In this dissertation, we examine an advection diffusion model for insects inhabiting a spatially heterogeneous environment and moving toward a more favorable environment. We first study the effects of adding a term describing drift or advection toward a favorable environment to diffusion models for population dynamics. The diffusion model is a basic linear two-dimensional diffusion equation describing local dispersal of species. The mathematical advection terms are taken to be Fickian and describe directed movement of the population toward the favorable environment. For this model, the landscape is composed of one homogeneous habitat patch embedded in a spatially heterogeneous environment and the boundary of the habitat inhabited by the population acts as a lethal edge. We also derived the mean occupancy time and the boundary flux of the habitat patch. The diffusion rate and advection parameters of the advection diffusion model are estimated based on mean occupancy time and boundary flux. We then introduce two methods for the identification of these coefficients in the model as well as the capture rate. These two new methods have some advantages over other methods of estimating those parameters, including reduced computational cost and ease of use in the field. We further examine the statistical properties of new methods through simulation, and discuss how mean occupancy time and boundary flux could be estimated in field experiments.

Questões ambientais estão no centro das discussões nas últimas décadas. A poluição atmosférica, causada pela expansão pós-revolução industrial fez surgir a necessidade de aprender a descrever, usando modelos matemáticos, esse fenômeno. Com esse conhecimento pode-se propor soluções que mitiguem a poluição e os danos colaterais causados ao ambiente. A dispersão de poluentes modelada por soluções analíticas, a partir das equações de advecção-difusão oferecem um conhecimento sobre cada componente que constrói a equação, característica inexistente em outras abordagens, como a numérica. Entretanto ela era incapaz de descrever propriedades que se referem à turbulência, as estruturas coerentes, causadas por componentes não-lineares suprimidas por construção das equações governantes do modelo. Este trabalho estudou uma forma de recuperar características associadas à turbulência através de uma componente fundamental em estruturas coerentes, a fase. Essa é incluída no modelo que passa a descrever manifestações da turbulência em processos de dispersão através de flutuações de pequena escala na concentração da solução do modelo sesquilinear, que é determinístico-estocástico. No decorrer do trabalho há um estudo através de variações de parâmetros para compreender os efeitos da fase no modelo. Ele também foi aplicado ao experimento de Copenhagen e a dois cenários reais com a intenção de compreender o modelo frente à variáveis micrometeorológicas assim como aprimorá-lo para simular a dispersão de poluentes oriundos de fontes de forma realística. / Environmental issues have been at the center of discussions in the last few decades. Atmospheric pollution, caused by post-industrial revolution, has increased the necessity to describe, using mathematical models, this phenomenon. With this knowledge is possible to propose solutions mitigating the pollution and collateral damages caused in the environment. The pollutant dispersion modeled by analytical solutions, from advection-diffusion equations, offers a knowledge about each component that constructs the equation, a characteristic that does not exist in other approaches, such as numerical. However it was unable to describe properties that refer to turbulence, coherent structures, caused by nonlinear components suppressed by constructing the model governing equations. This work studied a way to recover characteristics associated with turbulence through a fundamental component in coherent structures, the phase. This is included in the model which describes manifestations of turbulence in the dispersion process through the presence of small-scale concentration fluctuations in the sesquilinear model, which is deterministicstochastic. In the course of this work there is a study through variations of parameters to understand the phase effects in the model. It was also applied to Copenhagen experiment and to two real scenarios with the intention of understanding the model regarding micrometeorological variables as well as improving it to simulate the pollutant dispersion from sources in a realistic way.

