Wavelet Methods For Integral Equations

Wavelet analysis, as one of the most exciting topics to emerge from mathematical research, has a wide range of engineering applications. Wavelet transforms complement the shortcomings of Fourier-based techniques because of their flexible time-frequency windows. Wavelets are widely applied in numerical analysis, signal processing, image processing and so on.This paper describes in detail the basic theory of wavelet, studies the application of wavelet analysis to integral equations. An improved numerical method based on the wavelet matrix transform method is presented and analyzed for a second kind of Fredholm integral equations with a logarithmic kernel. The method combines the Nystrom method for discretizing the integral equation with wavelet matrix transform method, followed by a preconditioned iterative method for solving the resulting dense and nonsymmetric linear system. The computational complexity is found to be reduced without sacrificing much accuracy of the solution. The computational complexity of solving electrodynamic problems using the wavelet basis transform is not reduced significantly, This drawback can be explained in terms of the oscillatory nature of the electrodynamic kernel and the constant-Q decomposition structure of the wavelet transform. Hence the adaptive wavelet packet transform is applied to sparsify moment matrix for the fast solution of electromagnetic integral equations. In the algorithm, a cost function is employed to adaptively select the optimal wavelet packet expansion/testing functions to achieve the maximum sparsity possible in the resulting transformed system. Multiwavelet bases for the fast solution of integral equations is studied, the subtleties in the differences of phrases-wavelet from wavelet-like transform is revealed, the algorithm for the construction of the wavelet-like basis matrix is improved, these effects are illustrated through examples. In the end, the reason for the sparsity of the matrix in the wavelet and wavelet-like domain is discussed qualitatively.