In this dissertation, we consider modified inexact Newton methods applied to second order nonlinear problems. In the implementation of Newton's method applied to problems with a large number of degrees of freedom, it is often necessary to solve the linear Jacobian system iteratively. Although a general theory for the convergence of modified inexact Newton's methods has been developed, its application to nonlinear problems from nonlinear PDE's is far from complete. The case where the nonlinear operator is a zeroth order perturbation of a fixed linear operator was considered in the paper written by Brown et al.. The goal of this dissertation is to show that one can develop modified inexact Newton's methods which converge at a rate independent of the number of unknowns for problems with higher order nonlinearities. To do this, we are required to first, set up the problem on a scale of Hilbert spaces, and second, to devise a special iterative technique which converges in a higher order Sobolev norm, i.e., H1+alpha(omega) \ H1 0(omega) with 0 < alpha < 1/2. We show that the linear system solved in Newton's method can be replaced with one iterative step provided that the initial iterate is close enough. The closeness criteria can be taken independent of the mesh size. In addition, we have the same convergence rates of the method in the norm of H1 0(omega) using the discrete Sobolev inequalities.

The GMRES method is a widely used iterative method for solving the linear systems, of the form Ax = b, especially for the solution of discretized partial differential equations. With an appropriate preconditioner, the solution of the linear system Ax = b can be achieved with less computational effort. Additive Schwarz Preconditioners have two good properties. First, they are easily parallelizable, since several smaller linear systems need to be solved: one system for each of the sub-domains, usually corresponding to the restriction of the differential operator to that subdomain. These are called local problems. Second, if a coarse problem is introduced, they are optimal in the sense that bounds on the convergence rate of the preconditioned iterative method are independent (or slowly dependent) on the finite element mesh size and the number of subproblems. We study certain cases where the same optimality can be obtained without a coarse grid correction. In another part of this thesis we consider inexact GMRES when applied to singular (or nearly singular) linear systems. This applies when instead of matrix-vector products with A, one uses A = A+E for some error matrix E which usually changes from one iteration to the next. Following a similar study by Simoncini and Szyld (2003) for nonsingular systems, we prescribe how to relax the exactness of the matrixvector product and still achieve the desired convergence. In addition, similar criteria is given to guarantee that the computed residual with the inexact method is close to the true residual. Furthermore, we give a new computable criteria to determine the inexactness of matrix-vector product using in inexact CG, and applied it onto control problems governed by parabolic partial differential equations. / Mathematics

With the booming development of WWW, many search engines have been developed to help users to find useful information from a great quantity of data. However, users may have different needs in different situations. Opposite to the Information Retrieval where users retrieve data actively, Information Filtering (IF) sends information from servers to passive users through broadcast mediums, rather than being searched by them. Therefore, each user has his (or her) profile stored in the database, where a profile records a set of interest items that can present his (or her) interests or habits. To efficiently store many user profiles in servers and filter irrelevant users, many signature-based index techniques are applied in IF systems. By using signatures, IF does not need to compare each item of profiles to filter out irrelevant ones. However, because signatures are incomplete information of profiles, it is very hard to answer the complex queries by using only the signatures. Therefore, a critical issue of the signature-based IF service is how to index the signatures of user profiles for an efficient filtering process. There are often two types of queries in the signature-based IF systems, the inexact filtering and the similarity search queries. In the inexact filtering, a query is an incoming document and it needs to find the profiles whose interest items are all included in the query. On the other hand, in the similarity search, a query is a user profile and it needs to find the users whose interest items are similar to the query user. In this thesis, we propose an ID-tree index strategy, which indexes signatures of user profiles by partitioning them into subgroups using a binary tree structure according to all of the different items among them. Basically, our ID-tree index strategy is a kind of the signature tree. In an ID-tree, each path from the root to a leaf node is the signature of the profile pointed by the leaf node. Because each profile is pointed only by one leaf node of the ID-tree, there will be no collision in the structure. In other words, there will be no two profiles assigned to the same signature. Moreover, only the different items among subgroups of profiles will be checked at one time to filter out irrelevant profiles for queries. Therefore, our strategy can answer the inexact filtering and the similarity search queries with less number of accessed profiles as compared to the previous strategies. Moreover, to build the index of signatures, it needs less time to batch a great deal of database profiles. From our simulation results, we show that our strategy can access less number of profiles to answer the queries than Chen's signature tree strategy for the inexact filtering and Aggarwal et al.'s SG-table strategy for the similarity search.