abstract: This thesis considers two problems in the control of robotic swarms. Firstly, it addresses a trajectory planning and task allocation problem for a swarm of resource-constrained robots that cannot localize or communicate with each other and that exhibit stochasticity in their motion and task switching policies. We model the population dynamics of the robotic swarm as a set of advection-diffusion- reaction (ADR) partial differential equations (PDEs). Specifically, we consider a linear parabolic PDE model that is bilinear in the robots' velocity and task-switching rates. These parameters constitute a set of time-dependent control variables that can be optimized and transmitted to the robots prior to their deployment or broadcasted in real time. The planning and allocation problem can then be formulated as a PDE-constrained optimization problem, which we solve using techniques from optimal control. Simulations of a commercial pollination scenario validate the ability of our control approach to drive a robotic swarm to achieve predefined spatial distributions of activity over a closed domain, which may contain obstacles. Secondly, we consider a mapping problem wherein a robotic swarm is deployed over a closed domain and it is necessary to reconstruct the unknown spatial distribution of a feature of interest. The ADR-based primitives result in a coefficient identification problem for the corresponding system of PDEs. To deal with the inherent ill-posedness of the problem, we frame it as an optimization problem. We validate our approach through simulations and show that reconstruction of the spatially-dependent coefficient can be achieved with considerable accuracy using temporal information alone. / Dissertation/Thesis / Masters Thesis Mechanical Engineering 2014

A presente dissertação apresenta uma análise da presença de componentes estocásticas na equação de advecção-difusão, e como estas influenciam a estabilidade da solução. Para tal, a equação de advecção-difusão determinística com fonte contínua idealizada é resolvida através da transformada de Fourier. Adiante, a equação determinística é combinada com componentes estocásticas na velocidade do vento, comprimento de rugosidade e coeficiente de difusão turbulenta vertical. Além disso, é considerada uma permeabilidade parcial nos contornos verticais, de modo que parte do poluente ultrapassa a camada limite atmosférica ou o solo, e outra parte reflete e retorna `a atmosfera. Os resultados obtidos foram validados com os dados do experimento de Hanford. / The present work presents an analysis of the presence of stochastic components in the advection-diffusion equation and how they influence the stability of the solution. For this purpose, the deterministic advection-diffusion equation with idealized continuous source is solved by Fourier transform. Further, the deterministic equation is combined with stochastic components in the wind speed, the roughness and the vertical eddy diffusion coefficient. Moreover, partial permeability is considered in the vertical contours, in the sense that part of the pollutant leaks out of the atmospheric boundary layer or into the soil, and a part is reflected back into the atmosphere. Results were validated with the Hanford experimental data.

This thesis provides a mathematical framework for the development of efficient control strategies that satisfy the charters of Integrated Pest Management (IPM) which aims to maintain pest population at a low impact level. This mathematical framework is based on a dynamical system approach and comprises the construction of mathematical models, their theoretical study, the development of adequate schemes for numerical solutions and reliable procedures for parameter identification. The first output of this thesis is the construction of trap-insect spatio-temporal models formulated via advection-diffusion-reaction processes. These models were used to simulate numerically trapping to compare with field data. As a result, practical protocols were identified to estimate pest-population size and distribution as well as its dispersal capacity and parameter values related to the attractiveness of the traps. The second major output of this thesis is the prediction of the impact of a specific control method: mating disruption using a female pheromone and trapping. A compartmental model, formulated via a system of ordinary differential equations, was built based on biological and mating behaviour knowledge of the pest. The theoretical analysis of the model yields threshold values for the dosage of the pheromone above which extinction of the population is ensured. The practical relevance of the results obtained in this thesis shows that mathematical modelling is an essential supplement to experiments in optimizing control strategies. / Thesis (PhD)--University of Pretoria, 2016. / Mathematics and Applied Mathematics / PhD / Unrestricted

No description available.

