Some numerical simulations of multi-scale physical phenomena consume a significant amount of computational resources, since their domains are discretized on high resolution meshes. An enormous wastage of these resources occurs in refinement of sections of the domain where computation of the solution does not require high resolutions. This problem is effectively addressed by adaptive mesh refinement (AMR), a technique of local refinement of a mesh only in sections where needed, thus allowing concentration of effort where it is required. Sections of the domain needing high resolution are generally determined by means of a criterion which may vary depending on the nature of the problem. Fairly straightforward criteria could include comparing the solution to a threshold or the gradient of a solution, that is, its local rate of change to a threshold. While the former criterion is not particularly rigorous and hardly ever represents a physical phenomenon of interest, it is simple to implement. However, the gradient criterion is not as simple to implement as a direct comparison of values, but it is still quick and a good indicator of the effectiveness of the AMR technique. The objective of this thesis is to arrive at an adaptive mesh refinement algorithm for a finite difference scheme using a quadtree decomposition approach. In the AMR algorithm developed, a mesh of increasingly fine resolution permits high resolution computation in sub-domains of interest and low resolution in others. In this thesis work, the gradient of the solution has been considered as the criterion determining the regions of the domain needing refinement. Initial tests using the AMR algorithm demonstrate that the paradigm adopted has considerable promise for a variety of research problems. The tests performed thus far depict that the quantity of computational resources consumed is significantly less while maintaining the quality of the solution. Analysis included comparison of results obtained with analytical solutions for four test problems, as well as a thorough study of a contemporary problem in solid mechanics.

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This dissertation deals with the problem of manipulating and storing an image using quadtrees. A quadtree is a tree in which each node has four ordered children or is a leaf. It can be used to represent an image via hierarchical decomposition. The image is broken into four regions. A region can be a solid color (homogeneous) or a mixture of colors (heterogeneous). If a region is heterogeneous it is broken into four subregions, and the process continues recursively until all subregions are homogeneous. The traditional quadtree suffers from dependence on the underlying grid. The grid coordinate system is implicit, and therefore fixed. The fixed coordinate system implies a rigid tree. A rigid tree cannot be translated, scaled, or rotated. Instead, a new tree must be built which is the result of one of these transformations. This dissertation introduces the independent quadtree. The independent quadtree is free of any underlying coordinate system. The tree is no longer rigid and can be easily translated, scaled, or rotated. Algorithms to perform these operations axe presented. The translation and rotation algorithms take constant time. The scaling algorithm has linear time in the number nodes in the tree. The disadvantage of independent quadtrees is the longer generation and display time. This dissertation also introduces an alternate method of hierarchical decomposition. This new method finds the largest homogeneous block with respect to the corners of the image. This block defines the division point for the decomposition. If the size of the block is below some cutoff point, it is deemed to be to small to make the overhead worthwhile and the traditional method is used instead. This new method is compared to the traditional method on randomly generated rectangles, triangles, and circles. The new method is shown to use significantly less space for all three test sets. The generation and display times are ambiguous. More time is taken for each node, but there are, on average, fewer nodes. The worst case is significantly worse.

The extension of Conditional Random Fields (CRF) from discrete states to continuous states will help remove the limitation of the number of states and allow new applications for CRF. In this work, our attempts to obtain a correct procedure to train continuous state conditional random fields through maximum likelihood are presented. By deducing the equations governing the extension of the CRF to continuous states it was possible to merge with the Particle Filter (PF) concept to obtain a formulation governing the training of continuous states CRFs by using particle filters. The results obtained indicated that this process is unsuitable because of the low convergence of the PF integration rate in the needed integrations replacing the summation in CRFs. So a change in concept to an adaptive integration scheme was made. Based on an extension of the Binary Space Partition (BSP) algorithm an adaptive integration process was devised with the aim of producing a more precise integration while retaining a less costly function evaluation than PF. This allowed us to train continuous states conditional random fields with some success. To verify the possibility of increasing the dimension of the states as a vector of continuous states a scalable version was also used to briefly assess its fitness in two-dimensions with quadtrees. This is an asymmetric two-dimensional space partition scheme. In order to increase the knowledge of the problem it would be interesting to have further information of the relevant features. A feature selection embedded method was used based on the lasso regulariser with the intention of pinpointing the most relevant feature functions indicating the relevant features.

