Regularization of ill-posed problems involving constant-coefficient pseudo-differential operators

Abstract

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