Cette étude est consacrée au problème inverse d’estimation de l’intensité et de la position d’une source de polluant. Nous inversons tout d’abord des mesures de concentration à l’intérieur d’une chambre à sable(extension des égouts souterrain) en utilisant la méthode d’identification modale pour estimer l’intensité d’émission du gaz H2S. Une approche originale est ensuite proposée pour estimer la position d’une source en utilisant la notion de transmittance entre deux capteurs. Des outils de régularisation sont utilisés afin d’estimer chaque transmittance, dont le produit de convolution avec la concentration d’un capteur choisi comme référence correspond à la réponse d’un capteur quelconque. Les transmittances estimées à partir des signaux de concentration simulés dans une configuration d’un écoulement tunnel avec source ponctuelle sont ensuite comparées à la sortie d’un modèle analytique 1D de l’équation de transport.L’utilisation d’un algorithme de minimisation non linéaire a permis d’estimer d’abord la vitesse et le coefficient de diffusion de l’écoulement et ensuite la position de la source, sous certaines hypothèses / This numerical study deals with the inverse problem of estimating the intensity and the positionof a pollutant source. The estimation of time-varying emission rates of pollutant sources of H2S in asewer chamber is first implemented through inversion of concentration measurements using the modal identification method. A specific method based on transmittance functions between sensors output isused to estimate the source position. Regularization tools are applied to estimate each transmittance,whose convolution product with the concentration of a reference sensor models the response of any sensor.Transmittances estimated from simulated concentration signals in a tunnel flow with a point source arethen compared to the output of 1D analytical model for the transport equation. A nonlinear minimizationalgorithm is used to estimate the velocity and the diffusion coefficient first and the source position next,under some assumption

Desde o importante trabalho de Barth e Frederickson (1990), um certo número de pesquisadores têm estudado o método de Volumes Finitos de alta-ordem k-exato, por exemplo o grupo do Prof. Ollivier-Gooch: Ollivier-Gooch e van Altena (2002), Nejat (2007), Michalak (2009), etc. Outras discretizações espaciais de alta-ordem bastante populares são o método Galerkin Descontínuo e o método de Diferença Espectral; processos iterativos que involucram estes esquemas tem sido acelerados, nos últimos anos, por métodos p-multigrid. Porém, esta aceleração não tem sido aplicada no contexto do método de Volumes Finitos de alta-ordem, pelo menos para conhecimento do autor desta tese. Por isso, o objetivo desta pesquisa é adaptar o p-multigrid desenvolvido por Liang et al. (2009b) no contexto da Diferença Espectral, para o ambiente dos Volumes Finitos estudado pelo Prof. Ollivier-Gooch. A pesquisa começa implementando o solver VF-RK, de Volumes Finitos com avanço Runge-Kutta, para resolver as equações de advecção-difusão e de Euler aplicados a problemas estacionários, por exemplo, o escoamento transônico ao redor do NACA 0012. Depois, estuda-se o método p-multigrid no contexto da Diferença Espectral; o p-multigrid acelera o processo iterativo comutando níveis polinomiais de alta e de baixa-ordem. Após esse estudo, a adaptação ao âmbito dos Volumes Finitos é realizada resultando num p-multigrid relativamente mais simples porque, em contraposição com o p-multigrid para Diferença Espectral, não precisa de operadores de restrição e prolongação para a comunicação entre diferentes níveis polinomiais. A pesquisa conclui com uma comparação com o método de Volumes Finitos de 4a ordem sem p-multigrid (solver VF-RK). Nesse sentido, implementa-se o solver pMG, baseado no p-multigrid proposto, para resolver os problemas estacionários considerados na primeira parte do trabalho; o smoother do p-multigrid é o esquema Runge-Kutta do código VF-RK, e cada problema estacionário é resolvido utilizando diferentes Vciclos procurando sempre soluções de 4a ordem. Os resultados indicam que o método p-multigrid proposto é mais eficiente que o método de Volumes Finitos de 4a ordem sem p-multigrid, isto é, os dois métodos oferecem a mesma precisão mas o primeiro pode levar menos de 50% do tempo de CPU do segundo. / Since Barth and Frederickson\'s important work (Barth e Frederickson, 1990), a number of researchers have studied high-order k-exact Finite Volume method, for example Prof. Ollivier-Gooch\'s group: Ollivier-Gooch e van Altena (2002), Nejat (2007), Michalak (2009), etc. Other quite popular high-order spatial discretizations are the Discontinuous Galerkin methods and the Spectral Difference methods; the iterative processes involving these schemes have been accelerated in recent years by p-multigrid methods. However, this acceleration has not been applied in the context of the high-order Finite Volume method, at least for the knowledge of the author of this thesis. Therefore, the objective of this research is to adapt the p-multigrid developed by Liang et al. (2009b) in the context of Spectral Difference methods, to the environment of Finite Volume studied by Prof. Ollivier-Gooch. This research begins by implementing the solver VF-RK, Finite Volume solver with Runge-Kutta advance, to compute the advection-diffusion equation and Euler equations applied to steady state problems, for example, the transonic flow around NACA 0012. Then, it is studied the p-multigrid method in the context of Spectral Difference schemes; p-multigrid accelerates the iterative process by switching polynomial levels of high- and low-order. After this study, the adaptation to the context of the Finite Volume scheme is performed resulting in a relatively simple p-multigrid because, in contrast to the p-multigrid for Spectral Difference schemes, it doesn\'t need restriction and prolongation operators for communication between different polynomial levels. The research concludes with a comparison with 4th order Finite Volume method without p-multigrid (solver VF-RK). Accordingly, the solver pMG, based on the proposed p-multigrid, is implemented to resolve the steady state problems considered in the first part of the work; the p-multigrid smoother is the Runge-Kutta scheme from VF-RK code, and each steady state problem is solved using different Vcycles, looking for 4th order solutions ever. The results indicate that the proposed p-multigrid method is more efficient than the 4th order Finite Volume method without p-multigrid: the two methods give the same accuracy but the first one can take less than 50% of second one\'s CPU time.

