In this paper, we discuss the global existence of solutions for Chemotaxis models with saturation growth. If the coe±cients of the equations are all positive smooth T-periodic functions, then the problem has a positive T-periodic solution, and meanwhile we discuss here the stability problems for the T-periodic solutions.

This dissertation presents and analyzes two new algorithms for sparse direct solution methods based on the use of local-global solution (LOGOS) modes. One of the new algorithms is a rigorous error control strategy for LOGOS-based matrix factorizations that utilize overlapped, localizing modes (OL-LOGOS) on a shifted grid. The use of OL-LOGOS modes is critical to obtaining asymptotically efficient factorizations from LOGOS-based methods. Unfortunately, the approach also introduces a non-orthogonal basis function structure. This can cause errors to accumulate across levels of a multilevel implementation, which has previously posed a barrier to rigorous error control for the OL-LOGOS factorization method. This limitation is overcome, and it is shown that it is possible to efficiently decouple the fundamentally non-orthogonal factorization subspaces in a manner that prevents multilevel error propagation. This renders the OL-LOGOS factorization error controllable in a relative RMS sense. The impact of the new, error-controlled OL-LOGOS factorization algorithm on computational resource utilization is discussed and several numerical examples are presented to illustrate the performance of the improved algorithm relative to previously reported results. The second algorithmic development considered is the development of efficient out-of-core (OOC) versions of the OL-LOGOS factorization algorithm that allow associated software tools to take advantage of additional resources for memory management. The proposed OOC algorithm incorporates a memory page definition that is tailored to match the flow of the OL-LOGOS factorization procedure. Efficiency of the function of the part is evaluated using a quantitative approach, because the tested massive storage device performances do not follow analytical results. The performance latency and the memory usage of the resulting OOC tools are compared with in-core performance results. Both the new error control algorithm and the OOC method have been incorporated into previously existing software tools, and the dissertation presents results for real-world simulation problems.

京都大学 / 0048 / 新制・課程博士 / 博士(人間・環境学) / 甲第20455号 / 人博第805号 / 新制||人||194(附属図書館) / 28||人博||805(吉田南総合図書館) / 京都大学大学院人間・環境学研究科共生人間学専攻 / (主査)教授 清水 扇丈, 教授 足立 匡義, 准教授 木坂 正史, 教授 森本 芳則 / 学位規則第4条第1項該当 / Doctor of Human and Environmental Studies / Kyoto University / DFAM

This dissertation proposes a modular fast direct (MFD) analysis method for a class of problems involving a large fixed platform region and a smaller, variable design region. A modular solution algorithm is obtained by first decomposing the problem geometry into platform and design regions. The two regions are effectively detached from one another using basic equivalence concepts. Equivalence principles allow the total system model to be constructed in terms of independent interaction modules associated with the platform and design regions. These modules include interactions with the equivalent surface that bounds the design region. This dissertation discusses how to analyze (fill and factor) each of these modules separately and how to subsequently compose the solution to the original system using the separately analyzed modules. The focus of this effort is on surface integral equation formulations of electromagnetic scattering from conductors and dielectrics. In order to treat large problems, it is necessary to work with sparse representations of the underlying system matrix and other, related matrices. Fortunately, a number of such representations are available. In the following, we will primarily use the adaptive cross approximation (ACA) to fill the multilevel simply sparse method (MLSSM) representation of the system matrix. The MLSSM provides a sparse representation that is similar to the multilevel fast multipole method. Solutions to the linear systems obtained using the modular analysis strategies described above are obtained using direct methods based on the local-global solution (LOGOS) method. In particular, the LOGOS factorization provides a data sparse factorization of the MLSSM representation of the system matrix. In addition, the LOGOS solver also provides an approximate sparse factorization of the inverse of the system matrix. The availability of the inverse eases the development of the MFD method. Because the behavior of the LOGOS factorization is critical to the development of the proposed MFD method, a significant part of this dissertation is devoted to providing additional analyses, improvements, and characterizations of LOGOS-based direct solution methods. These further developments of the LOGOS factorization algorithms and their application to the development of the MFD method comprise the most significant contributions of this dissertation.

This dissertation develops surface integral equations using constraint-based Helmholtz decompositions for electromagnetic modeling. This new approach is applied to the electric field integral equation (EFIE), and it incorporates a Helmholtz decomposition (HD) of the current. For this reason, the new formulation is referred to as the EFIE-hd. The HD of the current is accomplished herein via appropriate surface integral constraints, and leads to a stable linear system. This strategy provides accurate solutions for the electric and magnetic fields at both high and low frequencies, it allows for the use of a locally corrected Nyström (LCN) discretization method for the resulting formulation, it is compatible with the local global solution framework, and it can be used with non-conformal meshes. To address large-scale and complex electromagnetic problems, an overlapped localizing local-global (OL-LOGOS) factorization is used to factorize the system matrix obtained from an LCN discretization of the augmented EFIE (AEFIE). The OL-LOGOS algorithm provides good asymptotic performance and error control when used with the AEFIE. This application is used to demonstrate the importance of using a well-conditioned formulation to obtain efficient performance from the factorization algorithm.

