Analytic Kramer kernels, Lagrange-type interpolation series and de Branges spaces

The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. In particular, when the involved kernel is analytic in the sampling parameter it can be stated in an abstract setting of reproducing kernel Hilbert spaces of entire functions which includes as a particular case the classical Shannon sampling theory. This abstract setting allows us to obtain a sort of converse result and to characterize when the sampling formula associated with an analytic Kramer kernel can be expressed as a Lagrange-type interpolation series. On the other hand, the de Branges spaces of entire functions satisfy orthogonal sampling formulas which can be written as Lagrange-type interpolation series. In this work some links between all these ideas are established.

An optimization method for conceptual design in engineering systems based on high order singular value decomposition

Se presenta un nuevo método de diseño conceptual en Ingeniería Aeronáutica basado el uso de modelos reducidos, también llamados modelos sustitutos (‘surrogates’). Los ingredientes de la función objetivo se calculan para cada indiviudo mediante la utilización de modelos sustitutos asociados a las distintas disciplinas técnicas que se construyen mediante definiciones de descomposición en valores singulares de alto orden (HOSVD) e interpolaciones unidimensionales. Estos modelos sustitutos se obtienen a partir de un número limitado de cálculos CFD. Los modelos sustitutos pueden combinarse, bien con un método de optimización global de tipo algoritmo genético, o con un método local de tipo gradiente. El método resultate es flexible a la par que mucho más eficiente, computacionalmente hablando, que los modelos convencionales basados en el cálculo directo de la función objetivo, especialmente si aparecen un gran número de parámetros de diseño y/o de modelado. El método se ilustra considerando una versión simplificada del diseño conceptual de un avión. Abstract An optimization method for conceptual design in Aeronautics is presented that is based on the use of surrogate models. The various ingredients in the target function are calculated for each individual using surrogates of the associated technical disciplines that are constructed via high order singular value decomposition and one dimensional interpolation. These surrogates result from a limited number of CFD calculated snapshots. The surrogates are combined with an optimization method, which can be either a global optimization method such as a genetic algorithm or a local optimization method, such as a gradient-like method. The resulting method is both flexible and much more computationally efficient than the conventional method based on direct calculation of the target function, especially if a large number of free design parameters and/or tunablemodeling parameters are present. The method is illustrated considering a simplified version of the conceptual design of an aircraft empennage.