Tendo em vista o aumento considerável da poltúção do ar provocado em grande parte pela industrialização e o aumento da emissão de poluentes resultantes da queima de combustíveis fósseis por veículos automotores, o presente trabalho tem como objetivo melhorar a previsão e o entendimento da dispersão turbulenta atmosférica. Para tanto, apresenta-se, pela primeira vez, uma representação analít ica para a equação de advecção-difusão-reação tridimensional transiente, com perfil de vento e coeficientes de difusão tmbulenta dependentes da altura, que modelam a dispersão de poluentes na atmosfera. A solução da equação é obtida pela combinação do método GILTT ( Generalized Integral Laplace Transform Technique) com o método da Decomposição de Adomian modificado. Consideram-se dois casos para a aplicação do modelo: no primeiro modela-se a dispersão de um poluente secundário formado por uma reação fotoquímica e no segundo caso, utiliza-se o modelo para determinar o campo de concentração de um poluente que sofre perdas e ganhos devido a influência da radiação solar. Para poder realizar essas análises propôs-se uma parametrização para o termo de reação fotoquímica. São apresentados os resultados numéricos e estatísticos, comparandose com os dados da campanha experimental da Usina Termelétrica de Candiota e com os dados de medições realizadas pela Fundação Estadual de Proteção Ambiental Henrique Luiz Roessler (FEPAM). / In view of the considerable increase of air pollution caused largely by industrialization and the increase of emission pollutants resulting from burning of fossil fuels by motor vehicles, the present work aims to improve the prediction and understanding of atmospheric turbu- lent dispersion. Therefore, is presented, for the rst time, an analytical representation to the transient three-dimensional advection-diffusion-reaction equation, with wind pro le and turbulent diffusion coefficients dependent of height, modeling the dispersion of pollutants in the atmosphere. The solution of the equation is obtained by combining of the GILTT method (Generalized Integral Laplace Transform Technique) with the modi ed Adomian Decomposition method. It is considered two cases for the application of the model: in the rst is modeled the dispersion of a secondary pollutant formed by a photochemical reaction, and in the second case the model is used to determine the concentration eld of a pollutant that suffers losses and gains due to the in uence of solar radiation. To realise these analisis a parameterization for the photochemical reaction term is proposed. Numerical and statistical results are presented, comparing with the experimental campaign data of the thermoelectric plant of Candiota and with data from measurements performed by the \Funda c~ao Estadual de Prote c~ao Ambiental Henrique Luiz Roessler" (FEPAM).

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