In the thesis, the Wilcoxon approach to linear regression problems is combined with the fractal image compression to form a novel Wilcoxon fractal image compression. When the original image is corrupted by noise, we argue that the fractal image compression scheme should be insensitive to those outliers present in the corrupted image. This leads to the new concept of robust fractal image compression. The proposed Wilcoxon fractal image compression is the first attempt toward the design of robust fractal image compression. Four different numerical methods, i.e., steepest decent, line minimization based on quadratic interpolation, line minimization based on cubic interpolation, and least absolute deviation, will be proposed to solve the associated linear Wilcoxon regression problem. From the simulation results, it will be seen that, compared with the traditional fractal image compression, Wilcoxon fractal image compression has very good robustness against outliers caused by salt-and-pepper noise. However, it does not show great improvement of the robustness against outliers caused by Gaussian noise.

Methane is an important greenhouse gas that is relatively long-lived in the atmosphere, and wetlands are a major natural source of atmospheric methane. Methane emissions from wetlands are variable across both space and time at scales ranging from meters to continents and a comprehensive accounting of wetland methane efflux is critical for quantifying the atmospheric methane balance. Major uncertainties in quantifying methane efflux arise when measuring and modelling its physical and biological determinants, including water table depth, microtopography, soil temperature, the distribution of aerenchymous vegetation, and the distribution of mosses. Further complications arise with the nonlinear interaction between flux and derivers in highly-heterogeneous wetland landscape. A possible solution for quantifying wetland methane efflux at multiple scales in space (‘upscaling’) is repeated observations using remote sensing technology to acquire information about the land surface across time, space, and spectra. These scaling issues must be resolved to progress in our understanding of the role of wetlands in the global atmospheric methane budget from peatlands. In this thesis, data collected from multiple aircraft and satellite-based remote sensing platforms were investigated to characterize the fine scale spatial heterogeneity of a peatland in southwestern Scotland for the purpose of developing techniques for quantifying (‘upscaling’) methane efflux at multiple scales and space. Seasonal variation in pools such as expansion and contraction was simulated with the LiDAR data to investigate the expansion and contraction of the lakes and pools that could give an idea of increase or decrease in methane emissions. Concepts from information theory applied on the different data sets also revealed the relative loss in some features on peatland surface and relative gain on others and find a natural application for reducing bias in multi-scale spatial classification as well as quantifying the length scales (or scales) at which important surface features for methane fluxes are lost. Results from the wavelet analysis demonstrated the preservation of fine scale heterogeneity up to certain length scale and the pattern on peatland surface was preserved. Variogram techniques were also tested to determine sample size, range and orientation in the data set. All the above has implications on estimating methane budget from the peatland landscape and could reduce the bias in the overall flux estimates. All the methods used can also be applied to contrasting sites.

Recent developments show that derivative free methods are highly demanded by researches for solving optimization problems in various practical contexts. Although well-known optimization methods that employ derivative information can be very effcient, a derivative free method will be more effcient in cases where the objective function is nondifferentiable, the derivative information is not available or is not reliable. Derivative Free Optimization (DFO) is developed for solving small dimensional problems (less than 100 variables) in which the computation of an objective function is relatively expensive and the derivatives of the objective function are not available. Problems of this nature more and more arise in modern physical, chemical and econometric measurements and in engineering applications, where computer simulation is employed for the evaluation of the objective functions. In this thesis, we give an example of the implementation of DFO in an approach for optimizing stirrer configurations, including a parametrized grid generator, a flow solver, and DFO. A derivative free method, i.e., DFO is preferred because the gradient of the objective function with respect to the stirrer&rsquo / s design variables is not directly available. This nonlinear objective function is obtained from the flow field by the flow solver. We present and interpret numerical results of this implementation. Moreover, a contribution is given to a survey and a distinction of DFO research directions, to an analysis and discussion of these. We also state a derivative free algorithm used within a clustering algorithm in combination with non-smooth optimization techniques to reveal the effectiveness of derivative free methods in computations. This algorithm is applied on some data sets from various sources of public life and medicine. We compare various methods, their practical backgrounds, and conclude with a summary and outlook. This work may serve as a preparation of possible future research.

Orientador: Sandra Augusta Santos / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T00:20:47Z (GMT). No. of bitstreams: 1 Nascimento_IvanXavierMourado_M.pdf: 5587602 bytes, checksum: 769fbf124a59d55361b184a6ec802f66 (MD5) Previous issue date: 2014 / Resumo: Métodos de região de confiança formam uma classe de algoritmos iterativos amplamente utilizada em problemas de otimização não linear irrestrita para os quais as derivadas da função objetivo não estão disponíveis ou são imprecisas. Uma das abordagens clássicas desses métodos envolve a otimização de modelos polinomiais aproximadores para a função objetivo, construídos a cada iteração com base em conjuntos amostrais de pontos. Em um trabalho recente, Scheinberg e Toint [SIAM Journal on Optimization, 20 (6) (2010), pp. 3512-3532 ] mostram que apesar do controle do posicionamento dos pontos amostrais ser essencial para a convergência do método, é possível que tal controle ocorra de modo direto apenas no estágio final do algoritmo. Baseando-se nessas ideias e incorporando-as a um esquema algorítmico teórico, os autores investigam analiticamente uma curiosa propriedade de autocorreção da geometria dos pontos, a qual se evidencia nas iterações de insucesso. A convergência global do novo algoritmo é, então, obtida como uma consequência da geometria autocorretiva. Nesta dissertação estudamos o posicionamento dos pontos em métodos baseados em modelos quadráticos de interpolação e analisamos o desempenho computacional do algoritmo teórico proposto por Scheinberg e Toint, cujos parâmetros são determinados / Abstract: Trust-region methods are a class of iterative algorithms widely applied to nonlinear unconstrained optimization problems for which derivatives of the objective function are unavailable or inaccurate. One of the classical approaches involves the optimization of a polynomial model for the objective function, built at each iteration and based on a sample set. In a recent work, Scheinberg and Toint [SIAM Journal on Optimization, 20 (6) (2010), pp. 3512¿3532 ] proved that, despite being essential for convergence results, the improvement of the geometry (poisedness) of the sample set might occur only in the final stage of the algorithm. Based on these ideas and incorporating them into a theoretical algorithm framework, the authors investigate analytically an interesting self-correcting geometry mechanism of the interpolating set, which becomes evident at unsuccessful iterations. Global convergence for the new algorithm is then proved as a consequence of this self-correcting property. In this work we study the positioning of the sample points within interpolation-based methods that rely on quadratic models and investigate the computational performance of the theoretical algorithm proposed by Scheinberg and Toint, whose parameters are based upon either choices of previous works or numerical experiments / Mestrado / Matematica Aplicada / Mestre em Matemática Aplicada