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Thesis (Ph.D.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / This paper discusses the following problem of approximation theory: a continuous function f(x) is to approximated over an interval alpha </= x </= B by N not necessarily connected polynomials of a given degree n, in such a way that the maximum error magnitude is a minimum. Each polynomial is associated with one subinterval [uj, uj+1]. If the end points Ui were specified in advance, the problem would reduce to N independent problems of the same type, namely, the fitting of a single polynomial in the Chebyshev sense. Here, however, the end points are taken as unknowns and the principal problem is to determine them. The paper presents a proof of the existence of the best approximation and examples showing the solution, in general, is not unique. However, in the special case of approximation of convex functions by line segments, the solution is shown to be unique. Further in this case a simple characterization of the solution is obtained and it is shown that the problem may be reduced analytically to a stage where in order to determine Ui computationally, it is only necessary to solve a system of equations rather than minimize a function. Results obtained by a dynamic programming method using a digital computer (IBM 7090) are used for illustration. / 2999-01-01

Thesis (M.A.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / This paper deals with methods of Chebyshev approximation. In particular polynomial approximation of continuous functions on a finite interval are discussed. Chapter I deals with the existence and uniqueness of Chebyshev or C-polynomials. In addition, some properties of the extremal points of the error function are derived, where the error fUnction E(x) = f(x) - p(x), p(x) being the C-polynomial. Chapter II discusses a method for finding the C-polynomial of degree n--the exchange method. After choosing a set of n+2 distinct abscissas, or a reference set, the so-called levelled reference polynomial is computed by the method of divided differences or by using the approximation errors of this polynomial. A point xj of maximal error is obtained and introduced into a new reference. A new levelled reference polynomial is then computed. This process continues until a reference is gotten, whose reference deviation equals the maximal approximating error of the levelled reference polynomial. The reference deviation is the common absolute value of the levelled reference polynomial at each of the reference points. The levelled reference polynomial for this reference is then shown to be the desired C-polynomial. Chapter III deals with phase methods for constructing the a-polynomial. It is shown that under suitable restrictions, if a Pn, A and €(phi) can be found such that the basic relation f(cos phi) = Pn(cos phi) + A cos[(n+1)phi + E(phi)] is satisfied on the approximation interval, then Pn is the a-polynomial. Two methods for finding the amplitude A and the phase function €(phi) are discussed. The complex method assumes f to be analytic on a domain and uses Cauchy's integral formula to obtain new values of €(phi), starting with a set of initial values. These values in turn generate new values of Pn and A. The values of Pn as well as values of A and €(phi) at certain points are gotten through convergence of this iterative scheme. Then an interpolation formula is used to obtain Pn from its values at these points. The second method attempts to find A, €(phi) and Pn so as to satisfy the basic relation only on a discrete set of points. First, assuming €(phi) so small that cos €(phi) may be replaced by 1, an expression is obtained for Pn(cos phi). In the general case, a system of phase equations is given, from which €(phi), A and hence Pn may be obtained. Although these results are valid only on a discrete set of points in the approximation interval, the polynomial derived in this way represents a good approximation to f(x). / 2999-01-01

Thesis (MSc (Physics))--University of Stellenbosch, 2006. / The aim of this thesis is the semi-classical implementation of Wegner’s flow equations and comparison with the well-established Wentzel-Kramers-Brillouin method. We do this by converting operators, in particular the Hamiltonian, into scalar functions, while an isomorphism with the operator product is maintained by the introduction of the Moyal product. A flow equation in terms of these scalar functions is set up and then approximated by expanding it to first order in ~. We apply this method to two potentials, namely the quartic anharmonic oscillator and the symmetric double-well potential. Results obtained via the flow equations are then compared with those obtained from the WKB method.

In many digital signal processing applications, we often need some special function units which can compute complicated arithmetic functions such as reciprocal and logarithm. Conventionally, table-based arithmetic design strategy uses lookup tables to implement these kinds of function units. However, the table size will increase exponentially with respect to the required precision. In this thesis, we propose two methods to reduce the table size: bottom-up non-uniform segmentation and the approach which merges uniform piecewise interpolation and Newton-Raphson method. Experimental results show that we obtain significant table sizes reduction in most cases.