Two types of fuzzy linear programming i.e. fuzzy number linear programming and interval number linear programming are used for optimization problems. In interval form of linear programming we convert the inequalities from the feasible region, containing intervals as coefficients, to two groups of inequalities characterized by real, exact coefficients values. Then classical programming has been used to achieve an optimal solution in the feasible region. In fuzzy number linear programming, α‐cuts and LR forms of fuzzy numbers as coefficients have been used to find optimal solution in the feasible region. Finally the numerical examples and their solutions are attached to provide explanations of procedures.

Dynamical systems are mathematical models characterized by a set of differential or difference equations. Model reduction aims to replace the original system with a reduced system of significantly smaller dimension that still describes the important dynamics of the large-scale model. Interpolatory model reduction methods define a reduced model that interpolates the full model at selected interpolation points. The reduced model may be obtained through a Krylov reduction process or by using the Iterative Rational Krylov Algorithm (IRKA), which iterates this Krylov reduction process to obtain an optimal ℋ₂ reduced model. This dissertation studies interpolatory model reduction for first-order descriptor systems, second-order systems, and DAEs. The main computational cost of interpolatory model reduction is the associated linear systems. Especially in the large-scale setting, inexact solves become desirable if not necessary. With the introduction of inexact solutions, however, exact interpolation no longer holds. While the effect of this loss of interpolation has previously been studied, we extend the discussion to the preconditioned case. Then we utilize IRKA's convergence behavior to develop preconditioner updates. We also consider the interpolatory framework for DAEs and second-order systems. While interpolation results still hold, the singularity associated with the DAE often results in unbounded model reduction errors. Therefore, we present a theorem that guarantees interpolation and a bounded model reduction error. Since this theorem relies on expensive projectors, we demonstrate how interpolation can be achieved without explicitly computing the projectors for index-1 and Hessenberg index-2 DAEs. Finally, we study reduction techniques for second-order systems. Many of the existing methods for second-order systems rely on the model's associated first-order system, which results in computations of a 2𝑛 system. As a result, we present an IRKA framework for the reduction of second-order systems that does not involve the associated 2𝑛 system. The resulting algorithm is shown to be effective for several dynamical systems. / Ph. D.

Cette dernière décennie, la modélisation des villes 3D est devenue l'un des enjeux de la recherche multimédia et un axe important en reconnaissance d'objets. Dans cette thèse nous nous sommes intéressés à localiser différentes primitives, plus particulièrement les fenêtres, dans les façades de Paris. Dans un premier temps, nous présentons une analyse des façades et des différentes propriétés des fenêtres. Nous en déduisons et proposons ensuite un algorithme capable d'extraire automatiquement des hypothèses de fenêtres. Dans une deuxième partie, nous abordons l'extraction et la reconnaissance des primitives à l'aide d'appariement de graphes de contours. En effet une image de contours est lisible par l'oeil humain qui effectue un groupement perceptuel et distingue les entités présentes dans la scène. C'est ce mécanisme que nous avons cherché à reproduire. L'image est représentée sous la forme d'un graphe d'adjacence de segments de contours, valué par des informations d'orientation et de proximité des segments de contours. Pour la mise en correspondance inexacte des graphes, nous proposons plusieurs variantes d'une nouvelle similarité basée sur des ensembles de chemins tracés sur les graphes, capables d'effectuer les groupements des contours et robustes aux changements d'échelle. La similarité entre chemins prend en compte la similarité des ensembles de segments de contours et la similarité des régions définies par ces chemins. La sélection des images d'une base contenant un objet particulier s'effectue à l'aide d'un classifieur SVM ou kppv. La localisation des objets dans l'image utilise un système de vote à partir des chemins sélectionnés par l'algorithme d'appariement. / This last decade, modeling of 3D city became one of the challenges of multimedia search and an important focus in object recognition. In this thesis we are interested to locate various primitive, especially the windows, in the facades of Paris. At first, we present an analysis of the facades and windows properties. Then we propose an algorithm able to extract automatically window candidates. In a second part, we discuss about extraction and recognition primitives using graph matching of contours. Indeed an image of contours is readable by the human eye, which uses perceptual grouping and makes distinction between entities present in the scene. It is this mechanism that we have tried to replicate. The image is represented as a graph of adjacency of segments of contours, valued by information orientation and proximity to edge segments. For the inexact matching of graphs, we propose several variants of a new similarity based on sets of paths, able to group several contours and robust to scale changes. The similarity between paths takes into account the similarity of sets of segments of contours and the similarity of the regions defined by these paths. The selection of images from a database containing a particular object is done using a KNN or SVM classifier.