Submitted by Rosivalda Pereira (mrs.pereira@ufma.br) on 2017-06-13T17:22:58Z No. of bitstreams: 1 WashingtonMenezes.pdf: 349401 bytes, checksum: 40e24e24fcc72911a8a30e2a9d7b0f1b (MD5) / Made available in DSpace on 2017-06-13T17:22:58Z (GMT). No. of bitstreams: 1 WashingtonMenezes.pdf: 349401 bytes, checksum: 40e24e24fcc72911a8a30e2a9d7b0f1b (MD5) Previous issue date: 2016-08-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / In this work, we introduce results of local and global solution for a heat equation in RN with nonlocal nonlinearity in time. / Neste trabalho, apresentamos resultados de existência local e global para uma equação do calor em RN com não-linearidade não local no tempo.

Computational electromagnetic modeling involves generating system matrices by discretizing integral equations and solving the resulting system of linear equations. Many methods of solving the system of linear equations exist and one such method is the factorization of the matrix using the so called local-global solution (LOGOS) modes. Computer codes to perform the discretization of the integral equations, filling of the matrix, and the subsequent LOGOS factorization have previously been developed by others. However, these codes are limited to complex double precision arithmetic only. This thesis extends and expands the existing computer by creating a more general implementation that is able to analyze a problem not only in complex double precision but also in real double precision and both complex and real single precision. The existing code is expanded using "templates" in Fortran 90 and the resulting generic code is used test the performance of the LOGOS (both OL- and NL-LOGOS) factorization on matrices generated by discretization of the volume integral equation. As part of this effort, we demonstrate for the first time that the LOGOS factorization provides an O(N log N) complexity solution to the volume integral equation formulation of low-frequency electromagnetic problems.

Dizertační práce se věnuje funkcionálním diferenciálním rovnicím smíšeného typu. Poskytuje kritéria pro existenci globálních a semi-globálních řešení diferenciálních systémů smíšeného typu. Metody použité v teto práci spočívají v sestavení vhodných operátorů pro diferenciální rovnice a prokázání existence jejich pevných bodů. Tyto pevné body jsou potom použity ke konstrukci řešení rovnic s předcházením a zpožděním. V důkazech tvrzení jsou použity monotónní iterační metoda a Schauderovy-Tychonovovy věty o existenci pevného bodu. V obou případech jsou uvedeny také odhady řešení. Pokud je použita iterační metoda, lze tyto odhady zlepšit iterováním. Kromě toho jsou odvozena kritéria pro lineární rovnice a systémy a je uvedena řada přikladů. Dosažené výsledky lze aplikovat také pro obyčejné diferenciální rovnice nebo diferenciální rovnice se zpožděním či s předcházením argumentu.

