Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)


Electrical and Computer Engineering

First Advisor

Martin Feldman


We apply stochastic techniques towards the solution of the two-dimensional Laplace's equation in boundary value problems encountered in the calculation of electrostatic potentials in electron lenses and deflectors. The justification of these techniques arises from an astonishingly simple but far-reaching principle, which has been known for a long time but has been rarely used: the potential at any point in the interior of a charge-free region can be calculated by performing random walks starting at this point and terminating at the boundary of the region--the potential is then the average of the potential boundary values (assumed known) over the random walks. By an optimal combination of the stochastic Monte-Carlo and deterministic Relaxation methods, we show the advantages and competitiveness of our hybrid Monte-Carlo-Relaxation (MCR) technique compared to the conventional numerical techniques used in the previously mentioned problem. In order to enhance the performance of our method, we investigate the convergence, speed and accuracy of MCR versus traditional techniques. We also develop optimized computational techniques that we believe increase MCR's appeal to problems not previously considered amenable to Monte-Carlo type simulations as well as demonstrate its applicability in problems that are intractable by traditional relaxation or analytical techniques. We use MCR to simulate electrostatic lenses and detectors previously presented in the literature. Finally, we demonstrate the application of MCR towards the numerical solution of general elliptic problems in arbitrary domains and we present the generalization of the stochastic method to solve problems with space charge, namely Poisson's equation.