This thesis studies the so-called “criticality problem”, an important generalised eigenvalue problem arising in neutron transport theory. The smallest positive real eigenvalue of the problem contains valuable information about the status of the fission chain reaction in the nuclear reactor (i.e. the criticality of the reactor), and thus plays an important role in the design and safety of nuclear power stations. Because of the practical importance, efficient numerical methods to solve the criticality problem are needed, and these are the focus of this thesis. In the theory we consider the time-independent neutron transport equation in the monoenergetic homogeneous case with isotropic scattering and vacuum boundary conditions. This is an unsymmetric integro-differential equation in 5 independent variables, modelling transport, scattering, and fission, where the dependent variable is the neutron angular flux. We show that, before discretisation, the nonsymmetric eigenproblem for the angular flux is equivalent to a related eigenproblem for the scalar flux, involving a symmetric positive definite weakly singular integral operator(in space only). Furthermore, we prove the existence of a simple smallest positive real eigenvalue with a corresponding eigenfunction that is strictly positive in the interior of the reactor. We discuss approaches to discretise the problem and present discretisations that preserve the underlying symmetry in the finite dimensional form. The thesis then describes methods for computing the criticality in nuclear reactors, i.e. the smallest positive real eigenvalue, which are applicable for quite general geometries and physics. In engineering practice the criticality problem is often solved iteratively, using some variant of the inverse power method. Because of the high dimension, matrix representations for the operators are often not available and the inner solves needed for the eigenvalue iteration are implemented by matrix-free inneriterations. This leads to inexact iterative methods for criticality computations, for which there appears to be no rigorous convergence theory. The fact that, under appropriate assumptions, the integro-differential eigenvalue problem possesses an underlying symmetry (in a space of reduced dimension) allows us to perform a systematic convergence analysis for inexact inverse iteration and related methods. In particular, this theory provides rather precise criteria on how accurate the inner solves need to be in order for the whole iterative method to converge. The theory is illustrated with numerical examples on several test problems of physical relevance, using GMRES as the inner solver. We also illustrate the use of Monte Carlo methods for the solution of neutron transport source problems as well as for the criticality problem. Links between the steps in the Monte Carlo process and the underlying mathematics are emphasised and numerical examples are given. Finally, we introduce an iterative scheme (the so-called “method of perturbation”) that is based on computing the difference between the solution of the problem of interest and the known solution of a base problem. This situation is very common in the design stages for nuclear reactors when different materials are tested, or the material properties change due to the burn-up of fissile material. We explore the relation ofthe method of perturbation to some variants of inverse iteration, which allows us to give convergence results for the method of perturbation. The theory shows that the method is guaranteed to converge if the perturbations are not too large and the inner problems are solved with sufficiently small tolerances. This helps to explain the divergence of the method of perturbation in some situations which we give numerical examples of. We also identify situations, and present examples, in which the method of perturbation achieves the same convergence rate as standard shifted inverse iteration. Throughout the thesis further numerical results are provided to support the theory.

Newton's method is at the core of many algorithms used for solving nonlinear equations. A globalized Newton method is an implementation of Newton's method augmented with ``globalization procedures' intended to enhance the likelihood of convergence to a solution from an arbitrary initial guess. A Newton-GMRES method is an implementation of Newton's method in which the iterative linear algebra method GMRES is used to solve approximately the linear system that characterizes the Newton step. A globalized Newton-GMRES method combines both globalization procedures and the GMRES scheme to develop robust and efficient algorithms for solving nonlinear equations. The aim of this project is to describe the development of some globalized Newton-GMRES methods and to compare their performances on a few benchmark fluid flow problems.

In science and engineering, many applications require the solution of a sequence of linear systems. There are many ways to solve linear systems and we always look for methods that are faster and/or require less storage. In this dissertation, we focus on solving these systems with Krylov subspace methods and how to obtain effective preconditioners inexpensively. We first present an application for electronic structure calculation. A sequence of slowly changing linear systems is produced in the simulation. The linear systems change by rank-one updates. Properties of the system matrix are analyzed. We use Krylov subspace methods to solve these linear systems. Krylov subspace methods need a preconditioner to be efficient and robust. This causes the problem of computing a sequence of preconditioners corresponding to the sequence of linear systems. We use recycling preconditioners, which is to update and reuse existing preconditioner. We investigate and analyze several preconditioners, such as ILU(0), ILUTP, domain decomposition preconditioners, and inexact matrix-vector products with inner-outer iterations. Recycling preconditioners produces cumulative updates to the preconditioner. To reduce the cost of applying the preconditioners, we propose approaches to truncate the cumulative preconditioner updates, which is a low-rank matrix. Two approaches are developed. The first one is to truncate the low-rank matrix using the best approximation given by the singular value decomposition (SVD). This is effective if many singular values are close to zero. If not, based on the ideas underlying GCROT and recycling, we use information from an Arnoldi recurrence to determine which directions to keep. We investigate and analyze their properties. We also prove that both truncation approaches work well under suitable conditions. We apply our truncation approaches on two applications. One is the Quantum Monte Carlo (QMC) method and the other is a nonlinear second order partial differential equation (PDE). For the QMC method, we test both truncation approaches and analyze their results. For the PDE problem, we discretize the equations with finite difference method, solve the nonlinear problem by Newton's method with a line-search, and utilize Krylov subspace methods to solve the linear system in every nonlinear iteration. The preconditioner is updated by Broyden-type rank-one updates, and we truncate the preconditioner updates by using the SVD finally. We demonstrate that the truncation is effective. In the last chapter, we develop a matrix reordering algorithm that improves the diagonal dominance of Slater matrices in the QMC method. If we reorder the entire Slater matrix, we call it global reordering and the cost is O(N^3), which is expensive. As the change is geometrically localized and impacts only one row and a modest number of columns, we propose a local reordering of a submatrix of the Slater matrix. The submatrix has small dimension, which is independent of the size of Slater matrix, and hence the local reordering has constant cost (with respect to the size of Slater matrix). / Ph. D.

