Στη παρούσα διπλωματική εργασία ασχολούμαστε με την αποδοτική επίλυση ταινιακών και γενικών, αραιών γραμμικών συστημάτων σε παράλληλες αρχιτεκτονικές μέσω της οικογένειας αλγορίθμων Spike. Ζητούμενο είναι η βελτίωση (μείωση) του χρόνου επίλυσης μέσω τεχνικών επίλυσης γραμμικών συστημάτων με πολλά δεξιά μέλη. Πιο συγκεκριμένα, επικεντρωνόμαστε στην επίλυση της εξίσωσης μητρώου $AX=F$ (1) όπου $A\in \mathbb{R}^{n\times n}$ είναι το μητρώο συντελεστών και το οποίο είναι αραιό ή/και ταινιακό, $F\in \mathbb{R}^{n\times s}$ είναι ένα μητρώο με $s$ στήλες το οποίο ονομάζεται μητρώο δεξιών μελών και $X\in \mathbb{R}^{n\times s}$ είναι η λύση του συστήματος. Μια σημαντική μέθοδος για την παράλληλη επίλυση της παραπάνω εξίσωσης, είναι η μέθοδος Spike και οι παραλλαγές της. Η μέθοδος Spike βασίζεται στη τεχνική διαίρει και βασίλευε και αποτελείται από δυο φάσεις: α) επίλυση ανεξάρτητων υπο-προβλημάτων τοπικά σε κάθε επεξεργαστή, και β) επίλυση ενός πολύ μικρότερου προβλήματος το οποίο απαιτεί επικοινωνία μεταξύ των επεξεργαστών. Οι δύο φάσεις συνδυάζονται ώστε να παραχθεί η τελική λύση $X$. Η συνεισφορά της διπλωματικής εργασίας έγκειται στην επιτάχυνση της οικογένειας αλγορίθμων Spike για την επίλυση της εξίσωσης (1) μέσω της μελέτης, το σχεδιασμό και την υλοποίηση νέων, περισσότερο αποδοτικών αλγοριθμικών σχημάτων τα οποία βασίζονται σε τεχνικές επίλυσης γραμμικών συστημάτων με πολλά δεξιά μέλη. Αυτά τα νέα αλγοριθμικά σχήματα έχουν ως στόχο τη βελτίωση του χρόνου επίλυσης των γραμμικών συστημάτων καθώς και άλλα οφέλη όπως η αποδοτικότερη χρήση μνήμης. / In this thesis we focus on the efficient solution of general banded and general sparse linear systems on parallel architectures by exploiting the Spike family of algorithms. The equation of interest can be written in matrix form as $ AX = F $ (1) where $ A \ in \ mathbb {R} ^ {n \ times n} $ is the coefficient matrix, which is also sparse and / or banded, $ F \ in \ mathbb {R} ^ {n \ times s} $ is a matrix with $ s $ columns called matrix of the right hand sides and $ X \ in \ mathbb {R} ^ {n \ times s} $ is the solution of the system. An important method for the parallel solution of the above equation, is the Spike method and its variants. The Spike method is based on the divide and conquer technique and consists of two phases: a) solution of local, independent sub-problems in each processor, and b) solution of a much smaller problem which requires communication among the processors. The two phases are combined to produce the final solution $ X $. The contribution of this thesis is the acceleration of the Spike method for the solution of the matrix equation in (1) by studying, designing and implementing new, more efficient algorithmic schemes which are based on techniques used for the effective solution of linear systems with multiple right hand sides. These new algorithmic schemes were designed to improve the solving time of the linear systems as well as to provide other benefits such as more efficient use of memory.

