A Comparison Study on Wavelet Sparsification for Solving Acoustic Scattering Problems over a Wide Frequency Range
Maha A. Hassanein, Mohamed Hesham Farouk · International Journal for Computational Methods in Engineering Science and Mechanics · 2009
Solution of the integral equation of acoustic scattering problem results in an operator/matrix equation. The resulting matrices are complex and highly dense. The complexity of solving such equations exaggerates as the frequency of scattered acoustic wave increases. Wavelets can be used in sparsifying the problem matrix and accordingly reduce the solution complexity. In this work, two approaches to the sparsification process are used. The first one utilizes the properties of the discrete-wavelet transform (DWT) of a matrix while the other applies the expansion of an unknown function in terms of orthogonal wavelet bases within the method of moment. The matrix resulting from either technique is then thresholded to obtain a highly sparse matrix. The complexities of different forms of such sparsification operators are compared for the solution of the acoustic scattering from a hard sphere over a wide frequency range. A wavelet-based preconditioning is also employed to improve the accuracy of the sparsfied system of equations. The results show that the wavelet bases expansion gives a solution with lower complexity for low problem size. The use of DWT achieves higher accuracy than the wavelet-bases method with larger complexity. DWT preconditioners reduce the error resulting from the sparsification process on a slight increase in the computational load. Tests on larger problem sizes show that the iterative solution enhances the complexity.