| This thesis presents a parallelized Monte Carlo (MC) algorithm for the solution of the wave equation. This MC method does not require any discretization, and hence the memory requirements are lower than approaches based on discretization. Another advantage of the MC method is that it is inherently parallelizable and an almost linear increase in computational speed can be obtained with an increase in the number of processors. However, in spite of these advantages, the MC method has not been applied successfully to the solution of wave problems though the mathematical 'proof of principal' has already been established in literature. The absence of effective algorithms can be primarily attributed to the problem of resonance in frequency domain Green's functions (with homogeneous Dirichlet boundary conditions) for finite geometries at length scales which are multiples of half a wavelength, it has been demonstrated in previously published work (co-authored by my thesis supervisor, Prof. K. Chatterjee), that infinite-domain Green's functions can be used to obtain frequency-domain, solutions at multiple-wavelength scales. In this thesis, MC solutions to time-dependent problems in one-dimension are however obtained through the use of random walks in both space and time and through the use of a Green's function that is dependent on space and time. We have obtained excellent agreement between analytical solutions and MC results. The algorithm has been parallelized to demonstrate the almost linear rate of parallelization for a large number of processors. The ultimate objective of this research is the application of this algorithm to the full-wave analysis of IC interconnect structures at multi-GHz frequencies. |
