LECTURES ON NONLINEAR DISPERSIVE EQUATIONS I

CONTENTS J. Bona Derivation and some fundamental properties of nonlinear dispersive waves equations F. Planchon Schr\"odinger equations with variable coecients P. Rapha\"el On the blow up phenomenon for the L^2 critical non linear Schrodinger Equatio

Spatio-temporal integral equation methods with applications

Electromagnetic interactions are vital in many applications including physics, chemistry, material sciences and so on. Thus, a central problem in physical modeling is the electromagnetic analysis of materials. Here, we consider the numerical solution of the Maxwell equation for the evolution of the electromagnetic field given the charges, and the Newton or Schr\\"odinger equation for the evolution of particles. By combining integral equation techniques with new spectral deferred correction algorithms in time and hierarchical methods in space, we develop fast solvers for the calculation of electromagnetism with relaxations of the model in different scenarios. The dissertation consists of two parts, aiming to resolve the challenges in the temporal and spatial direction, respectively. In the first part, we study a new class of time stepping methods for time-dependent differential equations. The core algorithm uses the pseudo-spectral collocation formulation to discretize the Picard type integral equation reformulation, producing a highly accurate and stable representation, which is then solved via the deferred correction technique. By exploiting the mathematical properties of the formulation and the convergence procedure, we develop some new preconditioning techniques from different perspectives that are accurate, robust, and can be much more efficient than existing methods. As is typical of spectral methods, the solution to the discretization is spectral accurate and the time step-size is optimal, though the cost of solving the system can be high. Thus, the solver is particularly suited to problems where very accurate solutions are sought or large time-step is required, e.g., chaotic systems or long-time simulation. In the second part, we study the hierarchical methods with emphasis on the spatial integral equations. In the first application, we implement a parallel version of the adaptive recursive solver for two-point boundary value problem by Cilk multithreaded runtime system based on the integral equation formulation. In the second application, we apply the hierarchical method to two-layered media Helmholtz equations in the acoustic and electromagnetic scattering problems. With the method of images and integral representations, the spatially heterogeneous translation operators are derived with rigorous error analysis, and the information is then compressed and spread in a fashion similar to fast multipole methods. The preliminary results suggest that our approach can be faster than existing algorithms with several orders of magnitude. We demonstrate our solver on a number of examples and discuss various useful extensions. Preliminary results are favorable and show the viability of our techniques for integral equations. Such integral equation methods could well have a broad impact on many areas of computational science and engineering. We describe further applications in biology, chemistry, and physics, and outline some directions for future work.Doctor of Philosoph

Geometric Integrators for Schrödinger Equations

The celebrated Schrödinger equation is the key to understanding the dynamics of quantum mechanical particles and comes in a variety of forms. Its numerical solution poses numerous challenges, some of which are addressed in this work. Arguably the most important problem in quantum mechanics is the so-called harmonic oscillator due to its good approximation properties for trapping potentials. In Chapter 2, an algebraic correspondence-technique is introduced and applied to construct efficient splitting algorithms, based solely on fast Fourier transforms, which solve quadratic potentials in any number of dimensions exactly - including the important case of rotating particles and non-autonomous trappings after averaging by Magnus expansions. The results are shown to transfer smoothly to the Gross-Pitaevskii equation in Chapter 3. Additionally, the notion of modified nonlinear potentials is introduced and it is shown how to efficiently compute them using Fourier transforms. It is shown how to apply complex coefficient splittings to this nonlinear equation and numerical results corroborate the findings. In the semiclassical limit, the evolution operator becomes highly oscillatory and standard splitting methods suffer from exponentially increasing complexity when raising the order of the method. Algorithms with only quadratic order-dependence of the computational cost are found using the Zassenhaus algorithm. In contrast to classical splittings, special commutators are allowed to appear in the exponents. By construction, they are rapidly decreasing in size with the semiclassical parameter and can be exponentiated using only a few Lanczos iterations. For completeness, an alternative technique based on Hagedorn wavepackets is revisited and interpreted in the light of Magnus expansions and minor improvements are suggested. In the presence of explicit time-dependencies in the semiclassical Hamiltonian, the Zassenhaus algorithm requires a special initiation step. Distinguishing the case of smooth and fast frequencies, it is shown how to adapt the mechanism to obtain an efficiently computable decomposition of an effective Hamiltonian that has been obtained after Magnus expansion, without having to resolve the oscillations by taking a prohibitively small time-step. Chapter 5 considers the Schrödinger eigenvalue problem which can be formulated as an initial value problem after a Wick-rotating the Schrödinger equation to imaginary time. The elliptic nature of the evolution operator restricts standard splittings to low order, ¿ < 3, because of the unavoidable appearance of negative fractional timesteps that correspond to the ill-posed integration backwards in time. The inclusion of modified potentials lifts the order barrier up to ¿ < 5. Both restrictions can be circumvented using complex fractional time-steps with positive real part and sixthorder methods optimized for near-integrable Hamiltonians are presented. Conclusions and pointers to further research are detailed in Chapter 6, with a special focus on optimal quantum control.Bader, PK. (2014). Geometric Integrators for Schrödinger Equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/38716TESISPremios Extraordinarios de tesis doctorale

