A variational approach to 3D cylindrical geometry reconstruction from multiple views

[EN] In this paper we present a variational technique for the reconstruction of 3D cylindrical surfaces. Roughly speaking by a cylindrical surface we mean a surface that can be parameterized using the projection on a cylinder in terms of two coordinates, representing the displacement and angle in a cylindrical coordinate system respectively. The starting point for our method is a set of different views of a cylindrical surface, as well as a precomputed disparity map estimation between pair of images. The proposed variational technique is based on an energy minimization where we balance on the one hand the regularity of the cylindrical function given by the distance of the surface points to cylinder axis, and on the other hand, the distance between the projection of the surface points on the images and the expected location following the precomputed disparity map estimation between pair of images. One interesting advantage of this approach is that we regularize the 3D surface by means of a bi-dimensio al minimization problem. We show some experimental results for large stereo sequences.

A variational approach to 3D geometry reconstruction from two or multiple views

[EN] In the last years we have developed some methods for 3D reconstruction. First we began with the problem of reconstructing a 3D scene from a stereoscopic pair of images. We developed some methods based on energy functionals which produce dense disparity maps by preserving discontinuities from image boundaries. Then we passed to the problem of reconstructing a 3D scene from multiple views (more than 2). The method for multiple view reconstruction relies on the method for stereoscopic reconstruction. For every pair of consecutive images we estimate a disparity map and then we apply a robust method that searches for good correspondences through the sequence of images. Recently we have proposed several methods for 3D surface regularization. This is a postprocessing step necessary for smoothing the final surface, which could be afected by noise or mismatch correspondences. These regularization methods are interesting because they use the information from the reconstructing process and not only from the 3D surface. We have tackled all these problems from an energy minimization approach. We investigate the associated Euler-Lagrange equation of the energy functional, and we approach the solution of the underlying partial differential equation (PDE) using a gradient descent method.

Assimilation of Meteosat Second Generation (MSG) satellite data in a regional numerical weather prediction model using a one-dimensional variational approach

The quality of temperature and humidity retrievals from the infrared SEVIRI sensors on the geostationary Meteosat Second Generation (MSG) satellites is assessed by means of a one dimensional variational algorithm. The study is performed with the aim of improving the spatial and temporal resolution of available observations to feed analysis systems designed for high resolution regional scale numerical weather prediction (NWP) models. The non-hydrostatic forecast model COSMO (COnsortium for Small scale MOdelling) in the ARPA-SIM operational configuration is used to provide background fields. Only clear sky observations over sea are processed. An optimised 1D–VAR set-up comprising of the two water vapour and the three window channels is selected. It maximises the reduction of errors in the model backgrounds while ensuring ease of operational implementation through accurate bias correction procedures and correct radiative transfer simulations. The 1D–VAR retrieval quality is firstly quantified in relative terms employing statistics to estimate the reduction in the background model errors. Additionally the absolute retrieval accuracy is assessed comparing the analysis with independent radiosonde and satellite observations. The inclusion of satellite data brings a substantial reduction in the warm and dry biases present in the forecast model. Moreover it is shown that the retrieval profiles generated by the 1D–VAR are well correlated with the radiosonde measurements. Subsequently the 1D–VAR technique is applied to two three–dimensional case–studies: a false alarm case–study occurred in Friuli–Venezia–Giulia on the 8th of July 2004 and a heavy precipitation case occurred in Emilia–Romagna region between 9th and 12th of April 2005. The impact of satellite data for these two events is evaluated in terms of increments in the integrated water vapour and saturation water vapour over the column, in the 2 meters temperature and specific humidity and in the surface temperature. To improve the 1D–VAR technique a method to calculate flow–dependent model error covariance matrices is also assessed. The approach employs members from an ensemble forecast system generated by perturbing physical parameterisation schemes inside the model. The improved set–up applied to the case of 8th of July 2004 shows a substantial neutral impact.

New regularization method for EXAFS analysis: Application to uranium and plutonium sorption onto kaolinite