Η απεικόνιση και συμπίεση χωρικών δεδομένων έχει γίνει ένα θέμα έμφασης και προσοχής για τον τομέα των computer graphics και για εφαρμογές επεξεργασίας εικόνας. Τα quadtrees, σαν μία από τις ιεραρχικές δομές δεδομένων, βασιζόμενα στην αρχή του επαναλαμβανόμενου διαχωρισμού του χώρου, προσφέρουν πάντα μία συμπαγή και αποτελεσματική αντιπροσώπευση μίας εικόνας. Ο σκοπός αυτής της εργασίας είναι η ανάλυση του αλγορίθμου quadtree, και η σχεδίαση του αλγορίθμου με γλώσσα VHDL. Αναλύεται η μηχανή καταστάσεων και το datapath του υπό ανάπτυξη κυκλώματος. Επίσης παρουσιάζονται κυματομορφές των σημάτων της κάθε κατάστασης από την προσομοίωση που έγινε στο Modelsim. Για την επιβεβαίωση της λειτουργίας του κυκλώματος, έγινε σύνθεση του κυκλώματος με τη βοήθεια του προγράμματος ISE της Xilinx. Όσον αφορά την υλοποίηση, χρησιμοποιήθηκε το board της Xilinx Virtex-5 FPGA ML 507 και εξομοίωση λειτουργίας για εικόνα μεγέθους 64x64. Στο Παράρτημα παρουσιάζονται ο κώδικας Matlab της συνάρτησης qtdecomp που χρησιμοποιήθηκε για την κατανόηση του αλγόριθμου quadtree και καθώς και ο κώδικας Matlab που δημιουργήθηκε για να επαληθεύσει τα αποτελέσματα της προσομοίωσης με Modelsim. / Compression of spatial data has become a matter of emphasis and attention in the field of computer graphics and image processing applications. The quadtrees, as one of the hierarchical data structures, applies the principle of repetitive decomposition of space and offers a compact and efficient representation of an image. The purpose of this paper is to analyze the algorithm quadtree, and to design the algorithm with the language VHDL. In this paper the state machine is analyzed as well as the datapath of the circuit. Furthermore the waveforms of the control signals for each state are presented from the simulation done in the Modelsim environment. A confirmation of the operation of the circuit took also place.The circuit was composed with the help of the ISE of Xilinx. Regarding the implementation, the board of Xilinx Virtex-5 FPGA ML 507 is used and the emulation is done for a picture sized 64x64. The Annex outlines code of the MATLAB function qtdecomp used for understanding the quadtree algorithm and the Matlab code created to verify the simulation results of Modelsim.

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Na última década métodos sintáticos vêm sendo bastante empregados para reconhecimento de padrões em imagens. Uma revisão sistemática sobre o assunto indicou que há trabalhos sobre a utilização de gramáticas para reconhecimento de objetos específicos em imagens, reconhecimento de texturas, construção de objetos, segmentação de imagens, mudança de escala de imagens e reconhecimento de layouts de páginas e documentos. Foi percebido que apenas alguns dos trabalhos analisados apresentavam métodos de inferência gramatical. Percebida essa lacuna, este trabalho teve como objetivo avaliar a utilização de gramáticas estocásticas inferidas a partir de amostras de treinamento que continham as estruturas de quadtrees das imagens no problema de classificação de imagens. Para atingir tal objetivo foram utilizados e adaptados dois algoritmos de inferência gramatical e um algoritmo de estimação de probabilidades responsável por transformar as gramáticas livres de contexto inferidas em gramáticas livres de contexto estocásticas. Os resultados obtidos mostram que as técnicas propostas podem ser utilizadas na classificação de figuras geométricas. Entretanto, embora o desempenho com quadtrees não tenha sido excelente, ficou claro que o uso de gramáticas pode ser uma abordagem interessante para a classificação de imagens. / In the last decade syntactic methods have been widely used for pattern recognition in images. A systematic review on this subject indicated that there are works about the usage of grammars for recognizing specific objects in images, recognition of textures, construction of objects, image segmentation, image scaling, and recognition of layouts in documents. It was noticed that only some of the analyzed studies showed methods of grammatical inference. Perceived this gap, this study aimed to evaluate the usage of stochastic grammars inferred from training samples containing structures of quadtrees of images in image classification problem. To achieve this goal it was used and adapted two algorithms for grammatical inference and one algorithm of probability estimation responsible for transforming the inferred context-free grammars in stochastic context-free grammars. The results show that the proposed techniques can be used in the classification of geometric figures. However, although the performance with quadtrees has not been excellent, it was clear that the use of grammars can be an interesting approach for image classification.