Optical systems design using the SMS method and optimizations

Resumen El diseño de sistemas ópticos, entendido como un arte por algunos, como una ciencia por otros, se ha realizado durante siglos. Desde los egipcios hasta nuestros días los sistemas de formación de imagen han ido evolucionando así como las técnicas de diseño asociadas. Sin embargo ha sido en los últimos 50 años cuando las técnicas de diseño han experimentado su mayor desarrollo y evolución, debido, en parte, a la aparición de nuevas técnicas de fabricación y al desarrollo de ordenadores cada vez más potentes que han permitido el cálculo y análisis del trazado de rayos a través de los sistemas ópticos de forma rápida y eficiente. Esto ha propiciado que el diseño de sistemas ópticos evolucione desde los diseños desarrollados únicamente a partir de la óptica paraxial hasta lo modernos diseños realizados mediante la utilización de diferentes técnicas de optimización multiparamétrica. El principal problema con el que se encuentra el diseñador es que las diferentes técnicas de optimización necesitan partir de un diseño inicial el cual puede fijar las posibles soluciones. Dicho de otra forma, si el punto de inicio está lejos del mínimo global, o diseño óptimo para las condiciones establecidas, el diseño final puede ser un mínimo local cerca del punto de inicio y lejos del mínimo global. Este tipo de problemática ha llevado al desarrollo de sistemas globales de optimización que cada vez sean menos sensibles al punto de inicio de la optimización. Aunque si bien es cierto que es posible obtener buenos diseños a partir de este tipo de técnicas, se requiere de muchos intentos hasta llegar a la solución deseada, habiendo un entorno de incertidumbre durante todo el proceso, puesto que no está asegurado el que se llegue a la solución óptima. El método de las Superficies Múltiples Simultaneas (SMS), que nació como una herramienta de cálculo de concentradores anidólicos, se ha demostrado como una herramienta también capaz utilizarse para el diseño de sistemas ópticos formadores de imagen, aunque hasta la fecha se ha utilizado para el diseño puntual de sistemas de formación de imagen. Esta tesis tiene por objeto presentar el SMS como un método que puede ser utilizado de forma general para el diseño de cualquier sistema óptico de focal fija o v afocal con un aumento definido así como una herramienta que puede industrializarse para ayudar al diseñador a afrontar de forma sencilla el diseño de sistemas ópticos complejos. Esta tesis está estructurada en cinco capítulos: El capítulo 1, es un capítulo de fundamentos donde se presentan los conceptos fundamentales necesarios para que el lector, aunque no posea una gran base en óptica formadora de imagen, pueda entender los planteamientos y resultados que se presentan en el resto de capítulos El capitulo 2 aborda el problema de la optimización de sistemas ópticos, donde se presenta el método SMS como una herramienta idónea para obtener un punto de partida para el proceso de optimización. Mediante un ejemplo aplicado se demuestra la importancia del punto de partida utilizado en la solución final encontrada. Además en este capítulo se presentan diferentes técnicas que permiten la interpolación y optimización de las superficies obtenidas a partir de la aplicación del SMS. Aunque en esta tesis se trabajará únicamente utilizando el SMS2D, se presenta además un método para la interpolación y optimización de las nubes de puntos obtenidas a partir del SMS3D basado en funciones de base radial (RBF). En el capítulo 3 se presenta el diseño, fabricación y medidas de un objetivo catadióptrico panorámico diseñado para trabajar en la banda del infrarrojo lejano (8-12 μm) para aplicaciones de vigilancia perimetral. El objetivo presentado se diseña utilizando el método SMS para tres frentes de onda de entrada utilizando cuatro superficies. La potencia del método de diseño utilizado se hace evidente en la sencillez con la que este complejo sistema se diseña. Las imágenes presentadas demuestran cómo el prototipo desarrollado cumple a la perfección su propósito. El capítulo 4 aborda el problema del diseño de sistemas ópticos ultra compactos, se introduce el concepto de sistemas multicanal, como aquellos sistemas ópticos compuestos por una serie de canales que trabajan en paralelo. Este tipo de sistemas resultan particularmente idóneos para él diseño de sistemas afocales. Se presentan estrategias de diseño para sistemas multicanal tanto monocromáticos como policromáticos. Utilizando la novedosa técnica de diseño que en este capítulo se presenta el diseño de un telescopio de seis aumentos y medio. En el capítulo 5 se presenta una generalización del método SMS para rayos meridianos. En este capítulo se presenta el algoritmo que debe utilizarse para el diseño de cualquier sistema óptico de focal fija. La denominada optimización fase 1 se vi introduce en el algoritmo presentado de forma que mediante el cambio de las condiciones iníciales del diseño SMS que, aunque el diseño se realice para rayos meridianos, los