As groundwater models are being used increasingly in the area of resource allocation, there has been an increase in the level of complexity in an attempt to capture heterogeneity, complex geometries and detail in interaction between the model domain and the outside hydraulic influences. As models strive to represent the real world in ever increasing detail, there is a strong likelihood that the boundary conditions will become nonlinear. Nonlinearities exist in the groundwater flow equation even in simple models when watertable (unconfined) conditions are simulated. This thesis is concerned with how these nonlinearities are treated numerically, with particular focus on the MODFLOW groundwater flow software and the nonlinear nature of the unconfined condition simulation. One of the limitations of MODFLOW is that it employs a first order fixed point iterative scheme to linearise the nonlinear system that arises as a result of the finite difference discretisation process, which is well known to offer slow convergence rates for highly nonlinear problems. However, Newton's method can achieve quadratic convergence and is more effective at dealing with higher levels of nonlinearity. Consequently, the main objective of this research is to investigate the inclusion of Newton's method to the suite of computational tools in MODFLOW to enhance its flexibility in dealing with the increasing complexity of real world problems, as well as providing a more competitive and efficient solution methodology. Furthermore, the underpinning linear iterative solvers that MODFLOW currently utilises are targeted at symmetric systems and a consequence of using Newton's method would be the requirement to solve non-symmetric Jacobian systems. Therefore, another important aspect of this work is to investigate linear iterative solution techniques that handle such systems, including the newer Krylov style solvers GMRES and BiCGSTAB. To achieve these objectives a number of simple benchmark problems involving nonlinearities through the simulation of unconfined conditions were established to compare the computational performance of the existing MODFLOW solvers to the new solution strategies investigated here. One of the highlights of these comparisons was that Newton's method when combined with an appropriately preconditioned Krylov solver was on average greater than 40% more CPU time efficient than the Picard based solution techniques. Furthermore, a significant amount of this time saving came from the reduction in the number of nonlinear iterations due to the quadratic nature of Newton's method. It was also found that Newton's method benefited more from improved initial conditions than Picard's method. Of all the linear iterative solvers tested, GMRES required the least amount of computational effort. While the Newton method involves more complexity in its implementation, this should not be interpreted as prohibitive in its application. The results here show that the extra work does result in performance increase, and thus the effort is certainly worth it.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 94-95). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 94-95). Also available on the Internet.

Thesis (Doctoral)--Ruhr-Universität Bochum, 1982.

Inaug.-diss.--Zürich. / Part of the Cornell Digital Library Math Collection.

Soit k un corps quelconque. Dans cette th±se, on étudie le groupe de Picard des k-groupes algébriques unipotents (lisses et connexes).Tout k-groupe algébrique unipotent est extension itérée de formes du groupe additif; on va donc d'abord s'intéresser au groupe de Picard des formes du groupe additif. L'étude de ce groupe est faite avec une méthode géométrique qui permet de traiter le cas plus général des formes de la droite affine. On obtient ainsi une borne explicite sur la torsion du groupe de Picard desformes de la droite affine et sur la torsion de la composante neutre du foncteur de Picard de leur complétion régulière. De plus, on trouve une condition suffisante pour que le groupe de Picard d'une forme de la droite affinesoit non trivial et on construit des exemples de formes non triviales de la droite affine dont le groupe de Picard est trivial.Un k-groupe algébrique unipotent est une forme de l'espace affine. Afin d'étudier le groupe de Picard d'une forme X de l'espace affine avec une méthode géométrique, on définit un foncteur de Picard "restreint". On montre que si X admet une complétion régulière, alors le foncteur de Picard "restreint" est représentable par un k-groupe unipotent (lisse, non nécessairement connexe).Avec ce foncteur de Picard "restreint" et des raisonnements purement géométriques, on obtient que le groupe de Picard d'une forme unirationnelle de l'espace affine est fini. De plus, on généralise un résultat dû à B. Totaro: si k est séparablement clos, et si le groupe de Picard d'un k-groupe algébrique unipotent commutatif est non trivial, alors il admet une extension non triviale par le groupe multiplicatif. / Let k be any field. In this Ph.D. dissertation we study the Picard group of the (smooth connected) unipotent k-algebraic groups.As every unipotent algebraic group is an iterated extension of forms of the additive group, we will study the Picard group of the forms of the additive group. In fact we study the Picard group of forms of the additive group and the affine line simultaneously using a geometric method. We obtain anexplicit upper bound on the torsion of the Picard group of the forms of the affine line and their regular completion, and a sufficient condition for the Picard group of a form of the affine line to be nontrivial. We also give examples of nontrivial forms of the affine line with trivial Picard groups.In general, a unipotent k-algebraic group is a form of the affine n-space. In order to study the Picard group of a form X of the affine n-space with a geometric method, we define a "restricted" Picard functor; we show that if X admits a regular completion then the "restricted" Picard functor is representable by a unipotent k-algebraic group (smooth, not necessarly connected). With this "restricted" Picard functor and geometric arguments we show that the Picard group of a unirational form of the affine n-space is finite. Moreover we generalise a result of B. Totaro: if k is separablyclosed and if the Picard group of a unipotent k-algebraic group is nontrivial then it admits a nontrivial extension by the multiplicative group.

