Wavelet projection methods for solving pseudodifferential inverse problems (Q2875660)

The authors study wavelet-based solution methods to inverse problems \(Af=g\). In particular they consider operators with separable symbols \(a(x)\hat{b}(\omega)\) and therefore investigate inverse problems of type \(Bf=g_a\), where \(Bf(x)=\int \hat{b}(\omega)\hat{f}(\omega)e^{i\omega x}\) and the normalized data read \(g_a=g/a\).NEWLINENEWLINEThe proposed solution method is based on a suitable wavelet expansion of the data \(g_a=\sum_{jk}\psi_{jk}\) and results in a related expansion of \(f\) with respect to (approximated) ``preimages'' of wavelets satisfying \(B\mu_{jk}=\psi_{jk}\). The approximate solution of the inverse problems is obtained through a kind of dyadic band pass filtering of the data and thus stable with respect to noise. The authors perform some numerical experiments and relate their technique to wavelet-Galerkin-schemes as described in [\textit{V. Dicken} and \textit{P. Maass}, J. Inverse Ill-Posed Probl. 4, No. 3, 203--221 (1996; Zbl 0867.65026)].