The Use of Wavelets in the Operator Expansion Method for Time-Dependent Acoustic Obstacle Scattering

We present a generalization of the "operator expansion method" developed in [Mecocci et al., J. Acoust. Soc. Amer., 107 (2000), pp. 1825--1840]. Let $\Omega\subset{\bf R}^3$ be a bounded simply connected domain with locally Lipschitz boundary $\partial\Omega$. The boundary $\partial\Omega$ is characterized by a given boundary acoustic impedance not necessarily constant. The operator expansion method has been used to solve the exterior boundary value problem for the Helmholtz equation in ${\bf R}^3\setminus\Omega$ via a "perturbative series." This perturbative series is built using two auxiliary "reference" surfaces $\partial\Omega_c$ and $\partial\Omega_r$. The new formulation proposed here involves more general reference surfaces, and a more general choice of the coordinate system used to build the expansion than the choices in the Mecocci et al. article. Dense linear systems must be solved to compute the terms of the perturbative series associated with the operator expansion. These linear systems are large in cases of practical interest; however, an expansion reference surface well suited to the wavelet transform can be appropriately chosen. A suitable basis of wavelets [G. Beylkin, R. R. Coifman, and V. Rokhlin, Comm. Pure. Appl. Math., 44 (1991), pp. 141--183] is used. The use of this basis reduces the solution of the dense linear systems mentioned above to the solution of very sparse linear systems. The numerical method obtained combining these ideas to solve the exterior boundary value problem for the Helmholtz equation is very well suited for parallel computation and is a practical tool for solving the time-dependent scattering problem when the wave equation with suitable conditions is considered. We show some numerical experiments obtained using a parallel implementation of the computational method proposed. The speed-up factor obtained as a function of the number of processors used in the computation is shown. In the numerical experiments "realistic" objects are considered. The numerical results obtained are discussed from both qualitative and quantitative points of view.