The research group Applied Mathematics, headed by Prof. Karel int Hout and Prof. Wim Vanroose, department Mathematics and Computer Science, Universiteit Antwerpen and the Condensed Matter Theory group, headed by Prof Francois Peeters and Prof. Bart Partoens, department of Physics, collaborate in a project to develop efficient computational methods for the simulation and mathematical description of superconductors and quantum dots.
For this interdisciplinary project, we have an opening of a fulltime (m/f)
PhD scholarship Numerical Analysis / Computational Physics
Profile:
The candidate is expected to hold a masters degree in a subject related to the project and to have:
* Strong background in iterative methods for large linear and non-linear systems.
* Background in numerical analysis of computational methods.
* Skills in Scientific computing and programming.
We offer:
* an interdisciplinary work environment
* the positions start 1 Oktober 2007
* the scholarship is granted for an initial period of 2 years and will be extended after positive evaluation
Interested?
Electronic application, including CV and the names of two possible references should be send to Wim.Vanroose@ua.ac.be of Bart.Partoens@ua.ac.be
Project description
Iterative Krylov methods for linear and non-linear Schrödinger equations
The state of both the quantum dot and the mesoscopic superconductor is theoretically described by a wave function that fits the Schrödinger equation. The state of the quantum dot is an eigenstate of the linear Schrödinger equation and the state of the superconductor is the solution of the nonlinear Schrödinger equation, known as the Ginzburg-Landau equation. With the help of the wavefunction it is possible to calculate experimental observables such as the oscillator strengths of the transitions in quantum dots and the magnetization of a mesoscopic superconductor. In practice, we represent the wave function on a grid such that the Schrödinger equation is transformed into a system of equations. An accurate representation needs a few hundred grid points for each dimension. For 2D problems this leads to systems of equations with several ten thousands unknowns. In 3D, however, this becomes a few million. The time required to solve such a system with a conventional direct method typically scales as 3 N , with N the number of unknowns in the system. For 3D this leads to an enormous computing time to calculate a solution. For this reason, the theoretical treatments have mainly focused on 2D systems. A quantum dot, which is essentially a 3D system, has usually been approximated by a disc such that the system is reduced to only two dimensions, and computationally only flat superconductors were treated.
In the last 15 years, however, several methods have been developed in applied mathematics to solve large sparse linear systems that have a much better scaling with the number of variables. These methods are known collectively as iterative Krylov subspace methods and regularly solve systems with sizes of a few hundred thousand to several million. These methods, however, are not black-box routines that are called in a straight forward way. They require fine tuning to exploit the properties of each specific problem.
The aim of the project is to develop efficient computational methods, based on Krylov space methods, to solve the linear and non-linear Schrödinger equations. This will enable the theoretical methods to move from the approximate 2D models to the more realistic 3D description. The methods will be applied to practical physical problems: to solve the non-linear time-dependent and time-independent Ginzburg-Landau equations for the study of the vortex structure and dynamics in mesoscopic superconductors and to solve the linear Schrödinger equation for realistic self-assembled quantum dots.
Application Deadline: 01-Sep-07
For this interdisciplinary project, we have an opening of a fulltime (m/f)
PhD scholarship Numerical Analysis / Computational Physics
Profile:
The candidate is expected to hold a masters degree in a subject related to the project and to have:
* Strong background in iterative methods for large linear and non-linear systems.
* Background in numerical analysis of computational methods.
* Skills in Scientific computing and programming.
We offer:
* an interdisciplinary work environment
* the positions start 1 Oktober 2007
* the scholarship is granted for an initial period of 2 years and will be extended after positive evaluation
Interested?
Electronic application, including CV and the names of two possible references should be send to Wim.Vanroose@ua.ac.be of Bart.Partoens@ua.ac.be
Project description
Iterative Krylov methods for linear and non-linear Schrödinger equations
The state of both the quantum dot and the mesoscopic superconductor is theoretically described by a wave function that fits the Schrödinger equation. The state of the quantum dot is an eigenstate of the linear Schrödinger equation and the state of the superconductor is the solution of the nonlinear Schrödinger equation, known as the Ginzburg-Landau equation. With the help of the wavefunction it is possible to calculate experimental observables such as the oscillator strengths of the transitions in quantum dots and the magnetization of a mesoscopic superconductor. In practice, we represent the wave function on a grid such that the Schrödinger equation is transformed into a system of equations. An accurate representation needs a few hundred grid points for each dimension. For 2D problems this leads to systems of equations with several ten thousands unknowns. In 3D, however, this becomes a few million. The time required to solve such a system with a conventional direct method typically scales as 3 N , with N the number of unknowns in the system. For 3D this leads to an enormous computing time to calculate a solution. For this reason, the theoretical treatments have mainly focused on 2D systems. A quantum dot, which is essentially a 3D system, has usually been approximated by a disc such that the system is reduced to only two dimensions, and computationally only flat superconductors were treated.
In the last 15 years, however, several methods have been developed in applied mathematics to solve large sparse linear systems that have a much better scaling with the number of variables. These methods are known collectively as iterative Krylov subspace methods and regularly solve systems with sizes of a few hundred thousand to several million. These methods, however, are not black-box routines that are called in a straight forward way. They require fine tuning to exploit the properties of each specific problem.
The aim of the project is to develop efficient computational methods, based on Krylov space methods, to solve the linear and non-linear Schrödinger equations. This will enable the theoretical methods to move from the approximate 2D models to the more realistic 3D description. The methods will be applied to practical physical problems: to solve the non-linear time-dependent and time-independent Ginzburg-Landau equations for the study of the vortex structure and dynamics in mesoscopic superconductors and to solve the linear Schrödinger equation for realistic self-assembled quantum dots.
Application Deadline: 01-Sep-07
Comments