On the ill-posed analytic continuation problem: An order optimal regularization scheme

Abstract

The main focus of this paper is on studying an order optimal regularization scheme based on the Meyer wavelets method to solve the analytic continuation problem in the high-dimensional complex domain Ω:={x+iy∈CN:x∈RN,‖y‖≤‖y0‖,y,y0∈R+N}. This problem is exponentially ill-posed and suffers from the Hadamard's instability. Theoretically, we first provide an optimal conditional stability estimate for the proposed original problem. Applying the Meyer wavelets, an order optimal regularization scheme is then developed to stabilize the considered ill-posed problem. Some sharp error estimates of the Hölder-Logarithmic type controlled by the Sobolev scale under an a-priori information are also derived. The provided error estimates are of the order optimal in the sense of Tautenhahn. Finally, some different one- and two-dimensional examples are presented to confirm the efficiency and applicability of the proposed regularization scheme. The comparison results also show that the proposed method is more accurate than the other existing methods in the literature.