rayos skew tengan un comportamiento similar. Para probar la potencia del algoritmo desarrollado se presenta un conjunto de diseños con diferente número de superficies. La estabilidad y potencia del algoritmo se hace evidente al conseguirse por primera vez el diseño de un sistema de seis superficies diseñado por SMS. vii Abstract The design of optical systems, considered an art by some and a science by others, has been developed for centuries. Imaging optical systems have been evolving since Ancient Egyptian times, as have design techniques. Nevertheless, the most important developments in design techniques have taken place over the past 50 years, in part due to the advances in manufacturing techniques and the development of increasingly powerful computers, which have enabled the fast and efficient calculation and analysis of ray tracing through optical systems. This has led to the design of optical systems evolving from designs developed solely from paraxial optics to modern designs created by using different multiparametric optimization techniques. The main problem the designer faces is that the different optimization techniques require an initial design which can set possible solutions as a starting point. In other words, if the starting point is far from the global minimum or optimal design for the set conditions, the final design may be a local minimum close to the starting point and far from the global minimum. This type of problem has led to the development of global optimization systems which are increasingly less sensitive to the starting point of the optimization process. Even though it is possible to obtain good designs from these types of techniques, many attempts are necessary to reach the desired solution. This is because of the uncertain environment due to the fact that there is no guarantee that the optimal solution will be obtained. The Simultaneous Multiple Surfaces (SMS) method, designed as a tool to calculate anidolic concentrators, has also proved useful for the design of image-forming optical systems, although until now it has occasionally been used for the design of imaging systems. This thesis aims to present the SMS method as a technique that can be used in general for the design of any optical system, whether with a fixed focal or an afocal with a defined magnification, and also as a tool that can be commercialized to help designers in the design of complex optical systems. The thesis is divided into five chapters. Chapter 1 establishes the basics by presenting the fundamental concepts which the reader needs to acquire, even if he/she doesn‟t have extensive knowledge in the field viii of image-forming optics, in order to understand the steps taken and the results obtained in the following chapters. Chapter 2 addresses the problem of optimizing optical systems. Here the SMS method is presented as an ideal tool to obtain a starting point for the optimization process. The importance of the starting point for the final solution is demonstrated through an example. Additionally, this chapter introduces various techniques for the interpolation and optimization of the surfaces obtained through the application of the SMS method. Even though in this thesis only the SMS2D method is used, we present a method for the interpolation and optimization of clouds of points obtained though the SMS3D method, based on radial basis functions (RBF). Chapter 3 presents the design, manufacturing and measurement processes of a catadioptric panoramic lens designed to work in the Long Wavelength Infrared (LWIR) (8-12 microns) for perimeter surveillance applications. The lens presented is designed by using the SMS method for three input wavefronts using four surfaces. The powerfulness of the design method used is revealed through the ease with which this complex system is designed. The images presented show how the prototype perfectly fulfills its purpose. Chapter 4 addresses the problem of designing ultra-compact optical systems. The concept of multi-channel systems, such as optical systems composed of a series of channels that work in parallel, is introduced. Such systems are especially suitable for the design of afocal systems. We present design strategies for multichannel systems, both monochromatic and polychromatic. A telescope designed with a magnification of six-and-a-half through the innovative technique exposed in this chapter is presented. Chapter 5 presents a generalization of the SMS method for meridian rays. The algorithm to be used for the design of any fixed focal optics is revealed. The optimization known as phase 1 optimization is inserted into the algorithm so that, by changing the initial conditions of the SMS design, the skew rays have a similar behavior, despite the design being carried out for meridian rays. To test the power of the developed algorithm, a set of designs with a different number of surfaces is presented. The stability and strength of the algorithm become apparent when the first design of a system with six surfaces if obtained through the SMS method.

