In many areas of science one often has a given matrix, representing for example a measured data set and is required to find a matrix that is closest in a suitable norm to the matrix and possesses additionally a structure, inherited from the model used or coming from the application. We call these problems structured matrix nearness problems. We look at three different groups of these problems that come from real applications, analyze the properties of the corresponding matrix structure, and propose algorithms to solve them efficiently. The first part of this thesis concerns the nearness problem of finding the nearest k factor correlation matrix C(X) = diag(I_n -XX T)+XX T to a given symmetric matrix, subject to natural nonlinear constraints on the elements of the n x k matrix X, where distance is measured in the Frobenius norm. Such problems arise, for example, when one is investigating factor models of collateralized debt obligations (CDOs) or multivariate time series. We examine several algorithms for solving the nearness problem that differ in whether or not they can take account of the nonlinear constraints and in their convergence properties. Our numerical experiments show that the performance of the methods depends strongly on the problem, but that, among our tested methods, the spectral projected gradient method is the clear winner. In the second part we look at two two-sided optimization problems where the matrix of unknowns Y ε R {n x p} lies in the Stiefel manifold. These two problems come from an application in atomic chemistry where one is looking for atomic orbitals with prescribed occupation numbers. We analyze these two problems, propose an analytic optimal solution of the first and show that an optimal solution of the second problem can be found by solving a convex quadratic programming problem with box constraints and p unknowns. We prove that the latter problem can be solved by the active-set method in at most 2p iterations. Subsequently, we analyze the set of optimal solutions C}= {Y ε R n x p:Y TY=I_p,Y TNY=D} of the first problem for N symmetric and D diagonal and find that a slight modification of it is a Riemannian manifold. We derive the geometric objects required to make an optimization over this manifold possible. We propose an augmented Lagrangian-based algorithm that uses these geometric tools and allows us to optimize an arbitrary smooth function over C. This algorithm can be used to select a particular solution out of the latter set C by posing a new optimization problem. We compare it numerically with a similar algorithm that ,however, does not apply these geometric tools and find that our algorithm yields better performance. The third part is devoted to low rank nearness problems in the Q-norm, where the matrix of interest is additionally of linear structure, meaning it lies in the set spanned by s predefined matrices U₁,..., U_s ε {0,1} n x p. These problems are often associated with model reduction, for example in speech encoding, filter design, or latent semantic indexing. We investigate three approaches that support any linear structure and examine further the geometric reformulation by Schuermans et al. (2003). We improve their algorithm in terms of reliability by applying the augmented Lagrangian method and show in our numerical tests that the resulting algorithm yields better performance than other existing methods.

<p>A battery assembly experiences both mechanical and thermal loadings during its operation. It is critical to perform the thermomechanical analysis to propose a novel design for the highest efficiency.In this study,two main goals include mechanical characterization and deformation responses for a battery cell and assembly, as well as air-cooled concepts design and analysis.Initially, the cell dimensions were measured by cell-sectioning method, and then the mechanical properties were empirically measured by both 3-point flexural, and nanoindentation experiments. Moreover, three pairs of experiments and simulations were conducted to study mechanical behaviors on both a single cell and a battery assembly. They include (1) point-force loading for single, open cell; (2) internal pressurization for single, sealed cell; and (3) internal pressurization for battery assembly.Additionally, both parametric and experimental studies were executed to design, analyze,and validate air-cooled concepts based on the idea of microchannel heatsink. The proposed concepts have the features, which are integrated into the battery cell for generating the cooling channels. A series of thermomechanical simulations and a forced convection testbed were built for computationally and empirically analyzing the performances of the concepts. The results from the mechanical characterization showed a significant difference between the actual and nominal values of both cell dimensions and mechanical properties. Therefore, the effect of the manufacturing process to such values must be considered before inputting for analyzing the deformation responses. From the thermomechanical analyses, it was found that the mechanical loading might negatively influence the thermal performance if there were not enough mechanical supports from the air-cooling structure. The impact was minimal in the tapered-channel battery assembly. This configuration also significantly reduced the temperature difference on the cell compared with other concepts and the reference design.<br></p>

No description available.

