Regularization of ill-posed problems involving constant-coefficient pseudo-di?erential operators

Abstract This paper deals with the wavelet regularization for ill-posed problems involving linear constant-coefficient pseudo-differential operators. We concentrate on solving equations these operators, which are behaving badly in theory and practice. Since a wide range of inverse mathematical physics can be described rewritten by language it has gathered significant attention literature. Based general framework, we classify terms their degree ill-posedness into mildly, moderately , severely certain Sobolev scale. Using multi-resolution approximations, is shown that regularizers achieve order-optimal rates convergence operators special space both priori posteriori choice rules. Our strategy, however, turns out schemes yield comparable rates. In this setting, ultimately, provided some prototype examples our theoretical results correctly predict improved convergence.