<p>Numerical methods and algorithms for solving of partial differential equations, especially parallel algorithms, are an important research topic, given the very broad applicability range in all areas of science. Rapid advances of computer technology open up new possibilities for development of faster algorithms and numerical simulations of higher resolution. This is achieved through paralleliza-tion at different levels that&nbsp; practically all current computers support.</p><p>In this thesis we develop parallel algorithms for solving one kind of partial differential equations known as nonlinear Schr&ouml;dinger equation (NLSE) with a convolution integral kernel. Equations of this type arise in many fields of physics such as nonlinear optics, plasma physics and physics of ultracold atoms, as well as economics and quantitative&nbsp; finance. We focus on a special type of NLSE, the dipolar Gross-Pitaevskii equation (GPE), which characterizes the behavior of ultracold atoms in the state of Bose-Einstein condensation.</p><p>We present novel parallel algorithms for numerically solving GPE for a wide range of modern parallel computing platforms, from shared memory systems and dedicated hardware accelerators in the form of graphics processing units (GPUs), to&nbsp;&nbsp; heterogeneous computer clusters. For shared memory systems, we provide an algorithm and implementation targeting multi-core processors us-ing OpenMP. We also extend the algorithm to GPUs using CUDA toolkit and combine the OpenMP and CUDA approaches into a hybrid, heterogeneous al-gorithm that is capable of utilizing all&nbsp; available resources on a single computer. Given the inherent memory limitation a single&nbsp; computer has, we develop a distributed memory algorithm based on Message Passing Interface (MPI) and previous shared memory approaches. To maximize the performance of hybrid implementations, we optimize the parameters governing the distribution of data&nbsp; and workload using a genetic algorithm. Visualization of the increased volume of output data, enabled by the efficiency of newly developed algorithms, represents a challenge in itself. To address this, we integrate the implementations with the state-of-the-art visualization tool (VisIt), and use it to study two use-cases which demonstrate how the developed programs can be applied to simulate real-world systems.</p> / <p>Numerički metodi i algoritmi za re&scaron;avanje parcijalnih diferencijalnih jednačina, naročito paralelni algoritmi, predstavljaju izuzetno značajnu oblast istraživanja, uzimajući u obzir veoma &scaron;iroku primenljivost u svim oblastima nauke. Veliki napredak informacione tehnologije otvara nove mogućnosti za razvoj bržih al-goritama i&nbsp; numeričkih simulacija visoke rezolucije. Ovo se ostvaruje kroz para-lelizaciju na različitim nivoima koju poseduju praktično svi moderni računari. U ovoj tezi razvijeni su paralelni algoritmi za re&scaron;avanje jedne vrste parcijalnih diferencijalnih jednačina poznate kao nelinearna &Scaron;redingerova jednačina sa inte-gralnim konvolucionim kernelom. Jednačine ovog tipa se javljaju u raznim oblas-tima fizike poput nelinearne optike, fizike plazme i fizike ultrahladnih atoma, kao i u ekonomiji i kvantitativnim finansijama. Teza se bavi posebnim oblikom nelinearne &Scaron;redingerove jednačine, Gros-Pitaevski jednačinom sa dipol-dipol in-terakcionim članom, koja karakteri&scaron;e pona&scaron;anje ultrahladnih atoma u stanju Boze-Ajn&scaron;tajn kondenzacije.<br />U tezi su predstavljeni novi paralelni algoritmi za numeričko re&scaron;avanje Gros-Pitaevski jednačine za &scaron;irok spektar modernih računarskih platformi, od sis-tema sa deljenom memorijom i specijalizovanih hardverskih akceleratora u ob-liku grafičkih procesora, do heterogenih računarskih klastera. Za sisteme sa deljenom memorijom, razvijen je&nbsp; algoritam i implementacija namenjena vi&scaron;e-jezgarnim centralnim procesorima&nbsp; kori&scaron;ćenjem OpenMP tehnologije. Ovaj al-goritam je pro&scaron;iren tako da radi i u&nbsp; okruženju grafičkih procesora kori&scaron;ćenjem CUDA alata, a takođe je razvijen i&nbsp; predstavljen hibridni, heterogeni algoritam koji kombinuje OpenMP i CUDA pristupe i koji je u stanju da iskoristi sve raspoložive resurse jednog računara.<br />Imajući u vidu inherentna ograničenja raspoložive memorije koju pojedinačan računar poseduje, razvijen je i algoritam za sisteme sa distribuiranom memorijom zasnovan na Message Passing Interface tehnologiji i prethodnim algoritmima za sisteme sa deljenom memorijom. Da bi se maksimalizovale performanse razvijenih hibridnih implementacija, parametri koji određuju raspodelu podataka i računskog opterećenja su optimizovani kori&scaron;ćenjem genetskog algoritma. Poseban izazov je vizualizacija povećane količine izlaznih podataka, koji nastaju kao rezultat efikasnosti novorazvijenih algoritama. Ovo je u tezi re&scaron;eno kroz inte-graciju implementacija sa najsavremenijim alatom za vizualizaciju (VisIt), &scaron;to je omogućilo proučavanje dva primera koji pokazuju kako razvijeni programi mogu da se iskoriste za simulacije realnih sistema.</p>

