Narayan Kovvali

Pseudospectral method based on prolate spheroidal wave functions for semiconductor nanodevice simulation

We solve Schrodingers equation for semiconductor nanodevices by applying prolate spheroidal wave functions of order zero as basis functions in the pseudospectral method. When the functions involved in the problem are bandlimited, the prolate pseudospectral method outperforms the conventional pseudospectral methods based on trigonometric and orthogonal polynomials and related functions, asymptotically achieving similar accuracy using a factor of π/2 less unknowns than the latter. The prolate pseudospectral method also employs a more uniform spatial grid, achieving better resolution near the center of the domain.

Pseudospectral method based on prolate spheroidal wave functions for frequency-domain electromagnetic simulations

We apply prolate spheroidal wave functions of order zero as basis functions in the pseudospectral method for frequency-domain electromagnetic simulation problems. Like the traditional pseudospectral frequency-domain (PSFD) methods based on Chebyshev and Legendre polynomial series, the prolate PSFD method yields exponential order of accuracy. In terms of the number of samples utilized per wavelength, the prolate expansion is superior to the Chebyshev and Legendre polynomial series by a factor of /spl pi//2. In addition, the prolate PSFD method employs a more uniform spatial grid, achieving better resolution near the center of the domain.

Rapid Prolate Pseudospectral Differentiation and Interpolation with the Fast Multipole Method

Pseudospectral methods utilizing prolate spheroidal wave functions as basis functions have been shown to possess advantages over the conventional pseudospectral methods based on trigonometric and orthogonal polynomials. However, the spectral differentiation and interpolation steps of the prolate pseudospectral method involve matrix-vector products, which, if evaluated directly, entail O(N2) memory requirement and computational complexity (where N is the number of unknowns utilized for discretization and interpolation). In this work we show that the fast multipole method (FMM) can be used to reduce the memory requirement and computational complexity of the prolate pseudospectral method to O(N). Example simulation results demonstrate the speed and accuracy of the resulting fast prolate pseudospectral solver.

Volumetric fast multipole method for modeling Schrödinger's equation

A volume integral equation method is presented for solving Schrodingers equation for three-dimensional quantum structures. The method is applicable to problems with arbitrary geometry and potential distribution, with unknowns required only in the part of the computational domain for which the potential is different from the background. Two different Greens functions are investigated based on different choices of the background medium. It is demonstrated that one of these choices is particularly advantageous in that it significantly reduces the storage and computational complexity. Solving the volume integral equation directly involves O(N2) complexity. In this paper, the volume integral equation is solved efficiently via a multi-level fast multipole method (MLFMM) implementation, requiring O(NlogN) memory and computational cost. We demonstrate the effectiveness of this method for rectangular and spherical quantum wells, and the quantum harmonic oscillator, and present preliminary results of interest for multi-atom quantum phenomena.

Ridgelet-based implementation of multiresolution time domain

The theory of ridgelet-based analysis of time-domain wave propagation and scattering is developed. Some of the advantages of using ridgelets as compared to conventional wavelets are as follows. First, ridgelets often require less expansion coefficients (unknowns) to describe the fields. Second, radiation boundary conditions can be applied naturally, and therefore one need not employ a perfectly matched layer or other absorbing boundary. Moreover, plane-wave propagation (in arbitrary directions) is easily represented. In this paper, biorthogonal ridgelets are applied as basis functions to simulate time domain electromagnetic wave propagation and several numerical examples are presented.