In this paper, we analyse the conditioning properties of systems arising from microlocal discretizations. These systems use oscillating basis functions to model wave problems in harmonic regime. Facing severe condition numbers, we first interpret the difficulty as coming from an over-discretization, which creates evanescent waves. The first solution we investigate is to project the problem onto the orthogonal of these modes. This is done on a model problem but it is not a satisfactory solution on real-sized systems. Then we propose to transform the linear system by using a wavelet basis. It appears that this transformation discriminates strongly between small and big matrix coefficients. This allows us to threshold the transformed system to obtain a reduced one which is both better conditioned and smaller. The use of wavelets is original, since the transformation is done in the spectral domain thanks to the microlocal discretization. We finally obtain a method that uses between one and two degrees of freedom by wavelength to simulate scattering problems.
In this paper, we present a general method for approximating oscillatory integrals arising in some scattering problems. In particular, a microlocal basis is used to approximate wave functions, by discretizing their wavefront. Then, we exhibit a result concerning the approximation of the scattering operator, which allows one to retain only interactions between coefficients for which the product of the supports is near the wavefront of the kernel of the operator. Finally, we give an estimate of the algorithmic complexity of the method presented in the paper.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.