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No description available.

The solution of initial value problems is a fundamental component of many scientific simulations of physical phenomena. In many cases these initial value problems arise from a method of lines approach to solving partial differential equations, resulting in very large systems of equations that require the use of numerical time integration methods to solve. Many problems of scientific interest exhibit stiff behavior for which implicit methods are favorable, however standard implicit methods are computationally expensive. They require the solution of one or more large nonlinear systems at each timestep, which can be impractical to solve exactly and can behave poorly when solved approximately. The recently introduced ``lightly-implicit'' K-methods seek to avoid this issue by directly coupling the time integration methods with a Krylov based approximation of linear system solutions, treating a portion of the problem implicitly and the remainder explicitly. This work seeks to further two primary objectives: evaluation of these K-methods in large-scale parallel applications, and development of new linearly implicit methods for contexts where improvements can be made. To this end, Rosenbrock-Krylov methods, the first K-methods, are examined in a scalability study, and two new families of time integration methods are introduced: biorthogonal Rosenbrock-Krylov methods, and linearly implicit multistep methods. For the scalability evaluation of Rosenbrock-Krylov methods, two parallel contexts are considered: a GPU accelerated model and a distributed MPI parallel model. In both cases, the most significant performance bottleneck is the need for many vector dot products, which require costly parallel reduce operations. Biorthogonal Rosenbrock-Krylov methods are an extension of the original Rosenbrock-Krylov methods which replace the Arnoldi iteration used to produce the Krylov approximation with Lanczos biorthogonalization, which requires fewer vector dot products, leading to lower overall cost for stiff problems. Linearly implicit multistep methods are a new family of implicit multistep methods that require only a single linear solve per timestep; the family includes W- and K-method variants, which admit arbitrary or Krylov based approximations of the problem Jacobian while maintaining the order of accuracy. This property allows for a wide range of implementation optimizations. Finally, all the new methods proposed herein are implemented efficiently in the MATLODE package, a Matlab ODE solver and sensitivity analysis toolbox, to make them available to the community at large. / Doctor of Philosophy / Differential equations are a fundamental building block of the mathematical description of many physical phenomena. Thus, solving problems involving complex differential equations is necessary for construction of scientific models of these phenomena, which can then be used to make useful predictions, such as weather forecasts. Aside from some simplified cases, complex differential equations cannot be solved exactly. Time integration methods are a class of numerical algorithms used to compute approximate solutions to differential equations, by stepping a given initial solution forward in time, producing a new solution at each timestep. Time integration methods are generally categorized as either explicit or implicit methods. Explicit methods are simpler, but have significant restrictions on the size of timesteps for challenging differential equations. Implicit methods relax this timestep restriction, but are much more expensive to compute. The recently introduced ``lightly-implicit'' K-methods provide a way to fuse the advantages of both implicit and explicit methods, by effectively treating a portion of the problem implicitly and the remainder explicitly. This work seeks to further two primary objectives: evaluation of these K-methods on very large problems, and development of new time integration methods. To this end, Rosenbrock--Krylov methods, the first K-methods, are applied to a large-scale problem and examined in a scalability study, and two new families of time integration methods are introduced: biorthogonal Rosenbrock--Krylov methods, and linearly implicit multistep methods. Ultimately, the goal is to develop new methods which allow for the creation of larger, more detailed, and more accurate scientific models, in order to get better and faster predictions.

Nos livros das Odes Horácio empregou treze esquemas métricos distribuídos por poemas de temas diversos. A influência da tradição eólica representada pelos dois musicistas de Lesbos, Safo e Alceu, pautou a maioria das escolhas temáticas e formais do poeta. Odes compostas em metros asclepiadeus e jônicos kataV stivcon, em estrofes sáficas, alcaicas e asclepiadéias e em dísticos de formação variada (cola datílicos, sáficos, jâmbicos e trocaicos), mostram ritmos próprios, capazes de imprimir, no ânimo do ouvinte, sensações diferenciadas, de acordo com a natureza da seqüência métrica empregada. A teoria do ethos métrico leva em consideração o conceito da conveniência (Prevpon, decorum): conteúdo e forma em harmonia na criação poética. Os efeitos impressivos das medidas gregas, naturalizadas por Horácio, que deu feição datílica aos versos eólicos, fixou quantidades livres e disciplinou as estrofes, devem-se ao caráter psicagógico dos metros, herdado da antiga teoria musical. Desde Platão e Aristóteles, passando por Cícero, Demétrio, Dionísio de Halicarnasso, Longino e Quintiliano, registramse a preocupação de classificar os metros de acordo com sua adequação a cada tipo de composição e a censura de seu uso indiscriminado na prosa e na poesia. A análise rítmico-semântica de algumas odes de Horácio revelou o zelo do poeta em combinar forma e conteúdo e em selecionar palavras de composição sonora e formação métrica em harmonia com o sentido. Nas Odes a musicalidade do ritmo métrico tem implicações semânticas, realçando a expressão textual. / It has been observed in Horace\'s Odes books that he has employed thirteen metrical schemes distributed among thematic different poems. Aeolic tradition influence, represented by the two Lesbian musicians Sappho and Alcaeus, was responsible for most of the formal and thematic choices of the poet. Odes written in asclepiadean meters and ionic kataV stivcon, in sapphic, alcaic end asclepiadean strophes and in various distics (cola datctylics, sapphics, iambics and trochaics) show their own rhythms, which are able to impress different sensations to their listeners, according to the nature of the metrical sequence used. The theory of metrical ethos considers the convenience concept (Prevpon, decorum): subject and form harmonically living in poetic creation. The impressive effects of greek measures, (which were naturalized by Horace, gave dactylic features to the aeolic verses, fixed free amounts and regulated the strophes), are due to the psychagogic character of the meters, inherited by the old musical theory. Since Plato, Aristotle, Cicero, Demetrius, Dionysius of Halicarnassus, Longinus and Quintilianus, there is a worry at classifying meters according to their adequacy to each kind of composition and disapproval of its nonrestrictive use in prose and poetry. The rhythmic-semantics analysis of some odes from Horace revealed the poet care to combine form and subject and to select sonorous words and metrics in harmony with sense. In Odes, metrics rhythm musicality has semantic implications that highlight the textual expression.

