupling of the Eulerian and the Lagrangian representation of the flow, where the latter is defined by the trajectories of the particles of the fluid. The method leads to a sequence of uniquely determined approximate solutions with a high degree of regularity containing a convergent subsequence with limit function v such that v is a weak solution of the Navier-Stokes equations.

Sparse Representations of Multiple Signals

We discuss the problem of finding sparse representations of a class of signals. We formalize the problem and prove it is NP-complete both in the case of a single signal and that of multiple ones. Next we develop a simple approximation method to the problem and we show experimental results using artificially generated signals. Furthermore,we use our approximation method to find sparse representations of classes of real signals, specifically of images of pedestrians. We discuss the relation between our formulation of the sparsity problem and the problem of finding representations of objects that are compact and appropriate for detection and classification.

Método numérico-analítico generalizado para estimação do campo eletromagnético de linhas de transmissão de energia elétrica utilizando a teoria dos elementos finitos

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Eficiência de estimadores, geradores e algoritmos na simulação de dados diários de precipitação pluviométrica utilizando a distribuição gama

Pós-graduação em Agronomia (Energia na Agricultura) - FCA

Configurações magnéticas colineares e não-colineares em currais de Fe, Cr e Mn sobre a Pt (111)

Neste trabalho investigamos as propriedades magnéticas de currais de Fe, Cr e Mn adsorvidos sobre a superfície de Pt(111) utilizando o método RS-LMTO-ASA (Real Space Linear Muffin Tin Orbital - Atomic Sphere Approximation), o qual é um método de primeiros princípios baseado na Teoria do Funcional da Densidade (DFT-Density Functional Theory), que permite o cálculo de estruturas magnéticas não-colineares. Obtivemos que os átomos de Fe apresentam momentos magnéticos elevados, da ordem de 3.5µB /átomo, e têm uma interação de troca entre primeiros vizinhos forte e ferro-magnética. Isto leva a um arranjo magnético colinear no curral. Para os currais de Mn e Cr encontramos que estes possuem elevado momento magnético, da ordem de 4.51µB /átomo e 4.15µB /átomo, respectivamente, e interações de troca entre primeiros vizinhos antiferro-magnéticas. Isto conduz a arranjos magnéticos colineares em currais simples, assim como interessantes ordenamentos não-colineares, tais como estruturas tipo vértice (skyrmions), para os currais com uma geometria particular onde o antiferromagnetismo se apresenta frustado.

Propriedades magnéticas de nanoestruturas adsorvidas em superfícies metálicas

Neste trabalho, utilizamos o método de primeiros princípios RS-LMTO-ASA (Real Space – Linear Muffin-Tin Orbital - Atomic Sphere Approximation) baseado na Teoria do Funcional da Densidade (DFT - Density Functional Theory) e implementado para o cálculo de estruturas magnéticas não-colineares, para investigar as propriedades magnéticas de nanoestruturas adsorvidas em superfícies metálicas. Consideramos aglomerados com diferentes geometrias e tamanhos como adátomos, dímeros, trímeros, nanofios e nanoestruturas de geometria triangular de Fe, Fe-Co e Fe-Pt adsorvidos sobre a superfície de Pt(111) e tratamos também nanoestruturas de Mn sobre a superfície de Ag(111). Mostramos que os nanofios de Fe-Co sobre a superfície de Pt(111) apresentam um ordenamento ferromagnético. Devido à redução do número de coordenação presente na superfície, os momentos de spin e orbital nos sítios de Fe e Co mostram-se elevados comparados com os respectivos valores dos momentos destes metais como bulk. Analisamos também como estes momentos variam em função da concentração destes elementos nos nanofios. Para os sistemas compostos por nanofios Fe-Pt adsorvidos em Pt(111), mostramos que é possível sintonizar as interações de troca entre os adátomos magnéticos Fe através da introdução de um diferente número de átomos Pt para ligá-los. Por exemplo, a interação de troca entre os adátomos de Fe pode ser consideravelmente aumentada pela introdução de cadeias de Pt que os conectem e tanto configurações ferromagnéticas, antiferromagnéticas ou não-colineares entre os adátomos de Fe podem ser estabilizadas, dependendo da espessura do espaçador Pt. Para os aglomerados Mn sobre a Ag(111) mostramos que a interação de troca entre os sítios de Mn depende não somente da distância entre os átomos, mas também do número de coordenação de cada sítio. Desta forma, verificamos um magnetismo não-colinear nestas nanoestruturas causado tanto por frustração geométrica, quanto pela competição de interações de curto e longo alcance. Nossos resultados estão em boa concordância com os resultados experimentais da literatura e com os resultados teóricos obtidos por outros métodos, quando existentes.