La construction d'un quotient, en topologie, est relativement simple; si $G$ est un groupe topologique agissant sur un espace topologique $X$, on peut considérer l'application naturelle de $X$ dans $X/G$, l'espace d'orbites muni de la topologie quotient. En géométrie algébrique, malheureusement, il n'est généralement pas possible de munir l'espace d'orbites d'une structure de variété. Dans le cas de l'action d'un groupe linéairement réductif $G$ sur une variété projective $X$, la théorie géométrique des invariants nous permet toutefois de construire un morphisme de variété d'un ouvert $U$ de $X$ vers une variété projective $X//U$, se rapprochant autant que possible d'une application quotient, au sens topologique du terme. Considérons par exemple $X\subseteq P^{n}$, une $k$-variété projective sur laquelle agit un groupe linéairement réductif $G$ et supposons que cette action soit induite par une action linéaire de $G$ sur $A^{n+1}$. Soit $\widehat{X}\subseteq A^{n+1}$, le cône affine au dessus de $\X$. Par un théorème de la théorie classique des invariants, il existe alors des invariants homogènes $f_{1},...,f_{r}\in C[\widehat{X}]^{G}$ tels que $$C[\widehat{X}]^{G}= C[f_{1},...,f_{r}].$$ On appellera le nilcone, que l'on notera $N$, la sous-variété de $\X$ définie par le locus des invariants $f_{1},...,f_{r}$. Soit $Proj(C[\widehat{X}]^{G})$, le spectre projectif de l'anneau des invariants. L'application rationnelle $$\pi:X\dashrightarrow Proj(C[f_{1},...,f_{r}])$$ induite par l'inclusion de $C[\widehat{X}]^{G}$ dans $C[\widehat{X}]$ est alors surjective, constante sur les orbites et sépare les orbites autant qu'il est possible de le faire; plus précisément, chaque fibre contient exactement une orbite fermée. Pour obtenir une application régulière satisfaisant les mêmes propriétés, il est nécessaire de jeter les points du nilcone. On obtient alors l'application quotient $$\pi:X\backslash N\rightarrow Proj(C[f_{1},...,f_{r}]).$$ Le critère de Hilbert-Mumford, dû à Hilbert et repris par Mumford près d'un demi-siècle plus tard, permet de décrire $N$ sans connaître les $f_{1},...,f_{r}$. Ce critère est d'autant plus utile que les générateurs de l'anneau des invariants ne sont connus que dans certains cas particuliers. Malgré les applications concrètes de ce théorème en géométrie algébrique classique, les démonstrations que l'on en trouve dans la littérature sont généralement données dans le cadre peu accessible des schémas. L'objectif de ce mémoire sera, entre autres, de donner une démonstration de ce critère en utilisant autant que possible les outils de la géométrie algébrique classique et de l'algèbre commutative. La version que nous démontrerons est un peu plus générale que la version originale de Hilbert \cite{hilbert} et se retrouve, par exemple, dans \cite{kempf}. Notre preuve est valide sur $C$ mais pourrait être généralisée à un corps $k$ de caractéristique nulle, pas nécessairement algébriquement clos. Dans la seconde partie de ce mémoire, nous étudierons la relation entre la construction précédente et celle obtenue en incluant les covariants en plus des invariants. Nous démontrerons dans ce cas un critère analogue au critère de Hilbert-Mumford (Théorème 6.3.2). C'est un théorème de Brion pour lequel nous donnerons une version un peu plus générale. Cette version, de même qu'une preuve simplifiée d'un théorème de Grosshans (Théorème 6.1.7), sont les éléments de ce mémoire que l'on ne retrouve pas dans la littérature. / The topological notion of a quotient is fairly simple. Given a topological group $G$ acting on a topological space $X$, one gets the natural application from $X$ to the quotient space $X/G$. In algebraic geometry, unfortunately, it is generally not possible to give the orbit space the structure of an algebraic variety. In the special case of a linearly reductive group acting on a projective variety $X$, the geometric invariant theory allows us to get a morphism of variety from an open $U$ of $X$ to a projective variety $X//G$, which is as close as possible to a quotient map, from a topological point of view. As an example, let $ X\subseteq P^{n}$ be a $k$-projective variety on which acts a linearly reductive group $G$. Suppose further that this action is induced by a linear action of $G$ on $A^{n+1}$ and let $\widehat{X}\subseteq A^{n +1}$ be the affine cone over $X$. By an important theorem of the classical invariants theory, there exist homogeneous invariants $f_{1},..., f_{r}\in C[\widehat{X}]^{G}$ such as $$\C[\widehat{X}]^{G}=\C[f_{1},...,f_{r}].