Nous nous intéressons dans cette thèse, à l’étude des méthodes itératives pour la résolutiond’équations matricielles de grande taille : Lyapunov, Sylvester, Riccati et Riccatinon symétrique.L’objectif est de chercher des méthodes itératives plus efficaces et plus rapides pour résoudreles équations matricielles de grande taille. Nous proposons des méthodes itérativesde type projection sur des sous espaces de Krylov par blocs Km(A, V ) = Image{V,AV, . . . ,Am−1V }, ou des sous espaces de Krylov étendus par blocs Kem(A, V ) = Image{V,A−1V,AV,A−2V,A2V, · · · ,Am−1V,A−m+1V } . Ces méthodes sont généralement plus efficaces et rapides pour les problèmes de grande dimension. Nous avons traité d'abord la résolution numérique des équations matricielles linéaires : Lyapunov, Sylvester, Stein. Nous avons proposé une nouvelle méthode itérative basée sur la minimisation de résidu MR et la projection sur des sous espaces de Krylov étendus par blocs Kem(A, V ). L'algorithme d'Arnoldi étendu par blocs permet de donner un problème de minimisation projeté de petite taille. Le problème de minimisation de taille réduit est résolu par différentes méthodes directes ou itératives. Nous avons présenté ainsi la méthode de minimisation de résidu basée sur l'approche global à la place de l'approche bloc. Nous projetons sur des sous espaces de Krylov étendus Global Kem(A, V ) = sev{V,A−1V,AV,A−2V,A2V, · · · ,Am−1V,A−m+1V }. Nous nous sommes intéressés en deuxième lieu à des équations matricielles non linéaires, et tout particulièrement l'équation matricielle de Riccati dans le cas continu et dans le cas non symétrique appliquée dans les problèmes de transport. Nous avons utilisé la méthode de Newtown et l'algorithme MINRES pour résoudre le problème de minimisation projeté. Enfin, nous avons proposé deux nouvelles méthodes itératives pour résoudre les équations de Riccati non symétriques de grande taille : la première basée sur l'algorithme d'Arnoldi étendu par bloc et la condition d'orthogonalité de Galerkin, la deuxième est de type Newton-Krylov, basée sur la méthode de Newton et la résolution d'une équation de Sylvester de grande taille par une méthode de type Krylov par blocs. Pour toutes ces méthodes, les approximations sont données sous la forme factorisée, ce qui nous permet d'économiser la place mémoire en programmation. Nous avons donné des exemples numériques qui montrent bien l'efficacité des méthodes proposées dans le cas de grandes tailles. / In this thesis, we focus in the studying of some iterative methods for solving large matrix equations such as Lyapunov, Sylvester, Riccati and nonsymmetric algebraic Riccati equation. We look for the most efficient and faster iterative methods for solving large matrix equations. We propose iterative methods such as projection on block Krylov subspaces Km(A, V ) = Range{V,AV, . . . ,Am−1V }, or block extended Krylov subspaces Kem(A, V ) = Range{V,A−1V,AV,A−2V,A2V, · · · ,Am−1V,A−m+1V }. These methods are generally most efficient and faster for large problems. We first treat the numerical solution of the following linear matrix equations : Lyapunov, Sylvester and Stein matrix equations. We have proposed a new iterative method based on Minimal Residual MR and projection on block extended Krylov subspaces Kem(A, V ). The extended block Arnoldi algorithm gives a projected minimization problem of small size. The reduced size of the minimization problem is solved by direct or iterative methods. We also introduced the Minimal Residual method based on the global approach instead of the block approach. We projected on the global extended Krylov subspace Kem(A, V ) = Span{V,A−1V,AV,A−2V,A2V, · · · ,Am−1V,A−m+1V }. Secondly, we focus on nonlinear matrix equations, especially the matrix Riccati equation in the continuous case and the nonsymmetric case applied in transportation problems. We used the Newton method and MINRES algorithm to solve the projected minimization problem. Finally, we proposed two new iterative methods for solving large nonsymmetric Riccati equation : the first based on the algorithm of extended block Arnoldi and Galerkin condition, the second type is Newton-Krylov, based on Newton’s method and the resolution of the large matrix Sylvester equation by using block Krylov method. For all these methods, approximations are given in low rank form, wich allow us to save memory space. We have given numerical examples that show the effectiveness of the methods proposed in the case of large sizes.