Nos livros das Odes Horácio empregou treze esquemas métricos distribuídos por poemas de temas diversos. A influência da tradição eólica representada pelos dois musicistas de Lesbos, Safo e Alceu, pautou a maioria das escolhas temáticas e formais do poeta. Odes compostas em metros asclepiadeus e jônicos kataV stivcon, em estrofes sáficas, alcaicas e asclepiadéias e em dísticos de formação variada (cola datílicos, sáficos, jâmbicos e trocaicos), mostram ritmos próprios, capazes de imprimir, no ânimo do ouvinte, sensações diferenciadas, de acordo com a natureza da seqüência métrica empregada. A teoria do ethos métrico leva em consideração o conceito da conveniência (Prevpon, decorum): conteúdo e forma em harmonia na criação poética. Os efeitos impressivos das medidas gregas, naturalizadas por Horácio, que deu feição datílica aos versos eólicos, fixou quantidades livres e disciplinou as estrofes, devem-se ao caráter psicagógico dos metros, herdado da antiga teoria musical. Desde Platão e Aristóteles, passando por Cícero, Demétrio, Dionísio de Halicarnasso, Longino e Quintiliano, registramse a preocupação de classificar os metros de acordo com sua adequação a cada tipo de composição e a censura de seu uso indiscriminado na prosa e na poesia. A análise rítmico-semântica de algumas odes de Horácio revelou o zelo do poeta em combinar forma e conteúdo e em selecionar palavras de composição sonora e formação métrica em harmonia com o sentido. Nas Odes a musicalidade do ritmo métrico tem implicações semânticas, realçando a expressão textual. / It has been observed in Horace\'s Odes books that he has employed thirteen metrical schemes distributed among thematic different poems. Aeolic tradition influence, represented by the two Lesbian musicians Sappho and Alcaeus, was responsible for most of the formal and thematic choices of the poet. Odes written in asclepiadean meters and ionic kataV stivcon, in sapphic, alcaic end asclepiadean strophes and in various distics (cola datctylics, sapphics, iambics and trochaics) show their own rhythms, which are able to impress different sensations to their listeners, according to the nature of the metrical sequence used. The theory of metrical ethos considers the convenience concept (Prevpon, decorum): subject and form harmonically living in poetic creation. The impressive effects of greek measures, (which were naturalized by Horace, gave dactylic features to the aeolic verses, fixed free amounts and regulated the strophes), are due to the psychagogic character of the meters, inherited by the old musical theory. Since Plato, Aristotle, Cicero, Demetrius, Dionysius of Halicarnassus, Longinus and Quintilianus, there is a worry at classifying meters according to their adequacy to each kind of composition and disapproval of its nonrestrictive use in prose and poetry. The rhythmic-semantics analysis of some odes from Horace revealed the poet care to combine form and subject and to select sonorous words and metrics in harmony with sense. In Odes, metrics rhythm musicality has semantic implications that highlight the textual expression.

No description available.

No description available.