Die Röntgenabsorptionsspektroskopie (Extended X-ray absorption fine structure (EXAFS) spectroscopy) ist eine wichtige Methode zur Speziation von Schwermetallen in einem weiten Bereich von umweltrelevanten Systemen. Um Strukturparameter wie Koordinationszahl, Atomabstand und Debye-Waller Faktoren für die nächsten Nachbarn eines absorbierenden Atoms zu bestimmen, ist es für experimentelle EXAFS-Spektren üblich, unter Verwendung von Modellstrukturen einen „Least-Squares-Fit“ durchzuführen. Oft können verschiedene Modellstrukturen mit völlig unterschiedlicher chemischer Bedeutung die experimentellen EXAFS-Daten gleich gut beschreiben. Als gute Alternative zum konventionellen Kurven-Fit bietet sich das modifizierte Tikhonov-Regularisationsverfahren an. Ergänzend zur Tikhonov-Standardvariationsmethode enthält der in dieser Arbeit vorgestellte Algorithmus zwei weitere Schritte, nämlich die Anwendung des „Method of Separating Functionals“ und ein Iterationsverfahren mit Filtration im realen Raum. Um das modifizierte Tikhonov-Regularisationsverfahren zu testen und zu bestätigen wurden sowohl simulierte als auch experimentell gemessene EXAFS-Spektren einer kristallinen U(VI)-Verbindung mit bekannter Struktur, nämlich Soddyit (UO2)2SiO4 x 2H2O, untersucht. Die Leistungsfähigkeit dieser neuen Methode zur Auswertung von EXAFS-Spektren wird durch ihre Anwendung auf die Analyse von Proben mit unbekannter Struktur gezeigt, wie sie bei der Sorption von U(VI) bzw. von Pu(III)/Pu(IV) an Kaolinit auftreten. Ziel der Dissertation war es, die immer noch nicht voll ausgeschöpften Möglichkeiten des modifizierten Tikhonov-Regularisationsverfahrens für die Auswertung von EXAFS-Spektren aufzuzeigen. Die Ergebnisse lassen sich in zwei Kategorien einteilen. Die erste beinhaltet die Entwicklung des Tikhonov-Regularisationsverfahrens für die Analyse von EXAFS-Spektren von Mehrkomponentensystemen, insbesondere die Wahl bestimmter Regularisationsparameter und den Einfluss von Mehrfachstreuung, experimentell bedingtem Rauschen, etc. auf die Strukturparameter. Der zweite Teil beinhaltet die Speziation von sorbiertem U(VI) und Pu(III)/Pu(IV) an Kaolinit, basierend auf experimentellen EXAFS-Spektren, die mit Hilfe des modifizierten Tikhonov-Regularisationsverfahren ausgewertet und mit Hilfe konventioneller EXAFS-Analyse durch „Least-Squares-Fit“ bestätigt wurden.

Fast and accurate numerical solutions in some problems of particle and radiation transport: synthetic acceleration for the method of short characteristics, Doppler-broadened scattering kernel, remote sensing of the cryosphere

The aim of this work is to present various aspects of numerical simulation of particle and radiation transport for industrial and environmental protection applications, to enable the analysis of complex physical processes in a fast, reliable, and efficient way. In the first part we deal with speed-up of numerical simulation of neutron transport for nuclear reactor core analysis. The convergence properties of the source iteration scheme of the Method of Characteristics applied to be heterogeneous structured geometries has been enhanced by means of Boundary Projection Acceleration, enabling the study of 2D and 3D geometries with transport theory without spatial homogenization. The computational performances have been verified with the C5G7 2D and 3D benchmarks, showing a sensible reduction of iterations and CPU time. The second part is devoted to the study of temperature-dependent elastic scattering of neutrons for heavy isotopes near to the thermal zone. A numerical computation of the Doppler convolution of the elastic scattering kernel based on the gas model is presented, for a general energy dependent cross section and scattering law in the center of mass system. The range of integration has been optimized employing a numerical cutoff, allowing a faster numerical evaluation of the convolution integral. Legendre moments of the transfer kernel are subsequently obtained by direct quadrature and a numerical analysis of the convergence is presented. In the third part we focus our attention to remote sensing applications of radiative transfer employed to investigate the Earth's cryosphere. The photon transport equation is applied to simulate reflectivity of glaciers varying the age of the layer of snow or ice, its thickness, the presence or not other underlying layers, the degree of dust included in the snow, creating a framework able to decipher spectral signals collected by orbiting detectors.

Approximating method of frames

In this paper, the well-known method of frames approach to the signal decomposition problem is reformulated as a certain bilevel goal-attainment linear least squares problem. As a consequence, a numerically robust variant of the method, named approximating method of frames, is proposed on the basis of a certain minimal Euclidean norm approximating splitting pseudo-iteration-wise method.

Space-time Reconstruction of Oceanic Sea States via Variational Stereo Methods

We present a remote sensing observational method for the measurement of the spatio-temporal dynamics of ocean waves. Variational techniques are used to recover a coherent space-time reconstruction of oceanic sea states given stereo video imagery. The stereoscopic reconstruction problem is expressed in a variational optimization framework. There, we design an energy functional whose minimizer is the desired temporal sequence of wave heights. The functional combines photometric observations as well as spatial and temporal regularizers. A nested iterative scheme is devised to numerically solve, via 3-D multigrid methods, the system of partial differential equations resulting from the optimality condition of the energy functional. The output of our method is the coherent, simultaneous estimation of the wave surface height and radiance at multiple snapshots. We demonstrate our algorithm on real data collected off-shore. Statistical and spectral analysis are performed. Comparison with respect to an existing sequential method is analyzed.