Large deviations for rare realizations of dynamical systems

A central problem in uncertainty quantification is how to characterize the impact that our incomplete knowledge about models has on the predictions we make from them. This question naturally lends itself to a probabilistic formulation, by making the unknown model parameters random with given statistics. In the following this approach is used in concert with tools from large deviation theory (LDT) and optimal control to estimate the probability that some observables in a dynamical system go above a large threshold after some time, given the prior statistical information about the system’s parameters and/or its initial conditions. It is established under which conditions the extreme events occur in a predictable way, as the minimizer of the LDT action functional, i.e. the instanton. In the first physical application, the appearance of rogue waves in a long-crested deep sea is investigated. First, the leading order equations are derived for the wave statistics in the framework of wave turbulence (WT), showing that the theory cannot go beyond Gaussianity, although it remains the main tool to understand the energetic transfers. It is shown how by applying our LDT method one can use the incomplete information contained in the spectrum (with the Gaussian statistics of WT) as prior and supplement this information with the governing nonlinear dynamics to reliably estimate the probability distribution of the sea surface elevation far in the tail at later times. Our results indicate that rogue waves occur when the system hits unlikely pockets of wave configurations that trigger large disturbances of the surface height. The rogue wave precursors in these pockets are wave patterns of regular height but with a very specific shape that is identified explicitly, thereby potentially allowing for early detection. Finally, the first experimental evidence of hydrodynamic instantons is presented using data collected in a long wave flume, elevating the instanton description to the role of a unifying theory of extreme water waves. Other applications of the method are illustrated: To the nonlinear Schrödinger equation with random initial conditions, relevant to fiber optics and integrable turbulence, and to a rod with random elasticity pulled by a time-dependent force. The latter represents an interesting nonequilibrium statistical mechanics setup with a strongly out-of-equilibrium transient (absence of local thermodynamic equilibrium) and a small number of degrees of freedom (small system), showing how the LDT method can be exploited to solve optimal-protocol problems

Particle and energy transport in strongly driven one-dimensional quantum systems

This Dissertation concerns the transport properties of a strongly–correlated one–dimensional system of spinless fermions, driven by an external electric field which induces the flow of charges and energy through the system. Since the system does not exchange information with the environment, the evolution can be accurately followed to arbitrarily long times by solving numerically the time–dependent Schrödinger equation, going beyond Kubo’s linear response theory. The thermoelectric response of the system is here characterized, using the ratio of the induced energy and particle currents, in the nonequilibrium state under the steady applied electric field. Even though the equilibrium response can be reached for vanishingly small driving, strong fields produce quantum–mechanical Bloch oscillations in the currents, which disrupt the proportionality of the currents. The effects of the driving on the local state of the ring are analyzed via the reduced density matrix of small subsystems. The local entropy density can be defined and shown to be consistent with the laws of thermodynamics for quasistationary evolution. Even integrable systems are shown to thermalize under driving, with heat being produced via the Joule effect by the flow of currents. The spectrum of the reduced density matrix is shown to be distributed according the Gaussian unitary ensemble predicted by random–matrix theory, both during driving and a subsequent relaxation. The first fully–quantum model of a thermoelectric couple is realized by connecting two correlated quantum wires. The field is shown to produce heating and cooling at the junctions according to the Peltier effect, by mapping the changes in the local entropy density. In the quasiequilibrium regime, a local temperature can be defined, at the same time verifying that the subsystems are in a Gibbs thermal state. The gradient of temperatures, established by the external field, is shown to counterbalance the flow of energy in the system, terminating the operation of the thermocouple. Strong applied fields lead to new nonequilibrium phenomena. At the junctions, observable Bloch oscillations of the density of charge and energy develop at the junctions. Moreover, in a thermocouple built out of Mott insulators, a sufficiently strong field leads to a dynamical transition reversing the sign of the charge carriers and the Peltier effect

Computational and Theoretical Developements for (Time Dependent) Density Functional Theory