Nous nous intéressons, dans cette thèse, à l'étude des méthodes d'extrapolation polynômiales et à l'application de ces méthodes dans l'accélération de méthodes de points fixes pour des problèmes donnés. L'avantage de ces méthodes d'extrapolation est qu'elles utilisent uniquement une suite de vecteurs qui n'est pas forcément convergente, ou qui converge très lentement pour créer une nouvelle suite pouvant admettreune convergence quadratique. Le développement de méthodes cycliques permet, deplus, de limiter le coût de calculs et de stockage. Nous appliquons ces méthodes à la résolution des équations de Navier-Stokes stationnaires et incompressibles, à la résolution de la formulation Kohn-Sham de l'équation de Schrödinger et à la résolution d'équations elliptiques utilisant des méthodes multigrilles. Dans tous les cas, l'efficacité des méthodes d'extrapolation a été montrée.Nous montrons que lorsqu'elles sont appliquées à la résolution de systèmes linéaires, les méthodes d'extrapolation sont comparables aux méthodes de sous espaces de Krylov. En particulier, nous montrons l'équivalence entre la méthode MMPE et CMRH. Nous nous intéressons enfin, à la parallélisation de la méthode CMRH sur des processeurs à mémoire distribuée et à la recherche de préconditionneurs efficaces pour cette même méthode.

Depuis quelques années, l'extension de l'utilisation de l'informatique dans tous les domaines de recherche scientifique et technique se traduit par un besoin croissant de puissance de calcul. Il est donc vital d'employer les microprocesseurs en parallèle. Le problème principal que nous cherchons à résoudre dans cette thèse est le calcul d'une forme canonique de très grandes matrices creuses à coefficients entiers, la forme normale de Smith. Par "très grandes", nous entendons un million d'inconnues et un million d'équations, c'est-à-dire mille milliards de variables. De tels systèmes sont même, en général, impossibles à stocker actuellement. Cependant, nous nous intéressons à des systèmes dans lesquels beaucoup de ces variables sont identiques et valent zéro; on parle dans ce cas de système creux. Enfin, nous voulons résoudre ces systèmes de manière exacte, c'est-à-dire que nous travaillons avec des nombres entiers ou dans une structure algébrique plus petite et autorisant toutes les opérations classiques, un corps fini. La reconstruction de la solution entière à partir des solutions plus petites est ensuite relativement aisée.

This dissertation concerns the model reduction of linear periodic descriptor systems both in continuous and discrete-time case. In this dissertation, mainly the projection based approaches are considered for model order reduction of linear periodic time varying descriptor systems. Krylov based projection method is used for large continuous-time periodic descriptor systems and balancing based projection technique is applied to large sparse discrete-time periodic descriptor systems to generate the reduce systems. For very large dimensional state space systems, both the techniques produce large dimensional solutions. Hence, a recycling technique is used in Krylov based projection methods which helps to compute low rank solutions of the state space systems and also accelerate the computational convergence. The outline of the proposed model order reduction procedure is given with more details. The accuracy and suitability of the proposed method is demonstrated through different examples of different orders. Model reduction techniques based on balance truncation require to solve matrix equations. For periodic time-varying descriptor systems, these matrix equations are projected generalized periodic Lyapunov equations and the solutions are also time-varying. The cyclic lifted representation of the periodic time-varying descriptor systems is considered in this dissertation and the resulting lifted projected Lyapunov equations are solved to achieve the periodic reachability and observability Gramians of the original periodic systems. The main advantage of this solution technique is that the cyclic structures of projected Lyapunov equations can handle the time-varying dimensions as well as the singularity of the period matrix pairs very easily. One can also exploit the theory of time-invariant systems for the control of periodic ones, provided that the results achieved can be easily re-interpreted in the periodic framework. Since the dimension of cyclic lifted system becomes very high for large dimensional periodic systems, one needs to solve the very large scale periodic Lyapunov equations which also generate very large dimensional solutions. Hence iterative techniques, which are the generalization and modification of alternating directions implicit (ADI) method and generalized Smith method, are implemented to obtain low rank Cholesky factors of the solutions of the periodic Lyapunov equations. Also the application of the solvers in balancing-based model reduction of discrete-time periodic descriptor systems is discussed. Numerical results are given to illustrate the effciency and accuracy of the proposed methods.

Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this thesis we focus on the approximation of A..=2b, where A 2 Rnn is a large, sparse symmetric positive definite matrix and b 2 Rn is a vector. In particular, we will focus on matrix function techniques for sampling from Gaussian Markov random fields in applied statistics and the solution of fractional-in-space partial differential equations. Gaussian Markov random fields (GMRFs) are multivariate normal random variables characterised by a sparse precision (inverse covariance) matrix. GMRFs are popular models in computational spatial statistics as the sparse structure can be exploited, typically through the use of the sparse Cholesky decomposition, to construct fast sampling methods. It is well known, however, that for sufficiently large problems, iterative methods for solving linear systems outperform direct methods. Fractional-in-space partial differential equations arise in models of processes undergoing anomalous diffusion. Unfortunately, as the fractional Laplacian is a non-local operator, numerical methods based on the direct discretisation of these equations typically requires the solution of dense linear systems, which is impractical for fine discretisations. In this thesis, novel applications of Krylov subspace approximations to matrix functions for both of these problems are investigated. Matrix functions arise when sampling from a GMRF by noting that the Cholesky decomposition A = LLT is, essentially, a `square root' of the precision matrix A. Therefore, we can replace the usual sampling method, which forms x = L..T z, with x = A..1=2z, where z is a vector of independent and identically distributed standard normal random variables. Similarly, the matrix transfer technique can be used to build solutions to the fractional Poisson equation of the form n = A..=2b, where A is the finite difference approximation to the Laplacian. Hence both applications require the approximation of f(A)b, where f(t) = t..=2 and A is sparse. In this thesis we will compare the Lanczos approximation, the shift-and-invert Lanczos approximation, the extended Krylov subspace method, rational approximations and the restarted Lanczos approximation for approximating matrix functions of this form. A number of new and novel results are presented in this thesis. Firstly, we prove the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and we give conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian. We then investigate a number of methods for approximating matrix functions of the form A..=2b and investigate stopping criteria for these methods. In particular, we derive a new method for restarting the Lanczos approximation to f(A)b. We then apply these techniques to the problem of sampling from a GMRF and construct a full suite of methods for sampling conditioned on linear constraints and approximating the likelihood. Finally, we consider the problem of sampling from a generalised Matern random field, which combines our techniques for solving fractional-in-space partial differential equations with our method for sampling from GMRFs.

The solution of large and sparse eigenvalue problems arising from the discretization of the diffusion equation is considered. The multigroup<p>diffusion equation is discretized by means of the Nodal expansion Method (NEM) [9, 10]. A new formulation of the higher order NEM variants revealing the true nature of the problem, that is, a generalized eigenvalue problem, is proposed. These generalized eigenvalue problems are solved using the Jacobi-Davidson (JD) method<p>[26]. The most expensive part of the method consists of solving a linear system referred to as correction equation. It is solved using Krylov subspace methods in combination with aggregation-based Algebraic Multigrid (AMG) techniques. In that context, a particular<p>aggregation technique used in combination with classical smoothers, referred to as oblique geometric coarsening, has been derived. Its particularity is that it aggregates unknowns that<p>are not coupled, which has never been done to our<p>knowledge. A modular code, combining JD with an AMG preconditioner, has been developed. The code comes with many options, that have been tested. In particular, the instability of the Rayleigh-Ritz [33] acceleration procedure in the non-symmetric case has been underlined. Our code has also been compared to an industrial code extracted from ARTEMIS. / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished

Nous nous intéressons dans cette thèse, à l'étude de certaines méthodes numériques de type krylov dans le cas symplectique, en utilisant la technique de blocs. Ces méthodes, contrairement aux méthodes classiques, permettent à la matrice réduite de conserver la structure Hamiltonienne ou anti-Hamiltonienne ou encore symplectique d'une matrice donnée. Parmi ces méthodes, nous nous sommes intéressés à la méthodes d'Arnoldi symplectique par bloc que nous appelons aussi bloc J-Arnoldi. Notre but essentiel est d’étudier cette méthode de façon théorique et numérique, sur la nouvelle structure du K-module libre ℝ²nx²s avec K = ℝ²sx²s où s ≪ n désigne la taille des blocs utilisés. Un deuxième objectif est de chercher une approximation de l'epérateur exp(A)V, nous étudions en particulier le cas où A est une matrice réelle Hamiltonnienne et anti-symétrique de taille 2n x 2n et V est une matrice rectangulaire ortho-symplectique de taille 2n x 2s sur le sous-espace de Krylov par blocs Km(A,V) = blockspan {V,AV,...,Am-1V}, en conservant la structure de la matrice V. Cette approximation permet de résoudre plusieurs problèmes issus des équations différentielles dépendants d'un paramètre (EDP) et des systèmes d'équations différentielles ordinaires (EDO). Nous présentons également une méthode de Lanczos symplectique par bloc, que nous nommons bloc J-Lanczos. Cette méthode permet de réduire une matrice structurée sous la forme J-tridiagonale par bloc. Nous proposons des algorithmes basés sur deux types de normalisation : la factorisation S R et la factorisation Rj R. Dans une dernière partie, nous proposons un algorithme qui généralise la méthode de Greville afin de déterminer la pseudo inverse de Moore-Penros bloc de lignes par bloc de lignes d'une matrice rectangulaire de manière itérative. Nous proposons un algorithme qui utilise la technique de bloc. Pour toutes ces méthodes, nous proposons des exemples numériques qui montrent l'efficacité de nos approches. / We study, in this thesis, some numerical block Krylov subspace methods. These methods preserve geometric properties of the reduced matrix (Hamiltonian or skew-Hamiltonian or symplectic). Among these methods, we interest on block symplectic Arnoldi, namely block J-Arnoldi algorithm. Our main goal is to study this method, theoretically and numerically, on using ℝ²nx²s as free module on (ℝ²sx²s, +, x) with s ≪ n the size of block. A second aim is to study the approximation of exp (A)V, where A is a real Hamiltonian and skew-symmetric matrix of size 2n x 2n and V a rectangular matrix of size 2n x 2s on block Krylov subspace Km (A, V) = blockspan {V, AV,...Am-1V}, that preserve the structure of the initial matrix. this approximation is required in many applications. For example, this approximation is important for solving systems of ordinary differential equations (ODEs) or time-dependant partial differential equations (PDEs). We also present a block symplectic structure preserving Lanczos method, namely block J-Lanczos algorithm. Our approach is based on a block J-tridiagonalization procedure of a structured matrix. We propose algorithms based on two normalization methods : the SR factorization and the Rj R factorization. In the last part, we proposea generalized algorithm of Greville method for iteratively computing the Moore-Penrose inverse of a rectangular real matrix. our purpose is to give a block version of Greville's method. All methods are completed by many numerical examples.