A mathematical framework for finite strain elastoplastic consolidation. Part 1: Balance laws, variational formulation, and linearization

A mathematical formulation for finite strain elasto plastic consolidation of fully saturated soil media is presented. Strong and weak forms of the boundary-value problem are derived using both the material and spatial descriptions. The algorithmic treatment of finite strain elastoplasticity for the solid phase is based on multiplicative decomposition and is coupled with the algorithm for fluid flow via the Kirchhoff pore water pressure. Balance laws are written for the soil-water mixture following the motion of the soil matrix alone. It is shown that the motion of the fluid phase only affects the Jacobian of the solid phase motion, and therefore can be characterized completely by the motion of the soil matrix. Furthermore, it is shown from energy balance consideration that the effective, or intergranular, stress is the appropriate measure of stress for describing the constitutive response of the soil skeleton since it absorbs all the strain energy generated in the saturated soil-water mixture. Finally, it is shown that the mathematical model is amenable to consistent linearization, and that explicit expressions for the consistent tangent operators can be derived for use in numerical solutions such as those based on the finite element method.

Contribution to the analysis and design of reflectarray antennas for reconfigurable beam applications at frequencies above 100 GHz using liquid crystal technology

El trabajo contenido en esta tesis doctoral está encuadrado en el desarrollo de antenas reconfigurables electrónicamente capaces de proporcionar prestaciones competitivas a las aplicaciones cada vez más comunes que operan a frecuencias superiores a 60 GHz. En concreto, esta tesis se centra en el estudio, diseño, e implementación de las antenas reflectarray, a las que se introduce la tecnología de cristal líquido como elemento característico con el que se consigue reconfigurabilidad de haz de forma electrónica. Desde un punto de vista muy general, se puede describir un cristal líquido como un material cuya permitividad eléctrica es variable y controlada por una excitación externa, que generalmente suele corresponderse con un campo eléctrico quasi-estático (AC). Las antenas reflectarray de cristal líquido se han escogido como objeto de estudio por varias razones. La primera de ellas tiene que ver con las ventajas que los reflectarrays, y en especial aquellos realizados en configuración planar, proporcionan con respecto a otras antenas de alta ganancia como los reflectores o los “phased-arrays”. En los reflectarrays, la alimentación a través de una fuente primaria común (característica de reflectores) y el elevado número de grados de libertad de las celdas que los componen (característica de arrays) hacen que estas antenas puedan proporcionar prestaciones eléctricas iguales o mejores que las anteriores, a un coste más reducido y con estructuras de antena más compactas. La segunda razón radica en la flexibilidad que ofrece el cristal líquido a ser confinado y polarizado en recintos de geometría variada, como consecuencia de su fluidez (propiedad de los líquidos). Por ello, la tecnología de cristal líquido permite que el propio elemento reconfigurable en las celdas de reflectarray se adapte a la configuración planar de manera que en sí mismo, el cristal líquido sea una o varias de las capas características de esta configuración. Esto simplifica de forma drástica la estructura y la fabricación de este tipo de antenas, incluso si se comparan con reflectarrays reconfigurables basados en otras tecnologías como diodos, MEMS, etc. Por tanto, su coste y desarrollo es muy reducido, lo que hace que se puedan fabricar reflectarrays reconfigurables eléctricamente grandes, a bajo coste, y en producción elevada. Un ejemplo claro de una estructura similar, y que ha tenido éxito comercial, son las pantallas de cristal líquido. La tercera razón reside en el hecho de que el cristal líquido es, hasta la fecha, de las pocas tecnologías capaces de ofrecer reconfigurabilidad del haz a frecuencias superiores a 60 GHz. De hecho, el cristal líquido permite reconfigurabilidad en un amplio margen de frecuencias, que va desde DC a frecuencias del espectro visible, incluyendo las microondas y los THz. Otras tecnologías, como los materiales ferroeléctricos, el grafeno o la tecnología CMOS “on chip” permiten también conmutar el haz en estas frecuencias. Sin embargo, la tecnología CMOS tiene un elevado coste y actualmente está limitada a frecuencias inferiores a 150 GHz, y aunque los materiales ferroeléctricos o el grafeno puedan conmutar a frecuencias más altas y en un rango más amplio, tienen serias dificultades que los hacen aún inmaduros. En el caso de los materiales ferroeléctricos, los elevados voltajes para conmutar el material los hacen poco atractivos, mientras que en el caso del grafeno, su modelado aún está en discusión, y todavía no se han arrojado resultados experimentales que validen su idoneidad. Estas tres razones hacen que los reflectarrays basados en cristal líquido sean atractivos para multitud de aplicaciones de haz reconfigurable a frecuencias superiores a 60 GHz. Aplicaciones como radar de escaneo de imágenes de alta resolución, espectroscopia molecular, radiómetros para observación atmosférica, o comunicaciones inalámbricas de alta frecuencia (WiGig) son algunas de ellas. La tesis está estructurada en tres partes. En la primera de ellas se describen las características más comunes de los cristales líquidos, centrándonos en detalle en aquellas propiedades ofrecidas por este material en fase nemática. En concreto, se estudiará la anisotropía dieléctrica (Ae) de los cristales líquidos uniaxiales, que son los que se emplean en esta tesis, definida como la diferencia entre la permitividad paralela (£//) y la perpendicular (e±): Ae = e,, - e±. También se estudiará la variación de este parámetro (Ae) con la