$$ The locus in $X$ of $f_{1},...,f_{r}$ is called the nullcone, noted $N$. Let $Proj(C[\widehat{X}]^{G})$ be the projective spectrum of the invariants ring. The rational map $$\pi:X\dashrightarrow Proj(C[f_{1},...,f_{r}])$$ induced by the inclusion of $C[\widehat{X}]^{G}$ in $C[\widehat{X}] $ is then surjective, constant on the orbits and separates orbits as much as possible, that is, the fibres contains exactly one closed orbit. A regular map is obtained by removing the nullcone; we then get a regular map $$\pi:X \backslash N\rightarrow Proj(C[f_{1},...,f_{r}])$$ which still satisfy the preceding properties. The Hilbert-Mumford criterion, due to Hilbert and revisited by Mumford nearly half-century later, can be used to describe $N$ without knowing the generators of the invariants ring. Since those are rarely known, this criterion had proved to be quite useful. Despite the important applications of this criterion in classical algebraic geometry, the demonstrations found in the literature are usually given trough the difficult theory of schemes. The aim of this master thesis is therefore, among others, to provide a demonstration of this criterion using classical algebraic geometry and of commutative algebra. The version that we demonstrate is somewhat wider than the original version of Hilbert \cite{hilbert}; a schematic proof of this general version is given in \cite{kempf}. Finally, the proof given here is valid for $C$ but could be generalised to a field $k$ of characteristic zero, not necessarily algebraically closed. In the second part of this thesis, we study the relationship between the preceding constructions and those obtained by including covariants in addition to the invariants. We give a Hilbert-Mumford criterion for covariants (Theorem 6.3.2) which is a theorem from Brion for which we prove a slightly more general version. This theorem, together with a simplified proof of a theorem of Grosshans (Theorem 6.1.7), are the elements of this thesis that can't be found in the literature.

Métodos do tipo Lagrangiano Aumentado são muito utilizados para minimização de funções sujeitas a restrições gerais. Nestes métodos, podemos separar o conjunto de restrições em dois grupos: restrições fáceis e restrições difíceis. Dizemos que uma restrição é fácil se existe um algoritmo disponível e eficiente para resolver problemas restritos a este tipo de restrição. Caso contrário, dizemos que a restrição é difícil. Métodos do tipo Lagrangiano aumentado resolvem, a cada iteração, problemas sujeitos às restrições fáceis, penalizando as restrições difíceis. Problemas de minimização com restrições lineares aparecem com freqüência, muitas vezes como resultados da aproximação de problemas com restrições gerais. Este tipo de problema surge também como subproblema de métodos do tipo Lagrangiano aumentado. Assim, uma implementação eficiente para resolver problemas com restrições lineares é relevante para a implementação eficiente de métodos para resolução de problemas de programação não-linear. Neste trabalho, começamos considerando fáceis as restrições de caixa. Introduzimos BETRA-ESPARSO, uma versão de BETRA para problemas de grande porte. BETRA é um método de restrições ativas que utiliza regiões de confiança para minimização em cada face e gradiente espectral projetado para sair das faces. Utilizamos BETRA (denso ou esparso) na resolução dos subproblemas que surgem a cada iteração de ALGENCAN (um método de lagrangiano aumentado). Para decidir qual algoritmo utilizar para resolver cada subproblema, desenvolvemos regras que escolhem um método para resolver o subproblema de acordo com suas características. Em seguida, introduzimos