In this thesis we present research on mathematical properties of methods for solv- ing symmetric systems of linear equations that arise in various optimization problem formulations and in methods for solving such problems. In the first and third paper (Paper A and Paper C), we consider the connection be- tween the method of conjugate gradients and quasi-Newton methods on strictly convex quadratic optimization problems or equivalently on a symmetric system of linear equa- tions with a positive definite matrix. We state conditions on the quasi-Newton matrix and the update matrix such that the search directions generated by the corresponding quasi-Newton method and the method of conjugate gradients respectively are parallel. In paper A, we derive such conditions on the update matrix based on a sufficient condition to obtain mutually conjugate search directions. These conditions are shown to be equivalent to the one-parameter Broyden family. Further, we derive a one-to-one correspondence between the Broyden parameter and the scaling between the search directions from the method of conjugate gradients and a quasi-Newton method em- ploying some well-defined update scheme in the one-parameter Broyden family. In paper C, we give necessary and sufficient conditions on the quasi-Newton ma- trix and on the update matrix such that equivalence with the method of conjugate gra- dients hold for the corresponding quasi-Newton method. We show that the set of quasi- Newton schemes admitted by these necessary and sufficient conditions is strictly larger than the one-parameter Broyden family. In addition, we show that this set of quasi- Newton schemes includes an infinite number of symmetric rank-one update schemes. In the second paper (Paper B), we utilize an unnormalized Krylov subspace frame- work for solving symmetric systems of linear equations. These systems may be incom- patible and the matrix may be indefinite/singular. Such systems of symmetric linear equations arise in constrained optimization. In the case of an incompatible symmetric system of linear equations we give a certificate of incompatibility based on a projection on the null space of the symmetric matrix and characterize a minimum-residual solu- tion. Further we derive a minimum-residual method, give explicit recursions for the minimum-residual iterates and characterize a minimum-residual solution of minimum Euclidean norm. / I denna avhandling betraktar vi matematiska egenskaper hos metoder för att lösa symmetriska linjära ekvationssystem som uppkommer i formuleringar och metoder för en mängd olika optimeringsproblem. I första och tredje artikeln (Paper A och Paper C), undersöks kopplingen mellan konjugerade gradientmetoden och kvasi-Newtonmetoder när dessa appliceras på strikt konvexa kvadratiska optimeringsproblem utan bivillkor eller ekvivalent på ett symmet- risk linjärt ekvationssystem med en positivt definit symmetrisk matris. Vi ställer upp villkor på kvasi-Newtonmatrisen och uppdateringsmatrisen så att sökriktningen som fås från motsvarande kvasi-Newtonmetod blir parallell med den sökriktning som fås från konjugerade gradientmetoden. I den första artikeln (Paper A), härleds villkor på uppdateringsmatrisen baserade på ett tillräckligt villkor för att få ömsesidigt konjugerade sökriktningar. Dessa villkor på kvasi-Newtonmetoden visas vara ekvivalenta med att uppdateringsstrategin tillhör Broydens enparameterfamilj. Vi tar också fram en ett-till-ett överensstämmelse mellan Broydenparametern och skalningen mellan sökriktningarna från konjugerade gradient- metoden och en kvasi-Newtonmetod som använder någon väldefinierad uppdaterings- strategi från Broydens enparameterfamilj. I den tredje artikeln (Paper C), ger vi tillräckliga och nödvändiga villkor på en kvasi-Newtonmetod så att nämnda ekvivalens med konjugerade gradientmetoden er- hålls. Mängden kvasi-Newtonstrategier som uppfyller dessa villkor är strikt större än Broydens enparameterfamilj. Vi visar också att denna mängd kvasi-Newtonstrategier innehåller ett oändligt antal uppdateringsstrategier där uppdateringsmatrisen är en sym- metrisk matris av rang ett. I den andra artikeln (Paper B), används ett ramverk för icke-normaliserade Krylov- underrumsmetoder för att lösa symmetriska linjära ekvationssystem. Dessa ekvations- system kan sakna lösning och matrisen kan vara indefinit/singulär. Denna typ av sym- metriska linjära ekvationssystem uppkommer i en mängd formuleringar och metoder för optimeringsproblem med bivillkor. I fallet då det symmetriska linjära ekvations- systemet saknar lösning ger vi ett certifikat för detta baserat på en projektion på noll- rummet för den symmetriska matrisen och karaktäriserar en minimum-residuallösning. Vi härleder även en minimum-residualmetod i detta ramverk samt ger explicita rekur- sionsformler för denna metod. I fallet då det symmetriska linjära ekvationssystemet saknar lösning så karaktäriserar vi en minimum-residuallösning av minsta euklidiska norm. / <p>QC 20150519</p>

