Comparison of Iteration Schemes for the Solution of the Multidimensional Schrödinger-Poisson Equations

We present a fast and robust iterative method for obtaining self-consistent solutions to the coupled system of Schrödinger's and Poisson's equations in quantum structures. A simple expression describing the dependence of the quantum electron density on the electrostatic potential is used to implement a predictor – corrector type iteration scheme for the solution of the coupled system of differential equations. This approach simplifies the software implementation of the nonlinear problem, and provides excellent convergence speed and stability. We demonstrate the algorithm by presenting an example for the calculation ofthe two-dimensional bound electron states within the cross-section of a GaAs-AlGaAs based quantum wire. For this example, six times fewer iterations are needed when our predictor – corrector approach is applied, compared to a corresponding underrelaxation algorithm

A Brief Review on Mathematical Tools Applicable to Quantum Computing for Modelling and Optimization Problems in Engineering

Since its emergence, quantum computing has enabled a wide spectrum of new possibilities and advantages, including its efficiency in accelerating computational processes exponentially. This has directed much research towards completely novel ways of solving a wide variety of engineering problems, especially through describing quantum versions of many mathematical tools such as Fourier and Laplace transforms, differential equations, systems of linear equations, and optimization techniques, among others. Exploration and development in this direction will revolutionize the world of engineering. In this manuscript, we review the state of the art of these emerging techniques from the perspective of quantum computer development and performance optimization, with a focus on the most common mathematical tools that support engineering applications. This review focuses on the application of these mathematical tools to quantum computer development and performance improvement/optimization. It also identifies the challenges and limitations related to the exploitation of quantum computing and outlines the main opportunities for future contributions. This review aims at offering a valuable reference for researchers in fields of engineering that are likely to turn to quantum computing for solutions. Doi: 10.28991/ESJ-2023-07-01-020 Full Text: PD

Exponential integrators: tensor structured problems and applications

The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as phi-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed

Topics in multiscale modeling: numerical analysis and applications

We explore several topics in multiscale modeling, with an emphasis on numerical analysis and applications. Throughout Chapters 2 to 4, our investigation is guided by asymptotic calculations and numerical experiments based on spectral methods. In Chapter 2, we present a new method for the solution of multiscale stochastic differential equations at the diffusive time scale. In contrast to averaging-based methods, the numerical methodology that we present is based on a spectral method. We use an expansion in Hermite functions to approximate the solution of an appropriate Poisson equation, which is used in order to calculate the coefficients in the homogenized equation. Extensions of this method are presented in Chapter 3 and 4, where they are employed for the investigation of the Desai—Zwanzig mean-field model with colored noise and the generalized Langevin dynamics in a periodic potential, respectively. In Chapter 3, we study in particular the effect of colored noise on bifurcations and phase transitions induced by variations of the temperature. In Chapter 4, we investigate the dependence of the effective diffusion coefficient associated with the generalized Langevin equation on the parameters of the equation. In Chapter 5, which is independent from the rest of this thesis, we introduce a novel numerical method for phase-field models with wetting. More specifically, we consider the Cahn—Hilliard equation with a nonlinear wetting boundary condition, and we propose a class of linear, semi-implicit time-stepping schemes for its solution.Open Acces

Macroscopic modeling of quantum effects in semiconductor devices

This dissertation explores the use of macroscopic quantum hydrodynamic (QHD) models as tools for investigating the transport of charge carriers in semiconductor devices in the regime where quantum effects are important. Chapter 1 provides a panoramic view of the field of carrier transport modeling in semiconductors. The essential differences between classical and quantum transport is brought out and a brief outline is given of the derivation of successively less detailed models from the fundamental starting points of the Boltzmann transport equation (BTE) for classical transport and the quantum distribution function (Wigner function, density matrix) based methods for quantum transport. A mention is made of the various quantum hydrodynamic models without going into the details of their derivation and applicability. Chapter 2 brings into focus the area of quantum hydrodynamic modeling of carrier transport. A detailed derivation using the method of moments is presented for each of the popular quantum hydrodynamic models currently being explored in the literature, namely the density-gradient method and the smooth quantum potential model. A summary is made of their limitations and these limitations are then shown as arising out of particular assumptions made in their derivations that could hamper their applicable regimes. Chapter 3 presents an analysis of the boundary layers near interfaces obtained in density-gradient theory. An integral equation for the density near such interfaces is obtained and this is used to analytically compare the DG solution with the solutions from one-electron quantum mechanics in non-degenerate conditions. Confinement in simple potential wells is then discussed using the macroscopic equations. Chapter 4 discusses the derivation of macroscopic equations to describe quantum mechanical tunneling through large barrier potentials. Using the approximate solutions of the Schr?dinger equation it is analytically shown that the density profile inside the barrier satisfies a second order differential equation, very similar to the Schr?dinger equation for a carrier at a suitably chosen average energy. Use of this is made to derive a consistent macroscopic treatment of tunneling transport in the insulating barrier. Chapter 5, the final chapter, summarizes the major contributions of this dissertation and concludes it with several suggestions for future research directions that can stem from this work