dois algoritmos de restrições ativas desenvolvidos para resolver problemas com restrições lineares (BETRALIN e GENLIN). Estes algoritmos utilizam, a cada iteração, o método do Gradiente Espectral Projetado Parcial quando decidem mudar o conjunto de restrições ativas. O método do gradiente Espectral Projetado Parcial foi desenvolvido especialmente para este propósito. Neste método, as projeções são computadas apenas em um subconjunto das restrições, com o intuito de torná-las mais eficientes. Por fim, tendo introduzido um método para minimização com restrições lineares, consideramos como fáceis as restrições lineares. Incorporamos BETRALIN e GENLIN ao arcabouço de Lagrangianos aumentados e verificamos experimentalmente a eficiência e eficácia destes métodos que trabalham explicitamente com restrições lineares e penalizam as demais. / Augmented Lagrangian methods are widely used to solve general nonlinear programming problems. In these methods, one can split the set of constraints in two groups: the set of easy and hard constraints. A constraint is called easy if there is an efficient method available to solve problems subject to that kind of constraint. Otherwise, the constraints are called hard. Augmented Lagrangian methods solve, at each iteration, problems subject to the set of easy constraints while penalizing the set of hard constraints. Linearly constrained problems appear frequently, sometimes as a result of a linear approximation of a problem, sometimes as an augmented Lagrangian subproblem. Therefore, an efficient method to solve linearly constrained problems is important for the implementation of efficient methods to solve nonlinear programming problems. In this thesis, we begin by considering box constraints as the set of easy constraints. We introduce a version of BETRA to solve large scale problems. BETRA is an active-set method that uses a trust-region strategy to work within the faces and spectral projected gradient to leave the faces. To solve each iteration\'s subproblem of ALGENCAN (an augmented Lagrangian method) we use either the dense or the sparse version of BETRA. We develope rules to decide which box-constrained inner solver should be used at each augmented Lagrangian iteration that considers the main characteristics of the problem to be solved. Then, we introduce two active-set methods to solve linearly constrained problems (BETRALIN and GENLIN). These methods use Partial Spectral Projected Gradient method to change the active set of constraints. The Partial Spectral Projected Gradient method was developed specially for this purpose. It computes projections onto a subset of the linear constraints, aiming to make the projections more efficient. At last, having introduced a linearly-constrained solver, we consider the set of linear constraints as the set of easy constraints. We use BETRALIN and GENLIN in the framework of augmented Lagrangian methods and verify, using numerical experiments, the efficiency and robustness of those methods that work with linear constraints and penalize the nonlinear constraints.

大部的類神經網路均為解決特定問題而設計，並非真正去模擬人腦的功能 ，在本論文中介紹一個模擬人類學習方式的類神經網路，稱為推理類神經 網路（The Reasoning Neural Network），其主要兩個組成為強記（ cram -ming）及推理（reasoning）部份，透過彈性的組合這兩個部份可 使類神經網路具有類似人類的學習程序。在本論文中介紹其中一個學習程 序並用四個實驗來評估推理類神經網路的績效，從實結果得知，推理類神 經網路能以合理的隱藏節點數（hidden nodes）達到學習的目標，並建立 一個網路內部表示方式（internal representation），及具有好的推理 能力（g eneralization ability）。 / Most of artification Neural Networks are designed to resolve spe -cific problems, rather than to model the brain. The Reasoning N -eural Network (RNN) that imitates the way of human learning is presented here. Two key components of RNN are the cramming and t -he reasoning. These components coulds be arranged flexibly to a -chieve the human-like learning procedure. One edition of the RNN used in experiments is introduces, and four different proble -ms are used to evaluate the RNN's performance. From simulation results, the RNN accomplishes the goal of learning with a reason -able number of hidden nodes, and evolves a good internal repres -entation and a generalization ability.