Optimal polygonal L1 linearization and fast interpolation of nonlinear systems

The analysis of complex nonlinear systems is often carried out using simpler piecewise linear representations of them. A principled and practical technique is proposed to linearize and evaluate arbitrary continuous nonlinear functions using polygonal (continuous piecewise linear) models under the L1 norm. A thorough error analysis is developed to guide an optimal design of two kinds of polygonal approximations in the asymptotic case of a large budget of evaluation subintervals N. The method allows the user to obtain the level of linearization (N) for a target approximation error and vice versa. It is suitable for, but not limited to, an efficient implementation in modern Graphics Processing Units (GPUs), allowing real-time performance of computationally demanding applications. The quality and efficiency of the technique has been measured in detail on two nonlinear functions that are widely used in many areas of scientific computing and are expensive to evaluate.

New method for designing distillation columns of multicomponent mixtures

This paper presents a series of calculation procedures for computer design of ternary distillation columns overcoming the iterative equilibrium calculations necessary in these kind of problems and, thus, reducing the calculation time. The proposed procedures include interpolation and intersection methods to solve the equilibrium equations and the mass and energy balances. The calculation programs proposed also include the possibility of rigorous solution of mass and energy balances and equilibrium relations.

FFT Interpolation From Nonuniform Samples Lying in a Regular Grid

This paper presents a method to interpolate a periodic band-limited signal from its samples lying at nonuniform positions in a regular grid, which is based on the FFT and has the same complexity order as this last algorithm. This kind of interpolation is usually termed “the missing samples problem” in the literature, and there exists a wide variety of iterative and direct methods for its solution. The one presented in this paper is a direct method that exploits the properties of the so-called erasure polynomial and provides a significant improvement on the most efficient method in the literature, which seems to be the burst error recovery (BER) technique of Marvasti’s The paper includes numerical assessments of the method’s stability and complexity.

Optimization of Chemical Processes Using Surrogate Models Based on a Kriging Interpolation

Superstructure approaches are the solution to the difficult problem which involves the rigorous economic design of a distillation column. These methods require complex initialization procedures and they are hard to solve. For this reason, these methods have not been extensively used. In this work, we present a methodology for the rigorous optimization of chemical processes implemented on a commercial simulator using surrogate models based on a kriging interpolation. Several examples were studied, but in this paper, we perform the optimization of a superstructure for a non-sharp separation to show the efficiency and effectiveness of the method. Noteworthy that it is possible to get surrogate models accurate enough with up to seven degrees of freedom.

Data and Code archive for the interpolation of surface ocean carbon dioxide

We have developed a statistical gap-filling method adapted to the specific coverage and properties of observed fugacity of surface ocean CO2 (fCO2). We have used this method to interpolate the Surface Ocean CO2 Atlas (SOCAT) v2 database on a 2.5°×2.5° global grid (south of 70°N) for 1985-2011 at monthly resolution. The method combines a spatial interpolation based on a 'radius of influence' to determine nearby similar fCO2 values with temporal harmonic and cubic spline curve-fitting, and also fits long term trends and seasonal cycles. Interannual variability is established using deviations of observations from the fitted trends and seasonal cycles. An uncertainty is computed for all interpolated values based on the spatial and temporal range of the interpolation. Tests of the method using model data show that it performs as well as or better than previous regional interpolation methods, but in addition it provides a near-global and interannual coverage.

Supporting interoperable interpolation: the INTAMAP approach

In many Environmental Information Systems the actual observations arise from a discrete monitoring network which might be rather heterogeneous in both location and types of measurements made. In this paper we describe the architecture and infrastructure for a system, developed as part of the EU FP6 funded INTAMAP project, to provide a service oriented solution that allows the construction of an interoperable, automatic, interpolation system. This system will be based on the Open Geospatial Consortium’s Web Feature Service (WFS) standard. The essence of our approach is to extend the GML3.1 observation feature to include information about the sensor using SensorML, and to further extend this to incorporate observation error characteristics. Our extended WFS will accept observations, and will store them in a database. The observations will be passed to our R-based interpolation server, which will use a range of methods, including a novel sparse, sequential kriging method (only briefly described here) to produce an internal representation of the interpolated field resulting from the observations currently uploaded to the system. The extended WFS will then accept queries, such as ‘What is the probability distribution of the desired variable at a given point’, ‘What is the mean value over a given region’, or ‘What is the probability of exceeding a certain threshold at a given location’. To support information-rich transfer of complex and uncertain predictions we are developing schema to represent probabilistic results in a GML3.1 (object-property) style. The system will also offer more easily accessible Web Map Service and Web Coverage Service interfaces to allow users to access the system at the level of complexity they require for their specific application. Such a system will offer a very valuable contribution to the next generation of Environmental Information Systems in the context of real time mapping for monitoring and security, particularly for systems that employ a service oriented architecture.