Propriedades magnéticas de nanoestruturas de metais de transição 3d adsorvidas na superfície de Pt(111)

Neste trabalho, utilizamos o método de primeiros princípios, RS-LMTO-ASA (“Real Space - Linear Muffin-Tin Orbital - Atomic Sphere Approximation”), baseado na Teoria do Funcional da Densidade (DFT) e implementado para o cálculo de estruturas magnéticas não-colineares, para investigar as propriedades magnéticas de nanoestruturas de metais de transição 3d (Cr, Mn, Fe, Co e Ni) adsorvidas na superfície de Pt(111). Diferentes geometrias como adátomos, dímeros, trímeros, fios lineares e zig-zag foram consideradas e, o tamanho dos aglomerados foi variado de 2 a 7 átomos. Mostramos que os aglomerados de Fe, Co e Ni sobre a superfície de Pt(111), para todas as geometrias simuladas, apresentam um ordenamento ferromagnético. Devido à redução do número de coordenação presente na superfície, os momentos de spin e orbital nos sítios de Fe, Co e Ni, para as diferentes geometrias, mostram-se elevados comparados com os respectivos valores dos momentos destes metais como bulk. Para os glomerados de Cr e Mn mostramos que a interação de troca antiferromagnética entre primeiros vizinhos leva a um ordenamento antiferromagnético colinear no caso de geometrias lineares. No entanto, se o antiferromagnetismo é frustrado por restrição geométrica imposta aos aglomerados pela superfície triangular do substrato, obtém-se um comportamento magnético não-colinear para aglomerados de Cr e Mn sobre a Pt(111). Nossos resultados estão em boa concordância com os resultados experimentais da literatura e com os resultados teóricos obtidos por outros métodos, quando existentes.

Determinação do fator de rigidez de junta aparafusada em tração utilizando o software Excel

When a bolted joint is loaded in tension with dynamically, part of this load is absorbed by the bolt and rest is absorbed by the joint material. What determines the portion that is to absorbed by the bolt is the joint stiffness factor. This factor influences the tension which corresponds to pre-load and the safety factor for fatigue failure, thus being an important factor in the design of bolted joints. In this work, three methods of calculating the stiffness factor are compared through a spreadsheet in Excel software. The ratio of initial pre-load and the safety factor for fatigue failure depending on the stiffness factor graph is generated. The calculations for each method show results with a small difference. It is therefore recommended that each project case is analyzed, and depending on its conditions and the range of stiffness values, the more or less rigid method about the safety factor for fatigue failure is chosen. In general, the approximation method provides consistent results and can be easily calculated

First-principles studies of complex magnetism in Mn nanostructures on the Fe(001) surface

The magnetic properties of Mn nanostructures on the Fe(001) surface have been studied using the noncollinear first-principles real space-linear muffin-tin orbital-atomic sphere approximation method within density-functional theory. We have considered a variety of nanostructures such as adsorbed wires, pyramids, and flat and intermixed clusters of sizes varying from two to nine atoms. Our calculations of interatomic exchange interactions reveal the long-range nature of exchange interactions between Mn-Mn and Mn-Fe atoms. We have found that the strong dependence of these interactions on the local environment, the magnetic frustration, and the effect of spin-orbit coupling lead to the possibility of realizing complex noncollinear magnetic structures such as helical spin spiral and half-skyrmion.

Stochastic models and dynamic measures for the characterization of bistable circuits

During the last few years, a great deal of interest has risen concerning the applications of stochastic methods to several biochemical and biological phenomena. Phenomena like gene expression, cellular memory, bet-hedging strategy in bacterial growth and many others, cannot be described by continuous stochastic models due to their intrinsic discreteness and randomness. In this thesis I have used the Chemical Master Equation (CME) technique to modelize some feedback cycles and analyzing their properties, including experimental data. In the first part of this work, the effect of stochastic stability is discussed on a toy model of the genetic switch that triggers the cellular division, which malfunctioning is known to be one of the hallmarks of cancer. The second system I have worked on is the so-called futile cycle, a closed cycle of two enzymatic reactions that adds and removes a chemical compound, called phosphate group, to a specific substrate. I have thus investigated how adding noise to the enzyme (that is usually in the order of few hundred molecules) modifies the probability of observing a specific number of phosphorylated substrate molecules, and confirmed theoretical predictions with numerical simulations. In the third part the results of the study of a chain of multiple phosphorylation-dephosphorylation cycles will be presented. We will discuss an approximation method for the exact solution in the bidimensional case and the relationship that this method has with the thermodynamic properties of the system, which is an open system far from equilibrium.In the last section the agreement between the theoretical prediction of the total protein quantity in a mouse cells population and the observed quantity will be shown, measured via fluorescence microscopy.