frecuencia, y el modelado electromagnético macroscópico más general que, extraído a partir de aquella, permite describir el cristal líquido para cada tensión de polarización en celdas de geometría planar. Este modelo es de suma importancia para garantizar precisión en el desfasaje proporcionado por las diferentes celdas reconfigurables para reflectarrays que se describirán en la siguiente parte de la tesis. La segunda parte de la tesis se centra en el diseño de celdas reflectarray resonantes basadas en cristal líquido. La razón por la que se escogen estos tipos de celdas reside en el hecho de que son las únicas capaces de proporcionar rangos de fase elevados ante la reducida anisotropía dieléctrica que ofrecen los cristales líquidos. El objetivo de esta parte trata, por tanto, de obtener estructuras de celdas reflectarray que sean capaces de proporcionar buenas prestaciones eléctricas a nivel de antena, mejorando sustancialmente las prestaciones de las celdas reportadas en el estado del arte, así como de desarrollar una herramienta de diseño general para aquellas. Para ello, se estudian las prestaciones eléctricas de diferentes tipos de elementos resonantes de cristal líquido que van, desde el más sencillo, que ha limitado el estado de la técnica hasta el desarrollo de esta tesis y que está formado por un sólo resonador, a elementos que constan de varios resonadores (multi-resonantes) y que pueden ser monocapa o multicapa. En un primer paso, el procedimiento de diseño de estas estructuras hace uso de un modelo convencional de cristal líquido que ha venido siendo usado en el estado del arte para este tipo de celdas, y que considera el cristal líquido como un material homogéneo e isótropo cuya permitividad varía entre (e/7) y (e±). Sin embargo, en esta parte de la tesis se demuestra que dicho modelado no es suficiente para describir de forma genérica el comportamiento del cristal líquido en las celdas tipo reflectarray. En la tesis se proponen procedimientos más exactos para el análisis y diseño basados en un modelo más general que define el cristal líquido como un material anisótropo e inhomogeneo en tres dimensiones, y se ha implementado una técnica que permite optimizar celdas multi-resonantes de forma eficiente para conseguir elevadas prestaciones en cuanto a ancho de banda, rango de fase, pérdidas, o sensibilidad al ángulo de incidencia. Los errores cometidos en el uso del modelado convencional a nivel de celda (amplitud y fase) se han analizado para varias geometrías, usando medidas de varios prototipos de antena que usan un cristal líquido real a frecuencias superiores a 100 GHz. Las medidas se han realizado en entorno periódico mediante un banco cuasi-óptico, que ha sido diseñado especialmente para este fin. Uno de estos prototipos se ha optimizado a 100 GHz para conseguir un ancho de banda relativamente elevado (10%), pérdidas reducidas, un rango de fase mayor de 360º, baja sensibilidad al ángulo de incidencia, y baja influencia de la inhomogeneidad transversal del cristal líquido en la celda. Estas prestaciones a nivel de celda superan de forma clara aquellas conseguidas por otros elementos que se han reportado en la literatura, de manera que dicho prototipo se ha usado en la última parte de la tesis para realizar diversas antenas de barrido. Finalmente, en esta parte se presenta una estrategia de caracterización de la anisotropía macroscópica a partir de medidas de los elementos de reflectarray diseñados en banco cuasi-óptico, obteniendo resultados tanto en las frecuencias de interés en RF como en AC, y comparándolas con aquellas obtenidas mediante otros métodos. La tercera parte de la tesis consiste en el estudio, diseño, fabricación y medida de antenas reconfigurables basadas en cristal líquido en configuraciones complejas. En reflectarrays pasivos, el procedimiento de diseño de la antena se limita únicamente al ajuste en cada celda de la antena de las dimensiones de las metalizaciones que se emplean para el control de fase, mediante procesos de optimización bien conocidos. Sin embargo, en el caso de reflectarrays reconfigurables basados en cristal líquido, resulta necesario un paso adicional, que consiste en calcular de forma adecuada las tensiones de control en cada celda del reflectarray para configurar la fase requerida en cada una de ellas, así como diseñar la estructura y los circuitos de control que permitan direccionar a cada elemento su tensión correspondiente. La síntesis de tensiones es por tanto igual o más importante que el diseño de la geometría de las celdas, puesto que éstas son las que están directamente relacionadas con la fase. En el estado del arte, existen varias estrategias de síntesis de tensiones que se basan en la caracterización experimental de la curva de fase respecto al voltaje. Sin embargo, esta caracterización sólo puede hacerse a un solo ángulo de incidencia y para unas determinadas dimensiones de celda, lo que produce que las tensiones sintetizadas sean diferentes de las adecuadas, y en definitiva que se alcancen errores de fase mayores de 70º. De esta forma, hasta la fecha, las prestaciones a nivel de antena que se han conseguido son reducidas en cuanto a ancho de banda, rango de escaneo o nivel de lóbulos secundarios. En esta última parte de la tesis, se introduce una nueva estrategia de síntesis de tensiones que es capaz de predecir mediante simulaciones, y con alta precisión, las tensiones que deben introducirse en cada celda teniendo en cuenta su ángulo de incidencia, sus dimensiones, la frecuencia, así como la señal de polarización definida por su frecuencia y forma de onda AC. Esta estrategia se basa en modelar cada uno de los estados de permitividad del cristal líquido como un sustrato anisótropo con inhomogeneidad longitudinal (1D), o en ciertos casos, como un tensor equivalente homogéneo. La precisión de ambos modelos electromagnéticos también se discute. Con el objetivo de obtener una herramienta eficiente de cálculo de tensiones, también se ha escrito e implementado una