Fast Numerical Algorithms for 3-D Scattering from PEC and Dielectric Random Rough Surfaces in Microwave Remote Sensing

abstract: We present fast and robust numerical algorithms for 3-D scattering from perfectly electrical conducting (PEC) and dielectric random rough surfaces in microwave remote sensing. The Coifman wavelets or Coiflets are employed to implement Galerkin’s procedure in the method of moments (MoM). Due to the high-precision one-point quadrature, the Coiflets yield fast evaluations of the most off-diagonal entries, reducing the matrix fill effort from O(N^2) to O(N). The orthogonality and Riesz basis of the Coiflets generate well conditioned impedance matrix, with rapid convergence for the conjugate gradient solver. The resulting impedance matrix is further sparsified by the matrix-formed standard fast wavelet transform (SFWT). By properly selecting multiresolution levels of the total transformation matrix, the solution precision can be enhanced while matrix sparsity and memory consumption have not been noticeably sacrificed. The unified fast scattering algorithm for dielectric random rough surfaces can asymptotically reduce to the PEC case when the loss tangent grows extremely large. Numerical results demonstrate that the reduced PEC model does not suffer from ill-posed problems. Compared with previous publications and laboratory measurements, good agreement is observed.Dissertation/ThesisDoctoral Dissertation Electrical Engineering 201

Application of Periodized Harmonic Wavelets towards Solution of Eigenvalue Problems for Integral Equations

This article deals with the application of the periodized harmonic wavelets for solution of integral equations and eigenvalue problems. The solution is searched as a series of products of wavelet coefficients and wavelets. The absolute error for a general case of the wavelet approximation was analytically estimated

Fast algorithms for integral equations.

by Wing-Fai Ng.Thesis (M.Phil.)--Chinese University of Hong Kong, 1996.Includes bibliographical references (leaves 7-8).Abstract --- p.1-2Introduction --- p.3-6References --- p.7-8Paper I --- p.9-32Paper II --- p.33-6

Post-Processing Techniques and Wavelet Applications for Hammerstein Integral Equations

This dissertation is focused on the varieties of numerical solutions of nonlinear Hammerstein integral equations. In the first part of this dissertation, several acceleration techniques for post-processed solutions of the Hammerstein equation are discussed. The post-processing techniques are implemented based on interpolation and extrapolation. In this connection, we generalize the results in [29] and [28] to nonlinear integral equations of the Hammerstein type. Post-processed collocation solutions are shown to exhibit better accuracy. Moreover, an extrapolation technique for the Galerkin solution of Hammerstein equation is also obtained. This result appears new even in the setting of the linear Fredholm equation. In the second half of this dissertation, the wavelet-collocation technique of solving nonlinear Hammerstein integral equation is discussed. The main objective is to establish a fast wavelet-collocation method for Hammerstein equation by using a \u27linearization\u27 technique. The sparsity in the Jacobian matrix takes place in the fast wavelet-collocation method for Hammerstein equation with smooth as well as weakly singular kernels. A fast algorithm is based upon the block truncation strategy which was recently proposed in [10]. A multilevel augmentation method for the linearized Hammerstein equation is subsequently proposed which further accelerates the solution process while maintaining the order of convergence. Numerical examples are given throughout this dissertation

Numerical solutions of integral equations by using CAS wavelets

Wavelets are mathematical functions that cut up data into different frequency components and study each component with a resolution matched to its scale. They have advantages over traditional Fourier methods in analyzing physical situations where the signal contains discontinuities and sharp spikes. Wavelets are developed independently in the field of mathematics, quantum physics, electrical engineering, and seismic geology. Interchange between this field during the last ten years have led to many new wavelet application such as image compression, turbulence, human vision, radar and earthquake prediction. In this we introduce a numerical method of solving integral equation by using CAS wavelets. This method is method upon CAS wavelet approximations. The properties of CAS wavelets are first presented. CAS wavelet approximations methods are then utilized to reduce the integral equations to the solution of algebraic equations

Development of 3D electromagnetic modeling tools for airborne vehicles

The main goal of this report is to advance the development of methodologies for scattering by airborne composite vehicles. Although the primary focus continues to be the development of a general purpose computer code for analyzing the entire structure as a single unit, a number of other tasks are also being pursued in parallel with this effort. One of these tasks discussed within is on new finite element formulations and mesh termination schemes. The goal here is to decrease computation time while retaining accuracy and geometric adaptability.The second task focuses on the application of wavelets to electromagnetics. Wavelet transformations are shown to be able to reduce a full matrix to a band matrix, thereby reducing the solutions memory requirements. Included within this document are two separate papers on finite element formulations and wavelets

An interpolation-based fast multipole method for higher order boundary elements on parametric surfaces

In this article, a black-box higher order fast multipole method for solving boundary integral equations on parametric surfaces in three spatial dimensions is proposed. Such piecewise smooth surfaces are the topic of recent studies in isogeometric analysis. Due to the exact surface representation, the rate of convergence of higher order methods is not limited by approximation errors of the surface. An element-wise clustering strategy yields a balanced cluster tree and an efficient numerical integration scheme for the underlying Galerkin method. By performing the interpolation for the fast multipole method directly on the reference domain, the cost complexity in the polynomial degree is reduced by one order. This gain is independent of the application of either (mathcal{H})- or (mathcal{H}^2)-matrices. In fact, several simplifications in the construction of (mathcal{H}^2)-matrices are pointed out, which are a by-product of the surface representation. Extensive numerical examples are provided in order to quantify and qualify the proposed method. In this article, a black-box higher order fast multipole method for solving boundary integral equations on parametric surfaces in three spatial dimensions is proposed. Such piecewise smooth surfaces are the topic of recent studies in isogeometric analysis. Due to the exact surface representation, the rate of convergence of higher order methods is not limited by approximation errors of the surface. An element-wise clustering strategy yields a balanced cluster tree and an efficient numerical integration scheme for the underlying Galerkin method. By performing the interpolation for the fast multipole method directly on the reference domain, the cost complexity in the polynomial degree is reducedby one order. This gain is independent of the application of either H - or H 2- matrices. In fact, several simplificationsin the construction of  H 2 -matrices are pointed out, which are a by-product of the surface representation. Extensive numerical examples are provided in order to quantify and qualify the proposed method