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Dans cette thèse, nous étudions les méthodes d'approximation et d'apprentissage qui fournissent des représentations parcimonieuses. Ces méthodes permettent d'analyser des bases de données très redondantes à l'aide de dictionnaires d'atomes appris. Etant adaptés aux données étudiées, ils sont plus performants en qualité de représentation que les dictionnaires classiques dont les atomes sont définis analytiquement. Nous considérons plus particulièrement des signaux multivariés résultant de l'acquisition simultanée de plusieurs grandeurs, comme les signaux EEG ou les signaux de mouvements 2D et 3D. Nous étendons les méthodes de représentations parcimonieuses au modèle multivarié, pour prendre en compte les interactions entre les différentes composantes acquises simultanément. Ce modèle est plus flexible que l'habituel modèle multicanal qui impose une hypothèse de rang 1. Nous étudions des modèles de représentations invariantes : invariance par translation temporelle, invariance par rotation, etc. En ajoutant des degrés de liberté supplémentaires, chaque noyau est potentiellement démultiplié en une famille d'atomes, translatés à tous les échantillons, tournés dans toutes les orientations, etc. Ainsi, un dictionnaire de noyaux invariants génère un dictionnaire d'atomes très redondant, et donc idéal pour représenter les données étudiées redondantes. Toutes ces invariances nécessitent la mise en place de méthodes adaptées à ces modèles. L'invariance par translation temporelle est une propriété incontournable pour l'étude de signaux temporels ayant une variabilité temporelle naturelle. Dans le cas de l'invariance par rotation 2D et 3D, nous constatons l'efficacité de l'approche non-orientée sur celle orientée, même dans le cas où les données ne sont pas tournées. En effet, le modèle non-orienté permet de détecter les invariants des données et assure la robustesse à la rotation quand les données tournent. Nous constatons aussi la reproductibilité des décompositions parcimonieuses sur un dictionnaire appris. Cette propriété générative s'explique par le fait que l'apprentissage de dictionnaire est une généralisation des K-means. D'autre part, nos représentations possèdent de nombreuses invariances, ce qui est idéal pour faire de la classification. Nous étudions donc comment effectuer une classification adaptée au modèle d'invariance par translation, en utilisant des fonctions de groupement consistantes par translation. / In this thesis, we study approximation and learning methods which provide sparse representations. These methods allow to analyze very redundant data-bases thanks to learned atoms dictionaries. Being adapted to studied data, they are more efficient in representation quality than classical dictionaries with atoms defined analytically. We consider more particularly multivariate signals coming from the simultaneous acquisition of several quantities, as EEG signals or 2D and 3D motion signals. We extend sparse representation methods to the multivariate model, to take into account interactions between the different components acquired simultaneously. This model is more flexible that the common multichannel one which imposes a hypothesis of rank 1. We study models of invariant representations: invariance to temporal shift, invariance to rotation, etc. Adding supplementary degrees of freedom, each kernel is potentially replicated in an atoms family, translated at all samples, rotated at all orientations, etc. So, a dictionary of invariant kernels generates a very redundant atoms dictionary, thus ideal to represent the redundant studied data. All these invariances require methods adapted to these models. Temporal shift-invariance is an essential property for the study of temporal signals having a natural temporal variability. In the 2D and 3D rotation invariant case, we observe the efficiency of the non-oriented approach over the oriented one, even when data are not revolved. Indeed, the non-oriented model allows to detect data invariants and assures the robustness to rotation when data are revolved. We also observe the reproducibility of the sparse decompositions on a learned dictionary. This generative property is due to the fact that dictionary learning is a generalization of K-means. Moreover, our representations have many invariances that is ideal to make classification. We thus study how to perform a classification adapted to the shift-invariant model, using shift-consistent pooling functions.