A precisão e acurácia dos métodos mais utilizados atualmente de processamento de dados de espectroscopia por Ressonância Magnética (MRS), baseados na Transformada de Fourier (FT), requerem supressão apropriada (o que está longe de ser trivial) e aquisições longas para a obtenção de alta resolução espectral. Além disso, a FT tem dificuldades quando faltam dados no domínio de tempo, como, por exemplo, pela redução do tempo de aquisição, e consequente número de pontos adquiridos. Isto pode ocorrer, também, por artefatos na aquisição ou, ainda, seja pela exclusão intencional dos primeiros pontos do sinal para a eliminação de ressonâncias largas que estão distorcendo a linha de base no domínio da frequência. Neste estudo, propomos a utilização do Método de Diagonalização na Base de Krylov (KBDM) como uma alternativa a FT para algumas de suas limitações. O método ajusta sinais de experimentos de Free Induction Decay (FID) por uma soma de funções harmônicas complexas, amortecidas exponencialmente, permitindo uma fácil manipulação dos seus parâmetros de caracterização. O KBDM é numericamente mais efetivo para análise de sinais truncados e tem diversos recursos que possibilitam remover picos de forma mais eficiente, como por exemplo, o pico residual da água. Além disso, foi introduzida a possibilidade de quantificação de dados de MRS com o método. Para avaliar a sensibilidade, eficiência e reprodutibilidade do método para quantificar e analisar sinais truncados, foi proposto fazer simulações de espectros clínicos e experimentos em phantoms que representassem o ambiente metabólico do cérebro, para MRS de próton de diferentes níveis de ruídos e para pequenas variações do N-acetil aspartato (NAA). Com estes estudos pôde se comprovar a viabilidade do método para processar dados de MRS e verificar seu potencial na complementação das técnicas atualmente empregadas, especialmente quando uma resolução espectral e temporal maior que o limite imposto pela Relação de Incerteza do formalismo de Fourier é necessária. Além disso, uma desejável facilidade de manipulação de picos específicos (por exemplo, exclusão e quantificação) é proporcionada pelo método. Como perspectivas animadoras deste trabalho esperamos a introdução do KBDM como uma técnica eficiente e coadjuvante ao Imageamento de Ressonância Magnética funcional (fMRI), auxiliando estudos de funções cerebrais, em sequências de MRS para identificar uma rápida variação das linhas associadas as atividades metabólicas dos cérebros. / The precision and accuracy of the most widely used methods to perform Magnetic Resonance Spectroscopy (MRS) data processing based on the Fourier Transform (FT), require appropriate suppression (which is far from trivial) and long acquisitions to obtain high spectral resolution. Furthermore, FT poses difficulty when there are missing data in the time domain. This occurs because of reduction of the acquisition time and consequently also in the number of acquired points, or because of artifacts during acquisition, or even intentional exclusion of the first signal points for the elimination of broad resonances that are producing the distorted baseline in the frequency domain. In this study, we propose the use of the Krylov Basis Diagonalization Method (KBDM) formalism as an alternative to some of FT limitations. The method adjusts signals of Free Induction Decay (FID) experiments with a sum of complex harmonic functions, exponentially damped, allowing easy manipulation of its characterization parameters. The KBDM is numerically more effective for truncated signal analysis and has several features that make it possible to remove peaks more efficiently, such as the residual water peak. Moreover, we introduced the possibility of quantification of MRS data with the described method. To evaluate the sensitivity, efficiency and reproducibility of the method for quantifying and analyzing truncated signals, and through the clinical spectra simulations and experiments in phantoms that would represent the brain metabolic environment, we proposed to perform proton MRS at different noise levels and with small variations of N- acetyl aspartate (NAA) metabolite. These studies allowed to prove the feasibility of the method to process MRS data and verified its potential in complementing techniques currently employed, especially when a greater temporal and spectral resolution is required, more than the limit imposed by the Uncertainty Relation of FT formalism. Furthermore, it is also a desirable effortless tool of handling specific peaks (e.g., exclusion and quantification). Exciting prospects from this work include the introduction of KBDM as an efficient and adjuvant technique to functional Magnetic Resonance Imaging (fMRI), for studying the brain functions, in MRS sequence to identify rapid variation in spectroscopic lines associated to metabolic activities in the brain.