La construction d'un quotient, en topologie, est relativement simple; si $G$ est un groupe topologique agissant sur un espace topologique $X$, on peut considérer l'application naturelle de $X$ dans $X/G$, l'espace d'orbites muni de la topologie quotient. En géométrie algébrique, malheureusement, il n'est généralement pas possible de munir l'espace d'orbites d'une structure de variété. Dans le cas de l'action d'un groupe linéairement réductif $G$ sur une variété projective $X$, la théorie géométrique des invariants nous permet toutefois de construire un morphisme de variété d'un ouvert $U$ de $X$ vers une variété projective $X//U$, se rapprochant autant que possible d'une application quotient, au sens topologique du terme. Considérons par exemple $X\subseteq P^{n}$, une $k$-variété projective sur laquelle agit un groupe linéairement réductif $G$ et supposons que cette action soit induite par une action linéaire de $G$ sur $A^{n+1}$. Soit $\widehat{X}\subseteq A^{n+1}$, le cône affine au dessus de $\X$. Par un théorème de la théorie classique des invariants, il existe alors des invariants homogènes $f_{1},...,f_{r}\in C[\widehat{X}]^{G}$ tels que $$C[\widehat{X}]^{G}= C[f_{1},...,f_{r}].$$ On appellera le nilcone, que l'on notera $N$, la sous-variété de $\X$ définie par le locus des invariants $f_{1},...,f_{r}$. Soit $Proj(C[\widehat{X}]^{G})$, le spectre projectif de l'anneau des invariants. L'application rationnelle $$\pi:X\dashrightarrow Proj(C[f_{1},...,f_{r}])$$ induite par l'inclusion de $C[\widehat{X}]^{G}$ dans $C[\widehat{X}]$ est alors surjective, constante sur les orbites et sépare les orbites autant qu'il est possible de le faire; plus précisément, chaque fibre contient exactement une orbite fermée. Pour obtenir une application régulière satisfaisant les mêmes propriétés, il est nécessaire de jeter les points du nilcone. On obtient alors l'application quotient $$\pi:X\backslash N\rightarrow Proj(C[f_{1},...,f_{r}]).$$ Le critère de Hilbert-Mumford, dû à Hilbert et repris par Mumford près d'un demi-siècle plus tard, permet de décrire $N$ sans connaître les $f_{1},...,f_{r}$. Ce critère est d'autant plus utile que les générateurs de l'anneau des invariants ne sont connus que dans certains cas particuliers. Malgré les applications concrètes de ce théorème en géométrie algébrique classique, les démonstrations que l'on en trouve dans la littérature sont généralement données dans le cadre peu accessible des schémas. L'objectif de ce mémoire sera, entre autres, de donner une démonstration de ce critère en utilisant autant que possible les outils de la géométrie algébrique classique et de l'algèbre commutative. La version que nous démontrerons est un peu plus générale que la version originale de Hilbert \cite{hilbert} et se retrouve, par exemple, dans \cite{kempf}. Notre preuve est valide sur $C$ mais pourrait être généralisée à un corps $k$ de caractéristique nulle, pas nécessairement algébriquement clos. Dans la seconde partie de ce mémoire, nous étudierons la relation entre la construction précédente et celle obtenue en incluant les covariants en plus des invariants. Nous démontrerons dans ce cas un critère analogue au critère de Hilbert-Mumford (Théorème 6.3.2). C'est un théorème de Brion pour lequel nous donnerons une version un peu plus générale. Cette version, de même qu'une preuve simplifiée d'un théorème de Grosshans (Théorème 6.1.7), sont les éléments de ce mémoire que l'on ne retrouve pas dans la littérature. / The topological notion of a quotient is fairly simple. Given a topological group $G$ acting on a topological space $X$, one gets the natural application from $X$ to the quotient space $X/G$. In algebraic geometry, unfortunately, it is generally not possible to give the orbit space the structure of an algebraic variety. In the special case of a linearly reductive group acting on a projective variety $X$, the geometric invariant theory allows us to get a morphism of variety from an open $U$ of $X$ to a projective variety $X//G$, which is as close as possible to a quotient map, from a topological point of view. As an example, let $ X\subseteq P^{n}$ be a $k$-projective variety on which acts a linearly reductive group $G$. Suppose further that this action is induced by a linear action of $G$ on $A^{n+1}$ and let $\widehat{X}\subseteq A^{n +1}$ be the affine cone over $X$. By an important theorem of the classical invariants theory, there exist homogeneous invariants $f_{1},..., f_{r}\in C[\widehat{X}]^{G}$ such as $$\C[\widehat{X}]^{G}=\C[f_{1},...,f_{r}].$$ The locus in $X$ of $f_{1},...,f_{r}$ is called the nullcone, noted $N$. Let $Proj(C[\widehat{X}]^{G})$ be the projective spectrum of the invariants ring. The rational map $$\pi:X\dashrightarrow Proj(C[f_{1},...,f_{r}])$$ induced by the inclusion of $C[\widehat{X}]^{G}$ in $C[\widehat{X}] $ is then surjective, constant on the orbits and separates orbits as much as possible, that is, the fibres contains exactly one closed orbit. A regular map is obtained by removing the nullcone; we then get a regular map $$\pi:X \backslash N\rightarrow Proj(C[f_{1},...,f_{r}])$$ which still satisfy the preceding properties. The Hilbert-Mumford criterion, due to Hilbert and revisited by Mumford nearly half-century later, can be used to describe $N$ without knowing the generators of the invariants ring. Since those are rarely known, this criterion had proved to be quite useful. Despite the important applications of this criterion in classical algebraic geometry, the demonstrations found in the literature are usually given trough the difficult theory of schemes. The aim of this master thesis is therefore, among others, to provide a demonstration of this criterion using classical algebraic geometry and of commutative algebra. The version that we demonstrate is somewhat wider than the original version of Hilbert \cite{hilbert}; a schematic proof of this general version is given in \cite{kempf}. Finally, the proof given here is valid for $C$ but could be generalised to a field $k$ of characteristic zero, not necessarily algebraically closed. In the second part of this thesis, we study the relationship between the preceding constructions and those obtained by including covariants in addition to the invariants. We give a Hilbert-Mumford criterion for covariants (Theorem 6.3.2) which is a theorem from Brion for which we prove a slightly more general version. This theorem, together with a simplified proof of a theorem of Grosshans (Theorem 6.1.7), are the elements of this thesis that can't be found in the literature.