En esta tesis se presentan avances computacionales y teoricos en la teoria de funcionales de la densidad (DFT) y en la teoria de funcionales de la densidad dependientes del tiempo (TDDFT). Hemos explorado una posible nueva ruta para la mejora de los funcionales de intercambio y correlacion (XCF) en DFT, comprobado y desarrollado propagadores numericos para TDDFT, y aplicado una combinacion de la teoria de control optimo con TDDFT.En los ultimos anos, DFT se ha convertido en el metodo mas utilizado en el area de estructura electronica gracias a su inigualable relacion entre coste y precision. Podemos usar DFT para calcular multitud de propiedades fisicas y quimicas de atomos, moleculas, nanoestructuras, y materia macroscopica. El factor principal que determina la precision que podemos alcanzar usando DFT es el XCF, un objeto desconocido para el cual se han propuesto cientos de aproximaciones distintas. Algunas de estas aproximaciones funcionan correctamente en ciertas situaciones, pero a dia de hoy no existe un XCF que pueda aplicarse con certeza sobre su validez a un sistema arbitrario. Mas aun, no hay una forma sistematica de refinar estos funcionales. Proponemos y exploramos, para sistemas unidimensionales, una nueva manera de estudiarlos y optimizarlos basada en establecer una relacion con la interaccion entre electrones.TDDFT es la extension de DFT a problemas dependientes del tiempo y problemas conestados excitados, y es tambien uno de los metodos mas populares (a veces el unico metodo que se puede poner en practica) en la comunidad de estructura electronica para tratar conellos. De nuevo, la razon detras de su popularidad reside en su relacion precision/coste computacional, que nos permite tratar sistemas mayores y mas complejos. Puede usarse en combinacion con la dinamica de Ehrenfest, un tipo de dinamica molecular no adiabatica.Hemos ido mas alla y hemos combinado TDDFT y la dinamica de Ehrenfest con la teoria de control optimo, creando un instrumento que nos permite, por ejemplo, predecir la forma de los pulsos laser que inducen una explosion de Coulomb en clusters de sodio. A pesar del buen rendimiento computacional de TDDFT en comparacion con otros metodos, hallamos que el coste de estos calculos era bastante elevado.Motivados por este hecho, tambien dedicamos una parte del trabajo de la tesis a la investigacion computacional. En particular, hemos estudiado e implementado familias de propagadores numericos que no se habian examinado en el contexto de TDDFT. Mas concretamente, metodos con varios pasos previos, formulas Runge-Kutta exponenciales, y las expansiones de Magnus sin conmutadores. Finalmente, hemos implementado modificaciones de estas expansiones de Magnus sin conmutadores para la propagacion de las ecuaciones clasico-cuanticas que resultan de la combinacion de la dinamica de Ehrenfest con TDDFT.In this thesis we present computational and theoretical developments for density functional theory (DFT) and time dependent density functional theory (TDDFT). We have explored a new possible route to improve exchange and correlation functionals (XCF) in DFT, tested and developed numerical propagators for TDDFT, and applied a combination of optimal control theory with TDDFT. In recent years, DFT has become the most used method in the electronic structure field thanks to its unparalleled precision/computational cost relationship. We can use DFT to accurately calculate many physical and chemical properties of atoms, molecules, nanostructures, and bulk materials. The main factor that determines the precision that we can obtain using DFT is the XCF, an unknown object for which hundreds of different approximations have been proposed. Some of these approximations work well enough for certain situations, but to this day there is no XCF that can be reliably applied to any arbitrary system. Moreover, there is no clear way for a systematic refinement of these functionals. We propose and explore, for one-dimensional systems, a new way to optimize them, based on establishing a relationship with the electron-electron interaction. TDDFT is the extension of DFT to time-dependent and excited-states problems, and it is also one of the most popular methods (sometimes the only practical one) in the electronic structure community to deal with them. Once again, the reason behind its popularity is its accuracy/computational cost ratio, which allows us to tackle bigger, more complex systems. It can be used in combination with Ehrenfest dynamics, a non-adiabatic type of molecular dynamics. We have furthermore combined both TDDFT and Ehrenfest dynamics with optimal control theory, a scheme that has allowed us, for example, to predict the shapes of the laser pulses that induce a Coulomb explosion in different sodium clusters. Despite the good numerical performance of TDDFT compared to other methods, we found that these computations were still quite expensive. Motivated by this fact, we have also dedicated a part of the thesis work to computational research. In particular, we have studied and implemented families of numerical propagators that had not been tested in the context of TDDFT. More concretely, linear multistep schemes, exponential Runge-Kutta formulas, and commutator-free Magnus expansions. Moreover, we have implemented modifications of these commutator-free Magnus methods for the propagation of the classical-quantum equations that result of combining Ehrenfest dynamics with TDDFT.<br /

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described