Pascal Filters

We introduce a new type of filter approximation method and call it the Pascal filter, which we construct from the Pascal polynomials. The roll-off characteristics of the Pascal, Butterworth, and the Chebyshev filters are compared.

Resistance heating of steel conductors of circular cross-section

The object of this thesis is to develop a method for calculating the losses developed in steel conductors of circular cross-section and at temperatures below 100oC, by the direct passage of a sinusoidally alternating current. Three cases are considered. 1. Isolated solid or tubular conductor. 2. Concentric arrangement of tube and solid return conductor. 3. Concentric arrangement of two tubes. These cases find applications in process temperature maintenance of pipelines, resistance heating of bars and design of bus-bars. The problems associated with the non-linearity of steel are examined. Resistance heating of bars and methods of surface heating of pipelines are briefly described. Magnetic-linear solutions based on Maxwell's equations are critically examined and conditions under which various formulae apply investigated. The conditions under which a tube is electrically equivalent to a solid conductor and to a semi-infinite plate are derived. Existing solutions for the calculation of losses in isolated steel conductors of circular cross-section are reviewed, evaluated and compared. Two methods of solution are developed for the three cases considered. The first is based on the magnetic-linear solutions and offers an alternative to the available methods which are not universal. The second solution extends the existing B/H step-function approximation method to small diameter conductors and to tubes in isolation or in a concentric arrangement. A comprehensive experimental investigation is presented for cases 1 and 2 above which confirms the validity of the proposed methods of solution. These are further supported by experimental results reported in the literature. Good agreement is obtained between measured and calculated loss values for surface field strengths beyond the linear part of the d.c. magnetisation characteristic. It is also shown that there is a difference in the electrical behaviour of a small diameter conductor or thin tube under resistance or induction heating conditions.

Variational Markov Chain Monte Carlo for Bayesian smoothing of non-linear niffusions

In this paper we develop set of novel Markov chain Monte Carlo algorithms for Bayesian smoothing of partially observed non-linear diffusion processes. The sampling algorithms developed herein use a deterministic approximation to the posterior distribution over paths as the proposal distribution for a mixture of an independence and a random walk sampler. The approximating distribution is sampled by simulating an optimized time-dependent linear diffusion process derived from the recently developed variational Gaussian process approximation method. Flexible blocking strategies are introduced to further improve mixing, and thus the efficiency, of the sampling algorithms. The algorithms are tested on two diffusion processes: one with double-well potential drift and another with SINE drift. The new algorithm's accuracy and efficiency is compared with state-of-the-art hybrid Monte Carlo based path sampling. It is shown that in practical, finite sample, applications the algorithm is accurate except in the presence of large observation errors and low observation densities, which lead to a multi-modal structure in the posterior distribution over paths. More importantly, the variational approximation assisted sampling algorithm outperforms hybrid Monte Carlo in terms of computational efficiency, except when the diffusion process is densely observed with small errors in which case both algorithms are equally efficient.

Bayesian error bars for regression

Regression problems are concerned with predicting the values of one or more continuous quantities, given the values of a number of input variables. For virtually every application of regression, however, it is also important to have an indication of the uncertainty in the predictions. Such uncertainties are expressed in terms of the error bars, which specify the standard deviation of the distribution of predictions about the mean. Accurate estimate of error bars is of practical importance especially when safety and reliability is an issue. The Bayesian view of regression leads naturally to two contributions to the error bars. The first arises from the intrinsic noise on the target data, while the second comes from the uncertainty in the values of the model parameters which manifests itself in the finite width of the posterior distribution over the space of these parameters. The Hessian matrix which involves the second derivatives of the error function with respect to the weights is needed for implementing the Bayesian formalism in general and estimating the error bars in particular. A study of different methods for evaluating this matrix is given with special emphasis on the outer product approximation method. The contribution of the uncertainty in model parameters to the error bars is a finite data size effect, which becomes negligible as the number of data points in the training set increases. A study of this contribution is given in relation to the distribution of data in input space. It is shown that the addition of data points to the training set can only reduce the local magnitude of the error bars or leave it unchanged. Using the asymptotic limit of an infinite data set, it is shown that the error bars have an approximate relation to the density of data in input space.