Projected sequential Gaussian processes: flexible interpolation for large data sets

Recently within the machine learning and spatial statistics communities many papers have explored the potential of reduced rank representations of the covariance matrix, often referred to as projected or fixed rank approaches. In such methods the covariance function of the posterior process is represented by a reduced rank approximation which is chosen such that there is minimal information loss. In this paper a sequential framework for inference in such projected processes is presented, where the observations are considered one at a time. We introduce a C++ library for carrying out such projected, sequential estimation which adds several novel features. In particular we have incorporated the ability to use a generic observation operator, or sensor model, to permit data fusion. We can also cope with a range of observation error characteristics, including non-Gaussian observation errors. Inference for the variogram parameters is based on maximum likelihood estimation. We illustrate the projected sequential method in application to synthetic and real data sets. We discuss the software implementation and suggest possible future extensions.

PSD modifications of FHRV due to interpolation and CTG storage rate

Cardiotocographic data provide physicians information about foetal development and permit to assess conditions such as foetal distress. An incorrect evaluation of the foetal status can be of course very dangerous. To improve interpretation of cardiotocographic recordings, great interest has been dedicated to foetal heart rate variability spectral analysis. It is worth reminding, however, that foetal heart rate is intrinsically an uneven series, so in order to produce an evenly sampled series a zero-order, linear or cubic spline interpolation can be employed. This is not suitable for frequency analyses because interpolation introduces alterations in the foetal heart rate power spectrum. In particular, interpolation process can produce alterations of the power spectral density that, for example, affects the estimation of the sympatho-vagal balance (computed as low-frequency/high-frequency ratio), which represents an important clinical parameter. In order to estimate the frequency spectrum alterations of the foetal heart rate variability signal due to interpolation and cardiotocographic storage rates, in this work, we simulated uneven foetal heart rate series with set characteristics, their evenly spaced versions (with different orders of interpolation and storage rates) and computed the sympatho-vagal balance values by power spectral density. For power spectral density estimation, we chose the Lomb method, as suggested by other authors to study the uneven heart rate series in adults. Summarising, the obtained results show that the evaluation of SVB values on the evenly spaced FHR series provides its overestimation due to the interpolation process and to the storage rate. However, cubic spline interpolation produces more robust and accurate results. © 2010 Elsevier Ltd. All rights reserved.

Projected sequential Gaussian processes:a C++ tool for interpolation of large datasets with heterogeneous noise

Heterogeneous datasets arise naturally in most applications due to the use of a variety of sensors and measuring platforms. Such datasets can be heterogeneous in terms of the error characteristics and sensor models. Treating such data is most naturally accomplished using a Bayesian or model-based geostatistical approach; however, such methods generally scale rather badly with the size of dataset, and require computationally expensive Monte Carlo based inference. Recently within the machine learning and spatial statistics communities many papers have explored the potential of reduced rank representations of the covariance matrix, often referred to as projected or fixed rank approaches. In such methods the covariance function of the posterior process is represented by a reduced rank approximation which is chosen such that there is minimal information loss. In this paper a sequential Bayesian framework for inference in such projected processes is presented. The observations are considered one at a time which avoids the need for high dimensional integrals typically required in a Bayesian approach. A C++ library, gptk, which is part of the INTAMAP web service, is introduced which implements projected, sequential estimation and adds several novel features. In particular the library includes the ability to use a generic observation operator, or sensor model, to permit data fusion. It is also possible to cope with a range of observation error characteristics, including non-Gaussian observation errors. Inference for the covariance parameters is explored, including the impact of the projected process approximation on likelihood profiles. We illustrate the projected sequential method in application to synthetic and real datasets. Limitations and extensions are discussed. © 2010 Elsevier Ltd.