Métodos do tipo Lagrangiano Aumentado são muito utilizados para minimização de funções sujeitas a restrições gerais. Nestes métodos, podemos separar o conjunto de restrições em dois grupos: restrições fáceis e restrições difíceis. Dizemos que uma restrição é fácil se existe um algoritmo disponível e eficiente para resolver problemas restritos a este tipo de restrição. Caso contrário, dizemos que a restrição é difícil. Métodos do tipo Lagrangiano aumentado resolvem, a cada iteração, problemas sujeitos às restrições fáceis, penalizando as restrições difíceis. Problemas de minimização com restrições lineares aparecem com freqüência, muitas vezes como resultados da aproximação de problemas com restrições gerais. Este tipo de problema surge também como subproblema de métodos do tipo Lagrangiano aumentado. Assim, uma implementação eficiente para resolver problemas com restrições lineares é relevante para a implementação eficiente de métodos para resolução de problemas de programação não-linear. Neste trabalho, começamos considerando fáceis as restrições de caixa. Introduzimos BETRA-ESPARSO, uma versão de BETRA para problemas de grande porte. BETRA é um método de restrições ativas que utiliza regiões de confiança para minimização em cada face e gradiente espectral projetado para sair das faces. Utilizamos BETRA (denso ou esparso) na resolução dos subproblemas que surgem a cada iteração de ALGENCAN (um método de lagrangiano aumentado). Para decidir qual algoritmo utilizar para resolver cada subproblema, desenvolvemos regras que escolhem um método para resolver o subproblema de acordo com suas características. Em seguida, introduzimos dois algoritmos de restrições ativas desenvolvidos para resolver problemas com restrições lineares (BETRALIN e GENLIN). Estes algoritmos utilizam, a cada iteração, o método do Gradiente Espectral Projetado Parcial quando decidem mudar o conjunto de restrições ativas. O método do gradiente Espectral Projetado Parcial foi desenvolvido especialmente para este propósito. Neste método, as projeções são computadas apenas em um subconjunto das restrições, com o intuito de torná-las mais eficientes. Por fim, tendo introduzido um método para minimização com restrições lineares, consideramos como fáceis as restrições lineares. Incorporamos BETRALIN e GENLIN ao arcabouço de Lagrangianos aumentados e verificamos experimentalmente a eficiência e eficácia destes métodos que trabalham explicitamente com restrições lineares e penalizam as demais. / Augmented Lagrangian methods are widely used to solve general nonlinear programming problems. In these methods, one can split the set of constraints in two groups: the set of easy and hard constraints. A constraint is called easy if there is an efficient method available to solve problems subject to that kind of constraint. Otherwise, the constraints are called hard. Augmented Lagrangian methods solve, at each iteration, problems subject to the set of easy constraints while penalizing the set of hard constraints. Linearly constrained problems appear frequently, sometimes as a result of a linear approximation of a problem, sometimes as an augmented Lagrangian subproblem. Therefore, an efficient method to solve linearly constrained problems is important for the implementation of efficient methods to solve nonlinear programming problems. In this thesis, we begin by considering box constraints as the set of easy constraints. We introduce a version of BETRA to solve large scale problems. BETRA is an active-set method that uses a trust-region strategy to work within the faces and spectral projected gradient to leave the faces. To solve each iteration\'s subproblem of ALGENCAN (an augmented Lagrangian method) we use either the dense or the sparse