Dermal wound healing involves a myriad of highly regulated and sophisticated mechanisms, which are coordinated and carried out via several specialized cell types. The dominant players involved in this process include platelets, neutrophils, macrophages and fibroblasts. These cells play a vital role in the repair of the wound by orchestrating tasks such as forming a fibrin clot to stanch blood flow, removing foreign organisms and cellular debris, depositing new collagen matrix and establishing the contractile forces which eventually bridge the void caused by the initial infraction.\\[5pt] \indent Our current understanding of these mechanisms has been primarily based upon animal models. Unfortunately, these models lack insight into pathologic conditions, which plague human beings, such as keloid scar or chronic ulcer formation. Consequently, investigators have proposed a number of {\it in vivo} techniques to study wound repair in humans in order to overcome this barrier. One approach, which has been devised to increase our level of understanding of these chronic conditions, involves the cutaneous placement of a small cylindrical structure within the appendage of a human test subject.\\[5pt] \indent Researches have designed a variety of these implantable structures to examine different aspects of wound healing in both healthy subjects and individuals that experience some trauma related condition. In each case, several implants are surgically positioned at multiple locations under sterile conditions. These structures are later removed at distinct time intervals at which point they are histologically analyzed and biochemically assayed to deduce the presence of biological markers involved in the repair process. Implantable structures used in this way are often referred to as Human Implantable Models or Systems.\\[5pt] \indent Clinical studies with implantable models open up tremendous opportunities in fields such as biomathematics because they provide an experimentally controlled setting that aids in the development and validation of mathematical models. Furthermore, experiments carried out with implants greatly simplify the mathematics required to describe the repair process because they minimize the modeling of complex features associated with healing such as wound geometry and the evolution of contractile forces.\\[5pt] \indent In this work, we present a notional mathematical model, which accounts for two fundamental processes involved in the repair of an acute dermal wound. These processes include the inflammatory response and fibroplasia. Our system describes each of these events through the time evolution of four primary species or variables. These include the density of initial damage, inflammatory cells, fibroblasts and deposition of new collagen matrix. Since it is difficult to populate the equations of our model with coefficients that have been empirically derived, we fit these constants by carrying out a large number of simulations until there is reasonable agreement between the time response of the variables of our system and those reported by the literature for normal healing. Once a suitable choice of parameters has been made, we then compare simulation results with data obtained from clinical investigations. While more data is desired, we have a promising first step toward describing the primary events of wound repair within the confines of an implantable system.

Dans ce mémoire, nous détaillons différents travaux que j'ai efféctués ces dernières années sur des problèmes d'analyse harmonique réelle et plus particulièrement d'analyse de Fourier en lien avec l'analyse des EDPs. Le coeur de l'analyse de Fourier réside dans la procédure suivante : afin d'étudier un objet mathématique (une fonction, un opérateur, un espace fonctionnel, ...), on le décompose en objets élémentaires, vérifiant certaines propriétés supplémentaires. Le but et la difficulté de l'analyse est alors d'utiliser des informations sur ces objets élémentaires et de les sommer (le plus finement possible) afin d'obtenir l'information désirée sur l'objet initial. Le qualificatif "de Fourier" fait ici référence au fait que l'opération de décomposition sera toujours associée à une décomposition temps-fréquence. Nous allons nous consacrer à deux cadres différents (mais pas complètement indépendants) d'application de cette "technique" : 1) L'analyse de Fourier Euclidienne bilinéaire : plus précisément, l'étude des continuités dans les espaces de Lebesgue d'opérateurs bilinéaires, définis en fréquence par la transformée de Fourier (principalement les multiplicateurs de Fourier bilinéaires). Ici, le caractère "Euclidien" renvoie à l'utilisation de la transformée de Fourier. Nous décrirons aussi une application pour l'analyse d'EDPs présentant une non-linéarité quadratique ainsi que l'obtention d'estimations bilinéaires de dispersion. 2) L'analyse de Fourier non Euclidienne fonctionnelle : le but sera de définir un cadre où la notion de "fréquence" n'est plus donnée par la transformée de Fourier mais nous travaillerons sur la notion plus générale "d'oscillations". Ceci permet de nous extraire du cadre Euclidien et de travailler dans un espace de type homogène, ou une variété Riemanienne. Nous essaierons alors d'étudier certaines propriétés d'espaces fonctionnels ainsi que certains opérateurs, définis par l'intermédiaire de ces oscillations. L'utilisation de le notion d'oscillations permet de travailler sur un espace ambiant plus général et permet aussi d'adapter l'analyse à un opérateur générant un semi-groupe. Ces deux cadres présentent un aspect commun : une "analyse temps-fréquence". Cette analyse est une situation particulière de l'analyse de Fourier où nous avons besoin de décomposer notre objet mathématique, non pas dans l'espace physique ou dans l'espace fréquentiel, mais simultanément dans les deux espaces, tout en respectant le principe d'incertitude d'Heisenberg. Dans le cadre de l'analyse de Fourier Euclidienne, la transformée de Fourier permet de donner une notion très pratique de "fréquence" et de décomposition fréquentielle. Dans le cadre de l'analyse fonctionnelle, la notion de "fréquence" sera donnée par des opérateurs, appelés "opérateurs d'oscillation". Nous verrons comment on peut alors transposer certaines techniques Euclidiennes à ce cadre et de ce fait étendre "l'analyse temps-fréquence" à des espaces non-Euclidiens très généraux (tels que ensembles fractals, variétés Riemaniennes, ...).

Thesis (Honors)--College of William and Mary, 2009. / Includes bibliographical references (leaves 60-63). Also available via the World Wide Web.