Approximate Approximations and a Boundary Point Method for the Linearized Stokes System

The method of approximate approximations, introduced by Maz'ya [1], can also be used for the numerical solution of boundary integral equations. In this case, the matrix of the resulting algebraic system to compute an approximate source density depends only on the position of a finite number of boundary points and on the direction of the normal vector in these points (Boundary Point Method). We investigate this approach for the Stokes problem in the whole space and for the Stokes boundary value problem in a bounded convex domain G subset R^2, where the second part consists of three steps: In a first step the unknown potential density is replaced by a linear combination of exponentially decreasing basis functions concentrated near the boundary points. In a second step, integration over the boundary partial G is replaced by integration over the tangents at the boundary points such that even analytical expressions for the potential approximations can be obtained. In a third step, finally, the linear algebraic system is solved to determine an approximate density function and the resulting solution of the Stokes boundary value problem. Even not convergent the method leads to an efficient approximation of the form O(h^2) + epsilon, where epsilon can be chosen arbitrarily small.

Approximate solutions and error estimates for a Stokes boundary value problem

The aim of this paper is the numerical treatment of a boundary value problem for the system of Stokes' equations. For this we extend the method of approximate approximations to boundary value problems. This method was introduced by V. Maz'ya in 1991 and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present paper we develop an approximation procedure for the solution of the interior Dirichlet problem for the system of Stokes' equations in two dimensions. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In a first step the unknown source density in the potential representation of the solution is replaced by approximate approximations. In a second step the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in a third step Nyström's method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.

Active regions for colour texture segmentation integrating region and boundary information

In this paper a colour texture segmentation method, which unifies region and boundary information, is proposed. The algorithm uses a coarse detection of the perceptual (colour and texture) edges of the image to adequately place and initialise a set of active regions. Colour texture of regions is modelled by the conjunction of non-parametric techniques of kernel density estimation (which allow to estimate the colour behaviour) and classical co-occurrence matrix based texture features. Therefore, region information is defined and accurate boundary information can be extracted to guide the segmentation process. Regions concurrently compete for the image pixels in order to segment the whole image taking both information sources into account. Furthermore, experimental results are shown which prove the performance of the proposed method

Image segmentation integrating colour, texture and boundary information

La tesis se centra en la Visión por Computador y, más concretamente, en la segmentación de imágenes, la cual es una de las etapas básicas en el análisis de imágenes y consiste en la división de la imagen en un conjunto de regiones visualmente distintas y uniformes considerando su intensidad, color o textura. Se propone una estrategia basada en el uso complementario de la información de región y de frontera durante el proceso de segmentación, integración que permite paliar algunos de los problemas básicos de la segmentación tradicional. La información de frontera permite inicialmente identificar el número de regiones presentes en la imagen y colocar en el interior de cada una de ellas una semilla, con el objetivo de modelar estadísticamente las características de las regiones y definir de esta forma la información de región. Esta información, conjuntamente con la información de frontera, es utilizada en la definición de una función de energía que expresa las propiedades requeridas a la segmentación deseada: uniformidad en el interior de las regiones y contraste con las regiones vecinas en los límites. Un conjunto de regiones activas inician entonces su crecimiento, compitiendo por los píxeles de la imagen, con el objetivo de optimizar la función de energía o, en otras palabras, encontrar la segmentación que mejor se adecua a los requerimientos exprsados en dicha función. Finalmente, todo esta proceso ha sido considerado en una estructura piramidal, lo que nos permite refinar progresivamente el resultado de la segmentación y mejorar su coste computacional. La estrategia ha sido extendida al problema de segmentación de texturas, lo que implica algunas consideraciones básicas como el modelaje de las regiones a partir de un conjunto de características de textura y la extracción de la información de frontera cuando la textura es presente en la imagen. Finalmente, se ha llevado a cabo la extensión a la segmentación de imágenes teniendo en cuenta las propiedades de color y textura. En este sentido, el uso conjunto de técnicas no-paramétricas de estimación de la función de densidad para la descripción del color, y de características textuales basadas en la matriz de co-ocurrencia, ha sido propuesto para modelar adecuadamente y de forma completa las regiones de la imagen. La propuesta ha sido evaluada de forma objetiva y comparada con distintas técnicas de integración utilizando imágenes sintéticas. Además, se han incluido experimentos con imágenes reales con resultados muy positivos.

A note on boundary layer friction in baroclinic cyclones

The interaction between extratropical cyclones and the underlying boundary layer has been a topic of recent discussion in papers by Adamson et al (2006) and Beare (2007). Their results emphasise different mechanisms through which the boundary layer dynamics may modify the growth of a baroclinic cyclone. By using different sea-surface temperature distributions and comparing the low-level winds, the differences are exposed and both of the proposed mechanisms appear to be acting within a single simulation.

Numerical simulation of baroclinic waves with a parameterized boundary layer

A dry three-dimensional baroclinic life cycle model is used to investigate the role of turbulent fluxes of heat and momentum within the boundary layer on mid-latitude cyclones. Simulations are performed of life cycles for two basic states, both with and without turbulent fluxes. The different basic states produce cyclones with contrasting frontal and mesoscale-flow structures. The analysis focuses on the generation of potential-vorticity (PV) in the boundary layer and its subsequent transport into the free troposphere. The dynamic mechanism through which friction mitigates a barotropic vortex is that of Ekman pumping. This has often been assumed to be also the dominant mechanism for baroclinic developments. The PV framework highlights an additional, baroclinic mechanism. Positive PV is generated baroclinically due to friction to the north-east of a surface low and is transported out of the boundary layer by a cyclonic conveyor belt flow. The result is an anomaly of increased static stability in the lower troposphere which restricts the growth of the baroclinic wave. The reduced coupling between lower and upper levels can be sufficient to change the character of the upper-level evolution of the mature wave. The basic features of the baroclinic damping mechanism are robust for different frontal structures, with and without turbulent heat fluxes, and for the range of surface roughness found over the oceans.

A case study of boundary layer ventilation by convection and coastal processes

It is often assumed that ventilation of the atmospheric boundary layer is weak in the absence of fronts, but is this always true? In this paper we investigate the processes responsible for ventilation of the atmospheric boundary layer during a nonfrontal day that occurred on 9 May 2005 using the UK Met Office Unified Model. Pollution sources are represented by the constant emission of a passive tracer everywhere over land. The ventilation processes observed include shallow convection, turbulent mixing followed by large-scale ascent, a sea breeze circulation and coastal outflow. Vertical distributions of tracer are validated qualitatively with AMPEP (Aircraft Measurement of chemical Processing Export fluxes of Pollutants over the UK) CO aircraft measurements and are shown to agree impressively well. Budget calculations of tracers are performed in order to determine the relative importance of these ventilation processes. Coastal outflow and the sea breeze circulation were found to ventilate 26% of the boundary layer tracer by sunset of which 2% was above 2 km. A combination of coastal outflow, the sea breeze circulation, turbulent mixing and large-scale ascent ventilated 46% of the boundary layer tracer, of which 10% was above 2 km. Finally, coastal outflow, the sea breeze circulation, turbulent mixing, large-scale ascent and shallow convection together ventilated 52% of the tracer into the free troposphere, of which 26% was above 2 km. Hence this study shows that significant ventilation of the boundary layer can occur in the absence of fronts (and thus during high-pressure events). Turbulent mixing and convection processes can double the amount of pollution ventilated from the boundary layer.

A Galerkin boundary element method for high frequency scattering by convex polygons

In this paper we consider the problem of time-harmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretization of a well-known second kind combined-layer-potential integral equation. We provide a proof that this equation and its adjoint are well-posed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.