herramienta de análisis basada en el Método de los Momentos en el Dominio Espectral (SD-MoM) para sustratos estratificados anisótropos, que se usa en cada iteración del procedimiento de síntesis para analizar cada una de las celdas de la antena. La síntesis de tensiones se ha diseñado además para reducir al máximo el efecto del rizado de amplitud en el diagrama de radiación, que es característico en los reflectarrays que están formados por celdas con pérdidas elevadas, lo que en sí, supone un avance adicional para la obtención de mejores prestaciones de antena. Para el cálculo de los diagramas de radiación empleados en el procedimiento de síntesis, se asume un análisis elemento a elemento considerando periodicidad local, y se propone el uso de un método capaz de modelar el campo incidente de forma que se elimine la limitación de la periodicidad local en la excitación. Una vez definida la estrategia adecuada de cálculo de las tensiones a aplicar al cristal líquido en cada celda, la estructura de direccionamiento de las mismas en la antena, y diseñados los circuitos de control, se diseñan, fabrican y miden dos prototipos diferentes de antena de barrido electrónico a 100 GHz usando las celdas anteriormente presentadas. El primero de estos prototipos es un reflectarray en configuración “single offset” con capacidad de escaneo en un plano (elevación o azimut). Aunque previamente se realizan diseños de antenas de barrido en 2D a varias frecuencias en el rango de milimétricas y sub-milimétricas, y se proponen ciertas estrategias de direccionamiento que permiten conseguir este objetivo, se desarrolla el prototipo con direccionamiento en una dimensión con el fin de reducir el número de controles y posibles errores de fabricación, y así también validar la herramienta de diseño. Para un tamaño medio de apertura (con un numero de filas y columnas entre 30 y 50 elementos, lo que significa un reflectarray con un número de elementos superior a 900), la configuración “single offset” proporciona rangos de escaneo elevados, y ganancias que pueden oscilar entre los 20 y 30 dBi. En concreto, el prototipo medido proporciona un haz de barrido en un rango angular de 55º, en el que el nivel de lóbulos secundarios (SLL) permanece mejor de -13 dB en un ancho de banda de un 8%. La ganancia máxima es de 19.4 dBi. Estas prestaciones superan de forma clara aquellas conseguidas por otros autores. El segundo prototipo se corresponde con una antena de doble reflector que usa el reflectarray de cristal líquido como sub-reflector para escanear el haz en un plano (elevación o azimut). El objetivo básico de esta geometría es obtener mayores ganancias que en el reflectarray “single offset” con una estructura más compacta, aunque a expensas de reducir el rango de barrido. En concreto, se obtiene una ganancia máxima de 35 dBi, y un rango de barrido de 12º. Los procedimientos de síntesis de tensiones y de diseño de las estructuras de las celdas forman, en su conjunto, una herramienta completa de diseño precisa y eficiente de antenas reflectarray reconfigurables basados en cristales líquidos. Dicha herramienta se ha validado mediante el diseño, la fabricación y la medida de los prototipos anteriormente citados a 100 GHz, que consiguen algo nunca alcanzado anteriormente en la investigación de este tipo de antenas: unas prestaciones competitivas y una predicción excelente de los resultados. El procedimiento es general, y por tanto se puede usar a cualquier frecuencia en la que el cristal líquido ofrezca anisotropía dieléctrica, incluidos los THz. Los prototipos desarrollados en esta tesis doctoral suponen también unas de las primeras antenas de barrido real a frecuencias superiores a 100 GHz. En concreto, la antena de doble reflector para escaneo de haz es la primera antena reconfigurable electrónicamente a frecuencias superiores a 60 GHz que superan los 25 dBi de ganancia, siendo a su vez la primera antena de doble reflector que contiene un reflectarray reconfigurable como sub-reflector. Finalmente, se proponen ciertas mejoras que aún deben se deben realizar para hacer que estas antenas puedan ser un producto completamente desarrollado y competitivo en el mercado. ABSTRACT The work presented in this thesis is focused on the development of electronically reconfigurable antennas that are able to provide competitive electrical performance to the increasingly common applications operating at frequencies above 60 GHz. Specifically, this thesis presents the study, design, and implementation of reflectarray antennas, which incorporate liquid crystal (LC) materials to scan or reconfigure the beam electronically. From a general point of view, a liquid crystal can be defined as a material whose dielectric permittivity is variable and can be controlled with an external excitation, which usually corresponds with a quasi-static electric field (AC). By changing the dielectric permittivity at each cell that makes up the reflectarray, the phase shift on the aperture is controlled, so that a prescribed radiation pattern can be configured. Liquid Crystal-based reflectarrays have been chosen for several reasons. The first has to do with the advantages provided by the reflectarray antenna with respect to other high gain antennas, such as reflectors or phased arrays. The RF feeding in reflectarrays is achieved by using a common primary source (as in reflectors). This arrangement and the large number of degrees of freedom provided by the cells that make up the reflectarray (as in arrays), allow these antennas to provide a similar or even better electrical performance than other low profile antennas (reflectors and arrays), but assuming a more reduced cost and compactness. The second reason is the flexibility of the liquid crystal to be confined in an arbitrary geometry due to its fluidity (property of liquids). Therefore, the liquid crystal is able to adapt to a planar geometry so that it is one or more of the typical layers of this configuration. This simplifies drastically both the