Modeling of a Heat-Induced Buckling of Plates Using the Mesh-free Method

In the process of engineering design of structural shapes, the flat plate analysis results can be generalized to predict behaviors of complete structural shapes. In this case, the purpose of this project is to analyze a thin flat plate under conductive heat transfer and to simulate the temperature distribution, thermal stresses, total displacements, and buckling deformations. The current approach in these cases has been using the Finite Element Method (FEM), whose basis is the construction of a conforming mesh. In contrast, this project uses the mesh-free Scan Solve Method. This method eliminates the meshing limitation using a non-conforming mesh. I implemented this modeling process developing numerical algorithms and software tools to model thermally induced buckling. In addition, convergence analysis was achieved, and the results were compared with FEM. In conclusion, the results demonstrate that the method gives similar solutions to FEM in quality, but it is computationally less time consuming.

Wind Resource Mapping Using Landscape Roughness and Spatial Interpolation Methods

Energy saving, reduction of greenhouse gasses and increased use of renewables are key policies to achieve the European 2020 targets. In particular, distributed renewable energy sources, integrated with spatial planning, require novel methods to optimise supply and demand. In contrast with large scale wind turbines, small and medium wind turbines (SMWTs) have a less extensive impact on the use of space and the power system, nevertheless, a significant spatial footprint is still present and the need for good spatial planning is a necessity. To optimise the location of SMWTs, detailed knowledge of the spatial distribution of the average wind speed is essential, hence, in this article, wind measurements and roughness maps were used to create a reliable annual mean wind speed map of Flanders at 10 m above the Earth’s surface. Via roughness transformation, the surface wind speed measurements were converted into meso- and macroscale wind data. The data were further processed by using seven different spatial interpolation methods in order to develop regional wind resource maps. Based on statistical analysis, it was found that the transformation into mesoscale wind, in combination with Simple Kriging, was the most adequate method to create reliable maps for decision-making on optimal production sites for SMWTs in Flanders (Belgium).

Investigating the use of LiDAR scanning as a method for the measurement of timber distortion features

Measuring the extent to which a piece of structural timber has distorted at a macroscopic scale is fundamental to assessing its viability as a structural component. From the sawmill to the construction site, as structural timber dries, distortion can render it unsuitable for its intended purposes. This rejection of unusable timber is a considerable source of waste to the timber industry and the wider construction sector. As such, ensuring accurate measurement of distortion is a key step in addressing ineffciencies within timber processing. Currently, the FRITS frame method is the established approach used to gain an understanding of timber surface profile. The method, while reliable, is dependent upon relatively few measurements taken across a limited area of the overall surface, with a great deal of interpolation required. Further, the process is unavoidably slow and cumbersome, the immobile scanning equipment limiting where and when measurements can be taken and constricting the process as a whole. This thesis seeks to introduce LiDAR scanning as a new, alternative approach to distortion feature measurement. In its infancy as a measurement technique within timber research, the practicalities of using LiDAR scanning as a measurement method are herein demonstrated, exploiting many of the advantages the technology has over current approaches. LiDAR scanning creates a much more comprehensive image of a timber surface, generating input data multiple magnitudes larger than that of the FRITS frame. Set-up and scanning time for LiDAR is also much quicker and more flexible than existing methods. With LiDAR scanning the measurement process is freed from many of the constraints of the FRITS frame and can be done in almost any environment. For this thesis, surface scans were carried out on seven Sitka spruce samples of dimensions 48.5x102x3000mm using both the FRITS frame and LiDAR scanner. The samples used presented marked levels of distortion and were relatively free from knots. A computational measurement model was created to extract feature measurements from the raw LiDAR data, enabling an assessment of each piece of timber to be carried out in accordance with existing standards. Assessment of distortion features focused primarily on the measurement of twist due to its strong prevalence in spruce and the considerable concern it generates within the construction industry. Additional measurements of surface inclination and bow were also made with each method to further establish LiDAR's credentials as a viable alternative. Overall, feature measurements as generated by the new LiDAR method compared well with those of the established FRITS method. From these investigations recommendations were made to address inadequacies within existing measurement standards, namely their reliance on generalised and interpretative descriptions of distortion. The potential for further uses of LiDAR scanning within timber researches was also discussed.