Impulsos são perturbações abruptas que ocorrem em curto espaço de tempo e podem ser consideradas instantâneas. E os mercados financeiros estão sujeitos a choques bruscos como mudanças de governos, quebra de empresas, entre outros. Assim, é natural considerarmos a ação de tais eventos na precificação de ativos financeiros. Nosso objetivo neste trabalho é obtermos uma formulação para a equação diferencial parcial de Black-Scholes com ação impulsiva de modo que os impulsos representem estes choques. Utilizaremos a teoria de integração não-absoluta em espaço de funções para obtenção desta formulação / Impulses describe the evolution of systems where the continuous development of a process is interrupted by abrupt changes of state. Financial markets are subject to extreme events or shocks as government changes, companies colapse, etc. Thus it seems natural to consider the action of these events in the valuation of derivative securities. The aim of this work is to obtain a formulation for the Black-Scholes equation with impulse action where the impulses can represent these shocks. We use the non-absolute integration theory in functional spaces to obtain such formulation

Nesse trabalho, estudamos a equação de Schrödinger não-linear com uma função potencial delta atrativa. As soluções para essa equação tem uma componente localizada e uma dispersiva. Além de estudar o comportamento das soluções dessa equação em espaços de Sobolev clássicos, mostramos algumas propriedades do grupo unitário em espaços Lp, L2 com peso, Sobolev com peso e assim obtemos alguns resultados de boa colocação local e global das soluções. O ponto central desta tese é mostrarmos a existência de uma variedade invariante centro que irá consistir de órbitas periódicas no tempo. / In this work, we study the nonlinear Schrodinger equation with an attractive delta function potential.The solutions to this equation have a localized and a dispersive component. In addition to studying the behavior of solutions of this equation in classical Sobolev space, we show some properties for the unitary group in Lp, weighted L2 and Sobolev spaces and so we get some results of local and global well-posedness of solutions. The central theme this thesis is to show the existence of a center invariant manifold, which will consist of time-periodic orbits.

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Ce manuscrit se décompose en deux parties principales, associées à deux types d'applications assez différentes. Dans la première partie qui comprend les deux premiers chapitres, je m'intéresse à des systèmes issus de problèmes de contrôle et d'estimation en physique quantique; dans la deuxième partie (troisième chapitre du manuscrit), j'étudie la propagation d'ondes électriques le long des fils classiques dans un réseau de lignes de transmission et je considère certains problèmes d'estimation de paramètres. Dans le premier chapitre nous étudions le problème de la planification de trajectoires pour des systèmes quantiques fermés modélisés par des équations de Schrödinger bilinéaire. Nous démontrons alors des résultats de la stabilisation approchée pour le cas d'une boite quantique infinie ainsi que pour le cas d'un potentiel décroissant. Dans les deux cas, le manque de pré-compacité des trajectoires dans des espaces fonctionnels appropriés nous oblige à proposer des méthodes de Lyapunov qui évitent des phénomènes de perte de masse à l'infini. Dans le deuxième chapitre nous étudions le problème de stabilisation de systèmes quantiques en observation. Cette observation nécessite l'ouverture du système à son environnement. Les modèles pertinents pour l'évolution de ce type de systèmes sont des modèles stochastiques basés sur des trajectoires de Monte-Carlo quantiques. Nous étudions alors certains problèmes de stabilisation qui parviennent de vraies expériences physiques. Enfin, dans le chapitre 3 nous considérons le problème d'estimation de paramètres pour un réseau de fils de câblage électrique. Dans ce but, nous étudions deux approches : l'approche temporelle et l'approche fréquentielle. Dans l'approche temporelle, nous considérons le réseau le plus simple qui consiste d'une seule ligne de transmission et nous proposons un algorithme d'identification pour l'équation d'onde associé qui est basé sur l'application des observateurs asymptotiques. Dans l'approche fréquentielle, nous considérons un réseau plus compliqué de la forme étoile. Nous proposons alors des résultats d'identifiabilité basés sur des techniques de l'inverse scattering.

No description available.

