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Concept# Basis function

Summary

In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors.
In numerical analysis and approximation theory, basis functions are also called blending functions, because of their use in interpolation: In this application, a mixture of the basis functions provides an interpolating function (with the "blend" depending on the evaluation of the basis functions at the data points).
Examples
Monomial basis for Cω
The monomial basis for the vector space of analytic functions is given by
{x^n \mid n\in\N}.
This basis is used in Taylor series, amongst others.
Monomial basis for polynomials
The monomial basis also forms a basis for the vector space of polynomials. After all, every polynomial can be written

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This thesis focuses on the development of novel multiresolution image approximations. Specifically, we present two kinds of generalization of multiresolution techniques: image reduction for arbitrary scales, and nonlinear approximations using other metrics than the standard Euclidean one. Traditional multiresolution decompositions are restricted to dyadic scales. As first contribution of this thesis, we develop a method that goes beyond this restriction and that is well suited to arbitrary scale-change computations. The key component is a new and numerically exact algorithm for computing inner products between a continuously defined signal and B-splines of any order and of arbitrary sizes. The technique can also be applied for non-uniform to uniform grid conversion, which is another approximation problem where our method excels. Main applications are resampling and signal reconstruction. Although simple to implement, least-squares approximations lead to artifacts that could be reduced if nonlinear methods would be used instead. The second contribution of the thesis is the development of nonlinear spline pyramids that are optimal for lp-norms. First, we introduce a Banach-space formulation of the problem and show that the solution is well defined. Second, we compute the lp-approximation thanks to an iterative optimization algorithm based on digital filtering. We conclude that l1-approximations reduce the artifacts that are inherent to least-squares methods; in particular, edge blurring and ringing. In addition, we observe that the error of l1-approximations is sparser. Finally, we derive an exact formula for the asymptotic Lp-error; this result justifies using the least-squares approximation as initial solution for the iterative optimization algorithm when the degree of the spline is even; otherwise, one has to include an appropriate correction term. The theoretical background of the thesis includes the modelisation of images in a continuous/discrete formalism and takes advantage of the approximation theory of linear shift-invariant operators. We have chosen B-splines as basis functions because of their nice properties. We also propose a new graphical formalism that links B-splines, finite differences, differential operators, and arbitrary scale changes.

The present work has been focused on the development of analytical and numerical techniques for the analysis of highly convoluted antennas and microwave devices including fractal shaped antennas. The accurate prediction of the frequency response of a high-iterated pre-fractal structure is frequently a very consuming task, in terms of computer resources. The techniques presented in this work try to ameliorate some of the bottlenecks in the solving process of fractal shaped or highly convoluted devices. First, in the frame of the Mixed Potential Integral Equation (MPIE) technique, a new set of basis functions for the discretization of the currents in the Method of Moments (MoM) solution is presented. The basis functions are defined over quadrangular domains and their aim is, on one hand, to allow a good representation of the current while preserving the main longitudinal direction existing in many practical surfaces, used as metalization in printed circuits, and on the other hand, to reduce the number of unknowns compared to a standard triangular mesh. The basis functions over quadrangular cells comprise as particular cases the classic rectangular and triangular rooftops. However, a new basis function over triangular domains is also included as a case derived from the general quadrangle, and has the particularity of being able to model the connection between two triangles with a common vertex, instead of the conventional attachment at the edge of the classic triangular pair. Second, different highly convoluted Euclidean structures have been studied in order to provide a benchmark for fractal devices performance. The considered structures are a meander line and a two-arm square spiral antenna. Both structures show miniaturization capabilities, the spiral being one of the outstanding shapes in terms of miniaturization keeping a reasonable frequency behavior. With the study of these two structures it has been shown that some properties, considered exclusive of the fractal shaped family, appear also in non-fractal shapes. Third, several analysis techniques based on a transmission line approach and specially suited to solve highly convoluted printed line devices have been developed. The aim is to have a simple and fast tool to allow a rapid analysis of complicated structures, providing reasonably accurate results taking into account the simplicity of the model. These methods have been applied to a set of fractal curves belonging to the family of the fractal tree. Some prototypes have been built in microstrip technology and measured to verify the validity of the method. Finally, in order to obtain a fast full-wave analysis of printed-line or wire-strip devices, a new technique is developed. The method takes advantage of the geometry of the structures presenting currents flowing mainly in the longitudinal direction. These 2D structures are then considered as 1D ones, thus, the cells are reduced to their axis or backbones. This approximation is valid for the limiting cases, namely, very narrow structures or interactions between far away cells. However, once the width compensation factor that is computed analytically is included, the approximation becomes valid for all the structures not having a width bigger than the standard mesh density limit in a 2D problem. Values of the error committed with respect to a classic 2D method are given, and the analysis of some line structures is performed, proving the validity of the proposed method.

In this thesis, we explore possible stabilisation methods for the reduce basis approximation of advection-diffusion problems, for which the advection term is dominating. The options we consider are mainly inspired by the Variational Multiscale method (VMS), which decomposes the solution of a variational problem into its coarse scale component, from a coarse scale space, and a fine scale component, from a fine scale space. Our stabilisation proposals are divided into three classes. The first one groups methods that rely on a stabilisation parameter. The second class uses VMS at the algebraic level to attempt stabilisation. Finally the third class is also inspired by VMS at the algebraic level, but with the additional constraint that the fine scale space is orthogonal to the coarse scale space. Numericals tests reported in this thesis show that the methods of the first class is not viable options as the best stabilisation parameter among those tested is the stabilisation parameter that is used at the high fidelity level. Although the stabilisation methods of the second class give accurate results when applied to stable problems, they were also dismissed by the numerical tests, as they did not improve the accuracy of the already stabilised problem. The third class also performs well when applied to stable problems. It has been shown in [7] one of those methods can improve accuracy. However in the current implementation, this result was not achieved here.

2016