We present a contrast source inversion (CSI) algorithm using truncated wavelet representations. Specifically, we represent the unknown contrast sources and the contrast function in terms of the wavelet basis functions. In order to reduce the number of wavelet coefficients for these unknowns, we apply a progressive multiscale truncation scheme based on the reconstructed contrast function. This approach increases the robustness of the CSI algorithm for noisy or limited data, and decreases the computation time as well as the memory usage. We tested the wavelet-domain CSI method using both synthetic and experimental data. The numerical experiments show that similar results with the regular spatial-domain CSI method are obtained when the number of (independent) measurement data points is comparable to the number of the unknowns in the contrast function. The advantages of the wavelet-domain CSI method become apparent as we deal with cases where the number of measurement data points is smaller than the number of unknowns in the contrast function.

https://iopscience.iop.org/article/10.1088/0266-5611/29/2/025015/pdf

A contrast source inversion method in the wavelet domain

A new approach is presented for the integral equation analysis of multilayer printed circuits, antennas and arrays. It is based on the introduction of new multiresolution vector functions with properties similar to those of the scalar wavelets, and flexible enough to accommodate for general shapes. The resulting method of moments (MoM) impedance matrix shows a good conditioning, that allows its safe sparsification and the efficient use of iterative methods for the solution of the linear system. The results of the analysis of various real-life structures show the advantage of the method.

Multiresolution analysis of printed antennas and circuits: a dual-isoscalar approach

In this paper, the semi-orthogonal compactly supported spline wavelets are used as basis functions for the efficient solution of the thin-wire electric field integral equation (EFIE) in frequency domain. The method of moments (MoM) is used via the Galerkin procedure. Conventional MoM directly applied to the EFIE, leads to dense matrix which often becomes computationally intractable when large-scale problems are approached. To overcome these difficulties, wavelets can be used as a basis set so obtaining the generation of a sparse matrix; this is due to the local supports and the vanishing moments properties of the wavelets. In the paper, this technique is applied to analyze electromagnetic transients in a lightning protection systems schematized as a thin-wire structure. The study is carried out in frequency domain; a discrete fast Fourier transform algorithm can be used to compute time profiles of the electromagnetic interesting quantities. The unknown longitudinal currents are expressed by using multiscale wavelet expansions. Thus, the thin-wire EFIE is converted into a matrix equation by the Galerkin method. Results for linear spline wavelets along with their comparison with conventional MoM that uses triangular basis functions and the point matching procedure are presented, for two case studies. Good agreement has been reached with a strong reduction of the computational complexity.

/pdf/an-advanced-numerical-model-in-solving-thin-wire-integral-1to5v4avso.pdf

An advanced numerical model in solving thin-wire integral equations by using semi-orthogonal compactly supported spline wavelets

Orthonormal wavelets have been successfully used as basis and testing functions for the integral equations and extremely sparse impedance matrices have been obtained. However, in many practical problems, the solution domain is confined in a bounded interval, while the wavelets are originally defined on the entire real line. To overcome this problem, periodic wavelets have been described in the literature. Nonetheless, the unknown functions must take on equal values at the endpoints of the bounded interval in order to apply periodic wavelets as the basis functions. We present the intervallic Coifman wavelets (coiflets) for the method of moments (MoM). The intervallic wavelets release the endpoints restrictions imposed on the periodic wavelets. The intervallic wavelets form an orthonormal basis and preserve the same multiresolution analysis (MRA) of other usual unbounded wavelets. The coiflets possesses a special property that their scaling functions have many vanishing moments. As a result, the zero entries of the matrices are identified directly, without using a truncation scheme with an artificially established threshold. Further, the majority of matrix elements are evaluated directly without performing numerical integration procedures such as Gaussian quadrature. For an n/spl times/n matrix, the number of actual numerical integrations is reduced from n/sup 2/ to the order of 3n(2L-1), when the coiflets of order L is employed. The construction of intervallic wavelets is presented. Numerical examples of scattering problems are discussed and the relative error of this method is studied analytically.

On the use of Coifman intervallic wavelets in the method of moments for fast construction of wavelet sparsified matrices

An effective hybrid boundary-element method (BEM) and wavelet-transform method (WTM) is proposed to analyze electromagnetic scattering from three-dimensional (3-D) open-ended cavities with arbitrary shapes. This hybrid technique formulates the original cavity problems by a magnetic field integral equation. The BEM is employed to establish the mapping between the original complex integral surface and the unit square. The WTM is used to reduce the density of the moment matrix. Since a surface integral equation has to be solved, the WTM requires a two-dimensional (2-D) wavelet basis to implement the numerical computation. The previous fast iterative algorithm for 2-D wavelets has been extended for efficiently constructing various 2-D wavelet basis functions by a tensorial product from two one-dimensional (1-D) regular multiresolution analyses. Unlike the conventional method of moments, the proposed hybrid technique can always obtain sparse moment matrix equations, which can be efficiently solved by sparse solvers. As the level scales for numerical discretization of cavities increase, larger compression rates can be obtained, which makes it possible for the hybrid BEM/WTM technique to efficiently solve scattering from large open-ended cavities with complex terminations. Numerical results are presented to demonstrate the merits of the proposed method.

A hybrid BEM/WTM approach for analysis of the EM scattering from large open-ended cavities

An effective numerical method based on wavelet matrix transforms for efficient solution of electromagnetic (EM) integral equations is proposed. Using the wavelet matrix transform produces highly sparse moment matrices which can be solved efficiently. A fast construction method for various orthonormal or nonorthonormal wavelet basis matrices is also given. It has been found that using nonsimilarity wavelet matrix transforms such as nonsimilarity nonorthonormal cardinal spline wavelet (NSNCSW) transform, one can obtain a much higher compression rate and much better accuracy of the approximate solutions than using similarity wavelet transforms such as Daubechies' (1992) orthonormal wavelet (DOW) transform. Numerical examples are given to show the validity and effectiveness of the method.

Zhonggui Xiang

Papers

An effective wavelet matrix transform approach for efficient solutions of electromagnetic integral equations