A new quantum transport simulator -- Schrödinger Equation Monte Carlo in Three Dimensions (SEMC-3D) -- has been developed for simulating the carrier transport in nanoscale 3D MOSFET geometries. SEMC-3D self-consistently solves: (1) the 1D quantum transport equations derived from the SEMC method with open boundary conditions and rigorous treatment of various scattering processes including phonon and surface roughness scattering, (2) the 2D Schrödinger equations of the device cross sections with close boundary conditions to obtain the spatially varying subband structure along the conduction channel, and (3) the 3D Poisson equation of the whole device. Therefore, SEMC-3D can provide a physically accurate and electrostatically selfconsistent approach to the quantum transport in the subbands of 3D nanoscale MOSFETs. SEMC-3D has been used to simulate Si nanowire (NW) nMOSFETs to both demonstrate the capabilities of SEMC-3D, itself, and to provide new insight into transport phenomena in nanoscale MOSFETs, particularly with regards to interplay among scattering, quantum confinement and transport, and strain. / text

In the mid-1920s, the great Albert Einstein proposed that at extremely low temperatures, a gas of bosonic particles will enter a new phase where a large fraction of them occupy the same quantum state. This state would bring many of the peculiar features of quantum mechanics, previously reserved for small samples consisting only of a few atoms or molecules, up to a macroscopic scale. This is what we today call a Bose-Einstein condensate. It would take physicists almost 70 years to realize Einstein's idea, but in 1995 this was finally achieved. The research on Bose-Einstein condensates has since taken many directions, one of the most exciting being to study their behavior when they are placed in optical lattices generated by laser beams. This has already produced a number of fascinating results, but it has also proven to be an ideal test-ground for predictions from certain nonlinear lattice models. Because on the other hand, nonlinear science, the study of generic nonlinear phenomena, has in the last half century grown out to a research field in its own right, influencing almost all areas of science and physics. Nonlinear localization is one of these phenomena, where localized structures, such as solitons and discrete breathers, can appear even in translationally invariant systems. Another one is the (in)famous chaos, where deterministic systems can be so sensitive to perturbations that they in practice become completely unpredictable. Related to this is the study of different types of instabilities; what their behavior are and how they arise. In this thesis we compare classical and quantum mechanical nonlinear lattice models which can be applied to BECs in optical lattices, and also examine how classical nonlinear concepts, such as localization, chaos and instabilities, can be transfered to the quantum world.

We numerically study the dynamics and interactions of vortex rings in the nonlinear Schrödinger equation (NLSE). Single ring dynamics for both bright and dark vortex rings are explored including their traverse velocity, stability, and perturbations resulting in quadrupole oscillations. Multi-ring dynamics of dark vortex rings are investigated, including scattering and merging of two colliding rings, leapfrogging interactions of co-traveling rings, as well as co-moving steady-state multi-ring ensembles. Simulations of choreographed multi-ring setups are also performed, leading to intriguing interaction dynamics. Due to the inherent lack of a close form solution for vortex rings and the dimensionality where they live, efficient numerical methods to integrate the NLSE have to be developed in order to perform the extensive number of required simulations. To facilitate this, compact high-order numerical schemes for the spatial derivatives are developed which include a new semi-compact modulus-squared Dirichlet boundary condition. The schemes are combined with a fourth-order Runge-Kutta time-stepping scheme in order to keep the overall method fully explicit. To ensure efficient use of the schemes, a stability analysis is performed to find bounds on the largest usable time step-size as a function of the spatial step-size. The numerical methods are implemented into codes which are run on NVIDIA graphic processing unit (GPU) parallel architectures. The codes running on the GPU are shown to be many times faster than their serial counterparts. The codes are developed with future usability in mind, and therefore are written to interface with MATLAB utilizing custom GPU-enabled C codes with a MEX-compiler interface. Reproducibility of results is achieved by combining the codes into a code package called NLSEmagic which is freely distributed on a dedicated website.

The time-dependent Schrödinger equation (TDSE) models the quantum nature of molecular processes.  Numerical simulations based on the TDSE help in understanding and predicting the outcome of chemical reactions. This thesis is dedicated to the derivation and analysis of efficient and reliable simulation tools for the TDSE, with a particular focus on models for the interaction of molecules with time-dependent electromagnetic fields. Various time propagators are compared for this setting and an efficient fourth-order commutator-free Magnus-Lanczos propagator is derived. For the Lanczos method, several communication-reducing variants are studied for an implementation on clusters of multi-core processors. Global error estimation for the Magnus propagator is devised using a posteriori error estimation theory. In doing so, the self-adjointness of the linear Schrödinger equation is exploited to avoid solving an adjoint equation. Efficiency and effectiveness of the estimate are demonstrated for both bounded and unbounded states. The temporal approximation is combined with adaptive spectral elements in space. Lagrange elements based on Gauss-Lobatto nodes are employed to avoid nondiagonal mass matrices and ill-conditioning at high order. A matrix-free implementation for the evaluation of the spectral element operators is presented. The framework uses hybrid parallelism and enables significant computational speed-up as well as the solution of larger problems compared to traditional implementations relying on sparse matrices. As an alternative to grid-based methods, radial basis functions in a Galerkin setting are proposed and analyzed. It is found that considerably higher accuracy can be obtained with the same number of basis functions compared to the Fourier method. Another direction of research presented in this thesis is a new algorithm for quantum optimal control: The field is optimized in the frequency domain where the dimensionality of the optimization problem can drastically be reduced. In this way, it becomes feasible to use a quasi-Newton method to solve the problem. / eSSENCE