La simulation numérique des phénomènes physiques est devenue un élément incontournable dans la boite à outils de l'ingénieur mécanicien. Des outils robustes et modulables, basés sur les méthodes classiques d'approximation, sont désormais couramment utilisés dans l'industrie. Cependant, ces outils nécessitent des moyens de calculs importants lorsqu'ils sont utilisés pour résoudre des problèmes complexes. Même si les progrès remarquables de l'industrie informatique rendent de tels moyens de calcul toujours plus abordables, il s'avère aujourd'hui nécessaire de proposer des méthodes d'approximation innovantes permettant de mieux exploiter les ressources informatiques disponibles. Les méthodes de réduction de modèle sont présentées comme un candidat idéal pour atteindre cet objectif. Parmi celles-ci, les méthodes basées sur la construction d'une approximation à variables séparées se sont révélées être très efficaces pour approcher la solution d'une grande variété de problèmes, réduisant les coûts numériques de plusieurs ordres de grandeur. Néanmoins, l'efficacité de ces méthodes dépend considérablement du problème traité. Aussi, on se propose ici d'évaluer l'intérêt d'une approximation à variables séparées espace-temps dans le cadre de problèmes académiques de dynamique transitoire. On définit tout d'abord la meilleure approximation (au sens d'un problème de minimisation) de la solution d'un problème transitoire, sous la forme d'une représentation à variables séparées espace-temps. Le calcul de cette approximation étant basé sur l'hypothèse que la solution du problème de référence est connue (méthode a posteriori), la suite du manuscrit est dédiée à la construction d'une telle approximation sans autres connaissances a priori sur la solution de référence, que les opérateurs du problème espace-temps dont elle est solution (méthode a priori). Un formalisme générique, basé sur une représentation tensorielle des opérateurs du problème espace-temps est alors introduit dans un cadre multichamps. On développe ensuite un solveur exploitant ce format générique, pour construire une approximation à variables séparées espace-temps de la solution d'un problème transitoire. Ce solveur est basé sur la décomposition généralisée propre de la solution (Proper Generalized Decomposition - PGD). Un état de l'art des algorithmes existants permet alors d'évaluer l'efficacité des définitions classiques de la PGD pour approcher la solution de problèmes académiques de dynamique transitoire. Les résultats obtenus mettant en défaut l'optimalité de la PGD la plus robuste, une nouvelle définition, récemment introduite dans la littérature, est appliquée dans un cadre multichamps à la résolution d'un problème d'élastodynamique 2D. Cette nouvelle définition, basée sur la minimisation du résidu dans une norme idéale, permet finalement d'obtenir une très bonne approximation de la meilleure approximation de rang donné, sans avoir à calculer un grand nombre de modes espace-temps. / Numerical simulation of physical phenomena has become an indispensable part of the mechanical engineer's toolbox. Robust and flexible tools, based on classical approximation methods, are now commonly used in industry. However, these tools require lots of computational resources to solve complex problems. Even if such resources are more and more affordable thanks to the remarkable progress in computer industry, it is now necessary to propose innovative approximation methods in order to better exploit the impressive amount of computational resources that are todays available. Reduced order modeling techniques are presented as ideal candidates to address this issue. Among these, methods based on the construction of low rank separated approximations have been shown to be very efficient to approach solutions of a wide variety of problems, reducing computational costs by several orders of magnitude. Nonetheless, efficiency of these methods significantly depends on the considered problem. In this manuscript, we propose to evaluate interest of space-time separated representations to approach solutions of academical transient dynamic problems. We first define the best space-time separated approximation (with respect to a minimization problem) of a given solution of a transient problem. The construction of this approximation being based on the hypothesis that the problem's solution is known (a posteriori method), the following of the manuscript is dedicated to the construction of such an approximation without any other knowledge on the reference solution than the operators of the space-time problem from which it is solution (a priori method). We then introduce a generic formalism, based on the tensor product structure of the operators of the space-time problem, in a multi-field framework. Next, this formalism is used to develop a generic solver that builds a separated approximation of a transient problem's solution, with the help of the Proper Generalized Decomposition (PGD). A state of art of existing algorithms is done, and efficiency of classical definitions of PGD to approach solutions of several academical transient dynamic problems is evaluated. Numerical results highlight the lack of optimality of the more robust PGD. Therefore, a new PGD definition, recently introduced in literature, is applied to solution of an elastodynamic problem in a multi-field framework. This new definition is based on minimization of an ideal residual norm and allows to find a very good approximation of the best approximation of a given rank, without having to calculate more space-time modes than needed.