Joseph D. Lakey
Appl. Res. Labs., Univ. of Texas at Austin, P.O. Box 8029, Austin, TX 78713-8029
Dept. of Math., Univ. of Texas at Austin, P.O. Box 8029, Austin, TX 78712
This paper will examine in brevity the matching pursuit algorithm proposed by Mallat and Zhang, which yields an adaptive signal decomposition that may be used to derive a phase plane representation of a signal. Based on these ideas, and motivated by the application of wavelets to linear differential operators, a new numerical method for solving differential equations is proposed which is based on performing a matching pursuit algorithm in an operated dictionary. Using this new algorithm the nonhomogeneous Helmholtz differential operator with zero and nonzero boundary conditions shall be examined. A classical differential equation arising from acoustics shall be presented and solved using this new method. Results are presented for the one-dimensional case, and the extension to higher dimensions is examined using the concept of tensor products to construct a dictionary for higher dimensional Hilbert spaces.