structure and manufacture of this type of antenna, even when compared with reconfigurable reflectarrays based on other technologies, such as diodes MEMS, etc. Therefore, the cost of developing this type of antenna is very small, which means that electrically large reconfigurable reflectarrays could be manufactured assuming low cost and greater productions. A paradigmatic example of a similar structure is the liquid crystal panel, which has already been commercialized successfully. The third reason lies in the fact that, at present, the liquid crystal is one of the few technologies capable of providing switching capabilities at frequencies above 60 GHz. In fact, the liquid crystal allows its permittivity to be switched in a wide range of frequencies, which are from DC to the visible spectrum, including microwaves and THz. Other technologies, such as ferroelectric materials, graphene or CMOS "on chip" technology also allow the beam to be switched at these frequencies. However, CMOS technology is expensive and is currently limited to frequencies below 150 GHz, and although ferroelectric materials or graphene can switch at higher frequencies and in a wider range, they have serious difficulties that make them immature. Ferroelectric materials involve the use of very high voltages to switch the material, making them unattractive, whereas the electromagnetic modelling of the graphene is still under discussion, so that the experimental results of devices based on this latter technology have not been reported yet. These three reasons make LC-based reflectarrays attractive for many applications that involve the use of electronically reconfigurable beams at frequencies beyond 60 GHz. Applications such as high resolution imaging radars, molecular spectroscopy, radiometers for atmospheric observation, or high frequency wireless communications (WiGig) are just some of them. This thesis is divided into three parts. In the first part, the most common properties of the liquid crystal materials are described, especially those exhibited in the nematic phase. The study is focused on the dielectric anisotropy (Ac) of uniaxial liquid crystals, which is defined as the difference between the parallel (e/7) and perpendicular (e±) permittivities: Ae = e,, - e±. This parameter allows the permittivity of a LC confined in an arbitrary volume at a certain biasing voltage to be described by solving a variational problem that involves both the electrostatic and elastic energies. Thus, the frequency dependence of (Ae) is also described and characterised. Note that an appropriate LC modelling is quite important to ensure enough accuracy in the phase shift provided by each cell that makes up the reflectarray, and therefore to achieve a good electrical performance at the antenna level. The second part of the thesis is focused on the design of resonant reflectarray cells based on liquid crystal. The reason why resonant cells have been chosen lies in the fact that they are able to provide enough phase range using the values of the dielectric anisotropy of the liquid crystals, which are typically small. Thus, the aim of this part is to investigate several reflectarray cell architectures capable of providing good electrical performance at the antenna level, which significantly improve the electrical performance of the cells reported in the literature. Similarly, another of the objectives is to develop a general tool to design these cells. To fulfill these objectives, the electrical yields of different types of resonant reflectarray elements are investigated, beginning from the simplest, which is made up of a single resonator and limits the state of the art. To overcome the electrical limitations of the single resonant cell, several elements consisting of multiple resonators are considered, which can be single-layer or multilayer. In a first step, the design procedure of these structures makes use of a conventional electromagnetic model which has been used in the literature, which considers that the liquid crystal behaves as homogeneous and isotropic materials whose permittivity varies between (e/7) y (e±). However, in this part of the thesis it is shown that the conventional modelling is not enough to describe the physical behaviour of the liquid crystal in reflectarray cells accurately. Therefore, a more accurate analysis and design procedure based on a more general model is proposed and developed, which defines the liquid crystal as an anisotropic three-dimensional inhomogeneous material. The design procedure is able to optimize multi-resonant cells efficiently to achieve good electrical performance in terms of bandwidth, phase range, losses, or sensitivity to the angle of incidence. The errors made when the conventional modelling (amplitude and phase) is considered have been also analysed for various cell geometries, by using measured results from several antenna prototypes made up of real liquid crystals at frequencies above 100 GHz. The measurements have been performed in a periodic environment using a quasi-optical bench, which has been designed especially for this purpose. One of these prototypes has been optimized to achieve a relatively large bandwidth (10%) at 100 GHz, low losses, a phase range of more than 360º, a low sensitivity to angle of incidence, and a low influence of the transversal inhomogeneity of the liquid crystal in the cell. The electrical yields of this prototype at the cell level improve those achieved by other elements reported in the literature, so that this prototype has been used in the last part of the thesis to perform several complete antennas for beam scanning applications. Finally, in this second part of the thesis, a novel strategy to characterise the macroscopic anisotropy using reflectarray cells is presented. The results in both RF and AC frequencies are compared with those obtained by other methods. The third part of the thesis consists on the study, design, manufacture and testing of LCbased reflectarray antennas in complex