Newton-Picard methods are iterative methods that work well for computing roots of nonlinear equations within a continuation framework. This project presents one of these methods and includes the results of a computation involving the Brusselator problem performed by an implementation of the method. This work was done in collaboration with Andrew Salinger at Sandia National Laboratories.

Les variétés de Picard sont des variétés de Shimura associées au groupe des similitudes unitaires d'un espace hermitien de dimension 3 sur un corps CM. Elles paramétrisent les classes d'isomorphismes de variétés abdéliennes munies de certaines structures supplémentaires. En particulier, il existe une variété abdélienne universelle sur une variété de Picard et plus généralement des familles de Kuga-Sato.A ces variétés sont attachés des groupes de cohomologie. L'un des intérêts de telles variétés est qu'il est possible de trouver des représentations automorphes dans les groupes de cohomologie qui lui sont attachés, en particulier dans les groupes de cohomologie intérieure. Selon le programme de Langlands, ces représentations correspondent conjecturalement à des motifs. Le résultat principal de cette thèse est la construction de facteurs directs du motif intérieur de certaines familles de Kuga-Sato sur des variétés de Picard, c'est-a-dire d'un analogue motivique de la cohomologie intérieure. Cela passe par une étude détaillée des poids du motif bord de ces familles. On en déduit l'existence de motifs associés à certaines représentations automorphes. / Picard varieties are Shimura varieties associated to the group of unitary similitudes of an hermitian space of dimension 3 over a CM eld. They parametrize isomorphism classes of abelian varieties with some additional data. In particular, there exists a universal abelian variety over a Picard variety and more generally Kuga-Sato families. Cohomology groups are attached to these varieties. Automorphic representations can be found in cohomology groups, more precisely in interior cohomology groups. Following Langlands' program, these representations correspond conjecturally to motives. The main result of this thesis is the construction of direct factors of the interior motive of certain Kuga-Sato families over a Picard variety, meaning a motivic analogue of interior cohomology. To prove this, we study the weights of the boundary motive of such families. We deduce from this the existence of a motive associated to certain automorphic representations.

Orientadores: Jose Manoel Balthazar / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica / Made available in DSpace on 2018-08-04T04:06:35Z (GMT). No. of bitstreams: 1 Peruzzi_NelsonJose_D.pdf: 9438459 bytes, checksum: 1e95acc28fd5e0b87f7b964ca5a2f34e (MD5) Previous issue date: 2005 / Resumo: Neste trabalho, apresenta-se um novo método numérico para aproximar matriz de transição de estados (STM) para sistemas com coeficientes periódicos no tempo. Este método, é baseado na expansão polinomial de Chebyshev, no método iterativo de Picard e na transformação de Lyapunov-Floquet (L-F) e aplica-se na análise da dinâmica e o controle de sistemas lineares e periódicos. Para o controle, aplicam-se dois projetos para eliminar o comportamento caótico de sistemas periódicos no tempo. O primeiro, usa o projeto de controle realimentado baseado na aplicação da transformação L-F, e o objetivo do controlador é conduzir a órbita do sistema para um ponto fixo ou para uma órbita periódica. No segundo, utiliza-se o controle não-linear para bifurcação, e o objetivo, neste caso, é modificar (atrasar ou eliminar) as características de uma bifurcação ao longo de sua rota para o caos. Como exemplo, aplicou-se, com sucesso, a técnica para análise e o controle da dinâmica: num pêndulo com excitação paramétrica, no oscilador de Duffing, no sistema de Rõssler e sistema pêndulo duplo invertido. O método, também, mostrou-se satisfatório na análise e controle de um sistema monotrilho não ideal / Abstract: In thiswork, a new numericalmethodto approximatestatetransitionmatrix(STM) for systems with time-periodic coefficients is presented. This method is based on the expansion Chebyshev polinomials,on the Picard iterationand on the Lyapunov-Floquet transfonnation(transfonnationL-F). It is applied to the dynamicalanalysis and control of linear periodic systems.For the control, two projectsto eliminatethe chaoticbehaviorof time periodic systemsare applied.The first one, uses the feedbackcontroldesignbased on the L-F transfonnation,and the controller'sobjectiveis to drive the orbit of the systemto an equilibriumpoint or a periodicorbit. fu the secondone, the non-lineal control for bifurcations used, and the objective,in this case, is to modify (to put back or to eliminate) the characteristicsof a bifurcation along its route to chaos. As example, the technique for dynamical analysis and control was applied, successfully, to a pendulum with parametric excitement, the Duffing's oscillator,the Rõssler's systemand the inverteddoublependulum The methodwas, also, to be shownsatisfactoryin the analysisand controlof a monorailnon-idealsystem / Doutorado / Mecanica dos Sólidos e Projeto Mecanico / Doutor em Engenharia Mecânica

In this thesis, we report three preprints [Li17a] [Li17b] and [HL17] the author wrote (the last one was written jointly with D. Hansen) during his pursuing of PhD at Columbia. We study smooth proper rigid varieties which admit formal models whose special fibers are projective. The main theorem asserts that the identity components of the associated rigid Picard varieties will automatically be proper. Consequently, we prove that non-archimedean Hopf varieties do not have a projective reduction. The proof of our main theorem uses the theory of moduli of semistable coherent sheaves. Combine known structure theorems for the relevant Picard varieties, together with recent advances in p-adic Hodge theory, We then prove several related results on the low-degree Hodge numbers of proper smooth rigid analytic varieties over p-adic fields.