Theory and application of frequency-selective wavelets
Orthonormal compactly supported wavelets have been successfully applied to generate sparse representations of piecewise-smooth functions, yielding fast numerical algorithms. The authors consider the case of case of piecewise oscillatory functions, and construct a variation of the original Daubechies family of wavelets which efficiently represents the oscillations. This new family is constructed by moving some of the zeros of the underlying symbol away from [pi], shifting the approximation properties of the wavelets. The zeros may be chosen to give a sparse representation of an oscillatory function whose spectrum is known. In this sense, these wavelets are frequency-selective. Existence, uniqueness, and regularity results are proved for this family of wavelets. A natural application is the numerical solution of the electric field integral equation in two spatial dimensions: The kernel is singular on the diagonal, and oscillatory within a narrow frequency spectrum away from the diagonal. Applying frequency selective wavelets with the discrete wavelet transform, the discrete equations are transformed into a sparse linear system which is economically solved by a multi-grid scheme based upon the discrete wavelet transform. Substantial computational savings are obtained over the same method using the original Daubechies family of wavelets, and a factor of 10 savings is obtained over standard LU-factorization.
- Research Organization:
- Washington Univ., Seattle, WA (United States)
- OSTI ID:
- 7296344
- Resource Relation:
- Other Information: Thesis (Ph.D.)
- Country of Publication:
- United States
- Language:
- English
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