Na última década métodos sintáticos vêm sendo bastante empregados para reconhecimento de padrões em imagens. Uma revisão sistemática sobre o assunto indicou que há trabalhos sobre a utilização de gramáticas para reconhecimento de objetos específicos em imagens, reconhecimento de texturas, construção de objetos, segmentação de imagens, mudança de escala de imagens e reconhecimento de layouts de páginas e documentos. Foi percebido que apenas alguns dos trabalhos analisados apresentavam métodos de inferência gramatical. Percebida essa lacuna, este trabalho teve como objetivo avaliar a utilização de gramáticas estocásticas inferidas a partir de amostras de treinamento que continham as estruturas de quadtrees das imagens no problema de classificação de imagens. Para atingir tal objetivo foram utilizados e adaptados dois algoritmos de inferência gramatical e um algoritmo de estimação de probabilidades responsável por transformar as gramáticas livres de contexto inferidas em gramáticas livres de contexto estocásticas. Os resultados obtidos mostram que as técnicas propostas podem ser utilizadas na classificação de figuras geométricas. Entretanto, embora o desempenho com quadtrees não tenha sido excelente, ficou claro que o uso de gramáticas pode ser uma abordagem interessante para a classificação de imagens. / In the last decade syntactic methods have been widely used for pattern recognition in images. A systematic review on this subject indicated that there are works about the usage of grammars for recognizing specific objects in images, recognition of textures, construction of objects, image segmentation, image scaling, and recognition of layouts in documents. It was noticed that only some of the analyzed studies showed methods of grammatical inference. Perceived this gap, this study aimed to evaluate the usage of stochastic grammars inferred from training samples containing structures of quadtrees of images in image classification problem. To achieve this goal it was used and adapted two algorithms for grammatical inference and one algorithm of probability estimation responsible for transforming the inferred context-free grammars in stochastic context-free grammars. The results show that the proposed techniques can be used in the classification of geometric figures. However, although the performance with quadtrees has not been excellent, it was clear that the use of grammars can be an interesting approach for image classification.

Projecting the ice sheets' contribution to sea-level rise is difficult because of the complexity of accurately modeling ice sheet dynamics for the full polar ice sheets, because of the uncertainty in key, unobservable parameters governing those dynamics, and because quantifying the uncertainty in projections is necessary when determining the confidence to place in them. This work presents the formulation and solution of the Bayesian inverse problem of inferring, from observations, a probability distribution for the basal sliding parameter field beneath the Antarctic ice sheet. The basal sliding parameter is used within a high-fidelity nonlinear Stokes model of ice sheet dynamics. This model maps the parameters "forward" onto a velocity field that is compared against observations. Due to the continental-scale of the model, both the parameter field and the state variables of the forward problem have a large number of degrees of freedom: we consider discretizations in which the parameter has more than 1 million degrees of freedom. The Bayesian inverse problem is thus to characterize an implicitly defined distribution in a high-dimensional space. This is a computationally demanding problem that requires scalable and efficient numerical methods be used throughout: in discretizing the forward model; in solving the resulting nonlinear equations; in solving the Bayesian inverse problem; and in propagating the uncertainty encoded in the posterior distribution of the inverse problem forward onto important quantities of interest. To address discretization, a hybrid parallel adaptive mesh refinement format is designed and implemented for ice sheets that is suited to the large width-to-height aspect ratios of the polar ice sheets. An efficient solver for the nonlinear Stokes equations is designed for high-order, stable, mixed finite-element discretizations on these adaptively refined meshes. A Gaussian approximation of the posterior distribution of parameters is defined, whose mean and covariance can be efficiently and scalably computed using adjoint-based methods from PDE-constrained optimization. Using a low-rank approximation of the covariance of this distribution, the covariance of the parameter is pushed forward onto quantities of interest.