configurations. Note that the design procedure of a passive reflectarray antenna just consists on finding out the dimensions of the metallisations of each cell (which are used for phase control), using well-known optimization processes. However, in the case of reconfigurable reflectarrays based on liquid crystals, an additional step must be taken into account, which consists of accurately calculating the control voltages to be applied to each cell to configure the required phase-shift distribution on the surface of the antenna. Similarly, the structure to address the voltages at each cell and the control circuitry must be also considered. Therefore, the voltage synthesis is even more important than the design of the cell geometries (dimensions), since the voltages are directly related to the phase-shift. Several voltage synthesis procedures have been proposed in the state of the art, which are based on the experimental characterization of the phase/voltage curve. However, this characterization can be only carried out at a single angle of incidence and at certain cell dimensions, so that the synthesized voltages are different from those needed, thus giving rise to phase errors of more than 70°. Thus, the electrical yields of the LCreflectarrays reported in the literature are limited in terms of bandwidth, scanning range or side lobes level. In this last part of the thesis, a new voltage synthesis procedure has been defined and developed, which allows the required voltage to be calculated at each cell using simulations that take into account the particular dimensions of the cells, their angles of incidence, the frequency, and the AC biasing signal (frequency and waveform). The strategy is based on the modelling of each one of the permittivity states of the liquid crystal as an anisotropic substrate with longitudinal inhomogeneity (1D), or in certain cases, as an equivalent homogeneous tensor. The accuracy of both electromagnetic models is also discussed. The phase errors made by using the proposed voltage synthesis are better than 7º. In order to obtain an efficient tool to analyse and design the reflectarray, an electromagnetic analysis tool based on the Method of Moments in the spectral domain (SD-MoM) has also written and developed for anisotropic stratified media, which is used at each iteration of the voltage synthesis procedure. The voltage synthesis is also designed to minimize the effect of amplitude ripple on the radiation pattern, which is typical of reflectarrays made up of cells exhibiting high losses and represents a further advance in achieving a better antenna performance. To calculate the radiation patterns used in the synthesis procedure, an element-by-element analysis is assumed, which considers the local periodicity approach. Under this consideration, the use of a novel method is proposed, which avoids the limitation that the local periodicity imposes on the excitation. Once the appropriate strategy to calculate the voltages to be applied at each cell is developed, and once it is designed and manufactured both the structure to address the voltages to the antenna and the control circuits, two complete LC-based reflectarray antennas that operate at 100 GHz have been designed, manufactured and tested using the previously presented cells. The first prototype consists of a single offset reflectarray with beam scanning capabilities on one plane (elevation and azimuth). Although several LC-reflectarray antennas that provide 2-D scanning capabilities are also designed, and certain strategies to achieve the 2-D addressing of the voltage are proposed, the manufactured prototype addresses the voltages in one dimension in order to reduce the number of controls and manufacturing errors, and thereby validating the design tool. For an average aperture size (with a number of rows and columns of between 30 and 50 elements, which means a reflectarray with more than 900 cells), the single offset configuration provides an antenna gain of between 20 and 30 dBi and a large scanning range. The prototype tested at 100 GHz exhibits an electronically scanned beam in an angular range of 55º and 8% of bandwidth, in which the side lobe level (SLL) remains better than -13 dB. The maximum gain is 19.4 dBi. The electrical performance of the antenna is clearly an improvement on those achieved by other authors in the state of the art. The second prototype corresponds to a dual reflector antenna with a liquid crystal-based reflectarray used as a sub-reflector for beam scanning in one plane (azimuth or elevation). The main objective is to obtain a higher gain than that provided by the single offset configuration, but using a more compact architecture. In this case, a maximum gain of 35 dBi is achieved, although at the expense of reducing the scanning range to 12°, which is inherent in this type of structure. As a general statement, the voltage synthesis and the design procedure of the cells, jointly make up a complete, accurate and efficient design tool of reconfigurable reflectarray antennas based on liquid crystals. The tool has been validated by testing the previously mentioned prototypes at 100 GHz, which achieve something never reached before for this type of antenna: a competitive electrical performance, and an excellent prediction of the results. The design procedure is general and therefore can be used at any frequency for which the liquid crystal exhibits dielectric anisotropy. The two prototypes designed, manufactured and tested in this thesis are also some of the first antennas that currently operate at frequencies above 100 GHz. In fact, the dual reflector antenna is the first electronically scanned dual reflector antenna at frequencies above 60 GHz (the operation frequency is 100 GHz) with a gain greater than 25 dBi, being in turn the first dual-reflector antenna with a real reconfigurable sub-reflectarray. Finally, some improvements that should be still investigated to make these antennas commercially competitive are proposed.