La bioinformática es la aplicación de las ciencias computacionales a la gestión y análisis de datos biológicos. A partir de 2005, con la aparición de los secuenciadores de ADN de nueva generación surge lo que se conoce como Next Generation Sequencing o NGS. Un único experimento biológico puesto en marcha en una máquina de secuenciación NGS puede producir fácilmente cientos de gigabytes o incluso terabytes de datos. Dependiendo de la técnica elegida este proceso puede realizarse en unas pocas horas o días. La disponibilidad de recursos locales asequibles, tales como los procesadores multinúcleo o las nuevas tarjetas gráfi cas preparadas para el cálculo de propósito general GPGPU (General Purpose Graphic Processing Unit ), constituye una gran oportunidad para hacer frente a estos problemas. En la actualidad, un tema abordado con frecuencia es el alineamiento de secuencias de ADN. En bioinformática, el alineamiento permite comparar dos o más secuencias de ADN, ARN, o estructuras primarias proteicas, resaltando sus zonas de similitud. Dichas similitudes podrían indicar relaciones funcionales o evolutivas entre los genes o proteínas consultados. Además, la existencia de similitudes entre las secuencias de un individuo paciente y de otro individuo con una enfermedad genética detectada podría utilizarse de manera efectiva en el campo de la medicina diagnóstica. El problema en torno al que gira el desarrollo de la tesis doctoral consiste en la localización de fragmentos de secuencia cortos dentro del ADN. Esto se conoce bajo el sobrenombre de mapeo de secuencia o sequence mapping. Dicho mapeo debe permitir errores, pudiendo mapear secuencias incluso existiendo variabilidad genética o errores de lectura en el mapeo. Existen diversas técnicas para abordar el mapeo, pero desde la aparición de la NGS destaca la búsqueda por pre jos indexados y agrupados mediante la transformada de Burrows-Wheeler [28] (o BWT en lo sucesivo). Dicha transformada se empleó originalmente en técnicas de compresión de datos, como es el caso del algoritmo bzip2. Su utilización como herramienta para la indización y búsqueda posterior de información es más reciente [22]. La ventaja es que su complejidad computacional depende únicamente de la longitud de la secuencia a mapear. Por otra parte, una gran cantidad de técnicas de alineamiento se basan en algoritmos de programación dinámica, ya sea Smith-Watterman o modelos ocultos de Markov. Estos proporcionan mayor sensibilidad, permitiendo mayor cantidad de errores, pero su coste computacional es mayor y depende del tamaño de la secuencia multiplicado por el de la cadena de referencia. Muchas herramientas combinan una primera fase de búsqueda con la BWT de regiones candidatas al alineamiento y una segunda fase de alineamiento local en la que se mapean cadenas con Smith-Watterman o HMM. Cuando estamos mapeando permitiendo pocos errores, una segunda fase con un algoritmo de programación dinámica resulta demasiado costosa, por lo que una búsqueda inexacta basada en BWT puede resultar más e ficiente. La principal motivación de la tesis doctoral es la implementación de un algoritmo de búsqueda inexacta basado únicamente en la BWT, adaptándolo a las arquitecturas paralelas modernas, tanto en CPU como en GPGPU. El algoritmo constituirá un método nuevo de rami cación y poda adaptado a la información genómica. Durante el periodo de estancia se estudiarán los Modelos ocultos de Markov y se realizará una implementación sobre modelos de computación funcional GTA (Aggregate o Test o Generate), así como la paralelización en memoria compartida y distribuida de dicha plataforma de programación funcional. / Salavert Torres, J. (2014). Inexact Mapping of Short Biological Sequences in High Performance Computational Environments [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/43721