Dans cette thèse, nous étudions certaines équations différentielles partielles avec mécanisme dissipatif, telles que l'équation de Boltzmann, l'équation de Landau et certains systèmes hyperboliques symétriques avec type de dissipation. L'existence globale de solutions ou les taux de dégradation optimaux des solutions pour ces systèmes sont envisagées dans les espaces de Sobolev ou de Besov. Les propriétés de lissage des solutions sont également étudiées. Dans cette thèse, nous prouvons principalement les quatre suivants résultats, voir les chapitres 3-6 pour plus de détails. Pour le premier résultat, nous étudions le problème de Cauchy pour le non linéaire inhomogène équation de Landau avec des molécules Maxwelliennes (= 0). Voir des résultats connus pour l'équation de Boltzmann et l'équation de Landau, leur existence globale de solutions est principalement prouvée dans certains espaces de Sobolev (pondérés) et nécessite un indice de régularité élevé, voir Guo [62], une série d'oeuvres d'Alexander Morimoto-Ukai-Xu-Yang [5, 6, 7, 9] et des références à ce sujet. Récemment, Duan-Liu-Xu [52] et Morimoto-Sakamoto [145] ont obtenu les résultats de l'existence globale de solutions à l'équation de Boltzmann dans l'espace critique de Besov. Motivés par leurs oeuvres, nous établissons l'existence globale de la solution dans des espaces de Besov spatialement critiques dans le cadre de perturbation. Précisément, si le datum initial est une petite perturbation de la distribution d'équilibre dans l'espace Chemin-Lerner eL 2v (B3=2 2;1 ), alors le problème de Cauchy de Landau admet qu'une solution globale appartient à eL 1t eL 2v (B3=2 2;1 ). Notre résultat améliore le résultat dans [62] et étend le résultat d'existence globale de l'équation de Boltzmann dans [52, 145] à l'équation de Landau. Deuxièmement, nous considérons le problème de Cauchy pour l'équation de Kac non-coupée spatialement inhomogène. Lerner-Morimoto-Pravda-Starov-Xu a considéré l'équation de Kac non-coupée spatialement inhomogène dans les espaces de Sobolev et a montré que le problème de Cauchy pour la fluctuation autour de la distribution maxwellienne admise S 1+ 1 2s 1+ 1 2s Propriétés de régularité Gelfand-Shilov par rapport à la variable de vélocité et propriétés de régularisation G1+ 1 2s Gevrey à la variable de position. Et les auteurs ont supposé qu'il restait encore à déterminer si les indices de régularité 1 + 1 2s étaient nets ou non. Dans cette thèse, si la donnée initiale appartient à l'espace de Besov spatialement critique, nous pouvons prouver que l'équation de Kac inhomogène est bien posée dans un cadre de perturbation. De plus, il est montré que la solution bénéficie des propriétés de régularisation de Gelfand-Shilov en ce qui concerne la variable de vitesse et des propriétés de régularisation de Gevrey en ce qui concerne la variable de position. Dans notre thèse, l'indice de régularité de Gelfand-Shilov est amélioré pour être optimal. Et ce résultat est le premier qui présente un effet de lissage pour l'équation cinétique dans les espaces de Besov. A propos du troisième résultat, nous considérons les équations de Navier-Stokes-Maxwell compressibles apparaissant dans la physique des plasmas, qui est un exemple concret de systèmes composites hyperboliques-paraboliques à dissipation non symétrique. On observe que le problème de Cauchy pour les équations de Navier-Stokes-Maxwell admet le mécanisme dissipatif de type perte de régularité. Par conséquent, une régularité plus élevée est généralement nécessaire pour obtenir le taux de dégradation optimal de L1(R3)-L2(R3) type, en comparaison avec cela pour l'existence globale dans le temps de solutions lisses. / In this thesis, we study some kinetic equations and some partial differential equations with dissipative mechanism, such as Boltzmann equation, Landau equation and some non-symmetric hyperbolic systems with dissipation type. Global existence of solutions or optimal decay rates of solutions for these systems are considered in Sobolev spaces or Besov spaces. Also the smoothing properties of solutions are studied. In this thesis, we mainly prove the following four results, see Chapters 3-6 for more details. For the _rst result, we investigate the Cauchy problem for the inhomogeneous nonlinear Landau equation with Maxwellian molecules ( = 0). See from some known results for Boltzmann equation and Landau equation, their global existence of solutions are mainly proved in some (weighted) Sobolev spaces and require a high regularity index, see Guo [62], a series works of Alexandre-Morimoto-Ukai-Xu-Yang [5, 6, 7, 9] and references therein. Recently, Duan-Liu-Xu [52] and Morimoto-Sakamoto [145] obtained the global existence results of solutions to the Boltzmann equation in critical Besov spaces. Motivated by their works, we establish the global existence of solutions for Landau equation in spatially critical Besov spaces in perturbation framework. Precisely, if the initial datum is a small perturbation of the equilibrium distribution in the Chemin-Lerner space eL 2v (B3=2 2;1 ), then the Cauchy problem of Landau equation admits a global solution belongs to eL 1t eL 2v (B3=2 2;1 ). Our results improve the result in [62] and extend the global existence result for Boltzmann equation in [52, 145] to Landau equation. Secondly, we consider the Cauchy problem for the spatially nhomogeneous non-cuto_ Kac equation. Lerner-Morimoto-Pravda-Starov-Xu [117] considered the spatially inhomogeneous non-cuto_ Kac equation in Sobolev spaces and showed that the Cauchy problem for the uctuation around the Maxwellian distribution admitted S 1+ 1 2s 1+ 1 2s Gelfand-Shilov regularity properties with respect to the velocity variable and G1+ 1 2s Gevrey regularizing properties with respect to the position variable. And the authors conjectured that it remained still open to determine whether the regularity indices 1+ 1 2s is sharp or not. In this thesis, if the initial datum belongs to the spatially critical Besov space eL 2v (B1=2 2;1 ), we prove the well-posedness to the inhomogeneous Kac equation under a perturbation framework. Furthermore, it is shown that the weak solution enjoys S 3s+1 2s(s+1) 3s+1 2s(s+1) Gelfand-Shilov regularizing properties with respect to the velocity variableand G1+ 1 2s Gevrey regularizing properties with respect to the position variable. In our results, the Gelfand-Shilov regularity index is improved to be optimal. And this result is the _rst one that exhibits smoothing e_ect for the kinetic equation in Besov spaces. About the third result, we consider compressible Navier-Stokes-Maxwell equations arising in plasmas physics, which is a concrete example of hyperbolic-parabolic composite systems with non-symmetric dissipation. It is observed that the Cauchy problem for Navier-Stokes-Maxwell equations admits the dissipative mechanism of regularity-loss type. Consequently, extra higher regularity is usually needed to obtain the optimal decay rate of L1(R3)-L2(R3) type, in comparison with that for the global-in-time existence of smooth solutions.