Reducing the condition number for microlocal discretization problems

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.