version of BETRA. We develope rules to decide which box-constrained inner solver should be used at each augmented Lagrangian iteration that considers the main characteristics of the problem to be solved. Then, we introduce two active-set methods to solve linearly constrained problems (BETRALIN and GENLIN). These methods use Partial Spectral Projected Gradient method to change the active set of constraints. The Partial Spectral Projected Gradient method was developed specially for this purpose. It computes projections onto a subset of the linear constraints, aiming to make the projections more efficient. At last, having introduced a linearly-constrained solver, we consider the set of linear constraints as the set of easy constraints. We use BETRALIN and GENLIN in the framework of augmented Lagrangian methods and verify, using numerical experiments, the efficiency and robustness of those methods that work with linear constraints and penalize the nonlinear constraints.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-25T13:08:29Z No. of bitstreams: 1 arquivototal.pdf: 1324370 bytes, checksum: 6a689c99393e6b9a2a7f27c49ef07a8d (MD5) / Made available in DSpace on 2017-08-25T13:08:29Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1324370 bytes, checksum: 6a689c99393e6b9a2a7f27c49ef07a8d (MD5) Previous issue date: 2017-02-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In thisworkwestudytheexistenceofgroundstatesforthefollowingclassofcoupled systems involvingnonlinearSchrödingerequations 8<: 􀀀 u + V1(x)u = f1(x; u) + (x)v;x 2 RN; 􀀀 v + V2(x)v = f2(x; v) + (x)u; x 2 RN; where thepotentials V1 : RN ! R, V2 : RN ! R are nonnegativeandrelatedwith the couplingterm : RN ! R by j (x)j < pV1(x)V2(x), forsome 0 < < 1. In the case N = 2, thenonlinearities f1 e f2 havecriticalexponentialgrowthinthesense of Trudinger-Moserinequality.Inthecase N 3, thenonlinearitiesarepolynomials with subcriticalandcriticalexponentintheSobolevsense.Westudyalsothefollowing class ofnonlocalcoupledsystems 8<: (􀀀 )1=2u + V1(x)u = f1(u) + (x)v;x 2 R; (􀀀 )1=2v + V2(x)v = f2(v) + (x)u; x 2 R; where (􀀀 )1=2 denotes thesquarerootoftheLaplacianoperatorandthenonlinearities havecriticalexponentialgrowth.Ourapproachisvariationalandbasedon minimization techniqueovertheNeharimanifold / Neste trabalhoestudamosaexistênciadegroundstatesparaaseguinteclassede sistemas acopladosenvolvendoequaçõesdeSchrödingernão-lineares 8<: 􀀀 u + V1(x)u = f1(x; u) + (x)v;x 2 RN; 􀀀 v + V2(x)v = f2(x; v) + (x)u; x 2 RN; onde ospotenciais V1 : RN ! R, V2 : RN ! R são não-negativoseestãorelacionados com otermodeacomplamento : RN ! R por j (x)j < pV1(x)V2(x), paraalgum 0 < < 1. Nocaso N = 2, asnão-linearidades f1 e f2 possuemcrescimentocrítico exponencialnosentidodadesigualdadedeTrudinger-Moser.Nocaso N 3, asnão- linearidades sãopolinômioscomexpoentesubcríticoecríticonosentidodeSobolev. Estudamos aindaaseguinteclassedesistemasacopladosnão-locais 8<: (􀀀 )1=2u + V1(x)u = f1(u) + (x)v;x 2 R; (􀀀 )1=2v + V2(x)v = f2(v) + (x)u; x 2 R; onde (􀀀 )1=2 denota ooperadorraízquadradadolaplacianoeasnão-linearidades possuemcrescimentocríticoexponencial.Nossaabordagemévariacionalebaseadana técnica deminimizaçãosobreavariedadedeNehari.

We derive basic properties of groups which can be approximated with matrices. These include closure of classes of such groups under group theoretic constructions including direct and inverse limits and free products. We show that metric ultraproducts of projective linear groups over fields of different characteristics are not isomorphic. We further prove that the lattice of normal subgroups in ultraproducts of compact simple groups is distributive. It is linearly ordered in the case of finite simple groups or Lie groups of bounded rank.