Two-dimensional isostatic meshes in the finite element method

In a Finite Element (FE) analysis of elastic solids several items are usually considered, namely, type and shape of the elements, number of nodes per element, node positions, FE mesh, total number of degrees of freedom (dot) among others. In this paper a method to improve a given FE mesh used for a particular analysis is described. For the improvement criterion different objective functions have been chosen (Total potential energy and Average quadratic error) and the number of nodes and dof's of the new mesh remain constant and equal to the initial FE mesh. In order to find the mesh producing the minimum of the selected objective function the steepest descent gradient technique has been applied as optimization algorithm. However this efficient technique has the drawback that demands a large computation power. Extensive application of this methodology to different 2-D elasticity problems leads to the conclusion that isometric isostatic meshes (ii-meshes) produce better results than the standard reasonably initial regular meshes used in practice. This conclusion seems to be independent on the objective function used for comparison. These ii-meshes are obtained by placing FE nodes along the isostatic lines, i.e. curves tangent at each point to the principal direction lines of the elastic problem to be solved and they should be regularly spaced in order to build regular elements. That means ii-meshes are usually obtained by iteration, i.e. with the initial FE mesh the elastic analysis is carried out. By using the obtained results of this analysis the net of isostatic lines can be drawn and in a first trial an ii-mesh can be built. This first ii-mesh can be improved, if it necessary, by analyzing again the problem and generate after the FE analysis the new and improved ii-mesh. Typically, after two first tentative ii-meshes it is sufficient to produce good FE results from the elastic analysis. Several example of this procedure are presented.

An iterative method for extracting energy-like quantities from protein structures.

We present a method (ENERGI) for extracting energy-like quantities from a data base of protein structures. In this paper, we use the method to generate pairwise additive amino acid "energy" scores. These scores are obtained by iteration until they correctly discriminate a set of known protein folds from decoy conformations. The method succeeds in lattice model tests and in the gapless threading problem as defined by Maiorov and Crippen [Maiorov, V. N. & Crippen, G. M. (1992) J. Mol. Biol. 227, 876-888]. A more challenging test of threading a larger set of test proteins derived from the representative set of Hobohm and Sander [Hobohm, U. & Sander, C. (1994) Protein Sci. 3, 522-524] is used as a "workbench" for exploring how the ENERGI scores depend on their parameter sets.

A variational approach to learning curves

We combine the replica approach from statistical physics with a variational approach to analyze learning curves analytically. We apply the method to Gaussian process regression. As a main result we derive approximative relations between empirical error measures, the generalization error and the posterior variance.

Variational inference for Student-t MLP models

This paper presents a novel methodology to infer parameters of probabilistic models whose output noise is a Student-t distribution. The method is an extension of earlier work for models that are linear in parameters to nonlinear multi-layer perceptrons (MLPs). We used an EM algorithm combined with variational approximation, the evidence procedure, and an optimisation algorithm. The technique was tested on two regression applications. The first one is a synthetic dataset and the second is gas forward contract prices data from the UK energy market. The results showed that forecasting accuracy is significantly improved by using Student-t noise models.

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.

An alternating boundary integral based method for a Cauchy problem for the Laplace equation in a quadrant

We study the Cauchy problem for the Laplace equation in a quadrant (quarter-plane) containing a bounded inclusion. Given the values of the solution and its derivative on the edges of the quadrant the solution is reconstructed on the boundary of the inclusion. This is achieved using an alternating iterative method where at each iteration step mixed boundary value problems are being solved. A numerical method is also proposed and investigated for the direct mixed problems reducing these to integral equations over the inclusion. Numerical examples verify the efficiency of the proposed scheme.