Complexity and Error Analysis of Numerical Methods for Wireless Channels, SDE, Random Variables, and Quantum Mechanics
Tid: On 2012-05-30 kl 10.15
Plats: Sal F3, Lindstedtsvägen 26
This thesis consists of four papers considering different aspects of stochastic process modeling, error analysis, and minimization of computational cost.
In Paper I, we construct a Multipath Fading Channel (MFC) model for wireless channels with noise introduced through scatterers flipping on and off. By coarse graining the MFC model a Gaussian process channel model is developed. Complexity and accuracy comparisons of the models are conducted.
In Paper II, we generalize a multilevel Forward Euler Monte Carlo method introduced by Giles  for the approximation of expected values depending on solutions of Itô stochastic differential equations. Giles’ work proposed and analyzed a Forward Euler Multilevel Monte Carlo (MLMC) method based on realizations on a hierarchy of uniform time discretizations and a coarse graining based control variates idea to reduce the computational cost required by a standard single level Forward Euler Monte Carlo method. This work is an extension of Giles’ MLMC method from uniform to adaptive time grids. It has the same improvement in computational cost and is applicable to a larger set of problems.
In paper III, we consider the problem to estimate the mean of a random variable by a sequential stopping rule Monte Carlo method. The performance of a typical second moment based sequential stopping rule is shown to be unreliable both by numerical examples and by analytical arguments. Based on analysis and approximation of error bounds we construct a higher moment based stopping rule which performs more reliably.
In paper IV, Born-Oppenheimer dynamics is shown to provide an accurate approximation of time-independent Schrödinger observables for a molecular system with an electron spectral gap, in the limit of large ratio of nuclei and electron masses, without assuming that the nuclei are localized to vanishing domains. The derivation, based on a Hamiltonian system interpretation of the Schrödinger equation and stability of the corresponding hitting time Hamilton-Jacobi equation for non ergodic dynamics, bypasses the usual separation of nuclei and electron wave functions, includes caustic states and gives a different perspective on the Born-Oppenheimer approximation, Schrödinger Hamiltonian systems and numerical simulation in molecular dynamics modeling at constant energy.
Ämnesområde: Numerisk Analys
Respondent: Håkon Hoel
Opponent: Professor Helge Holden, Institutt for matematiske fag, NTNU, Trondheim
Handledare: Professor Anders Szepessy