An a posteriori wavelet method for solving two kinds of ill-posed problems

ABSTRACT

The wavelet method based on the Meyer wavelet function and scaling function is a rather effective regularization method for solving some ill-posed problems. Recently, there are many works on this method limited to the a priori choice rule. The typical paper [H. Cheng and C.L. Fu, Wavelets and numerical pseudodifferential operator, Appl. Math. Model. 40 (2016), pp. 1776–1787] has systematically considered the a priori choice rule in the framework of the pseudodifferential operator (ΨDO). In this paper, we will systematically consider the a posteriori choice rule for two kinds of ill-posed problems in the framework of the ΨDO, and construct the convergence error estimates between the exact solution and its regularized approximation.

KEYWORDS:

• A posteriori choice rule
• pseudodifferential operator
• ill-posed problem
• regularization
• Meyer wavelet
• error estimate

2010 AMS Subject Classifications:

• 42C40
• 47A52
• 65J20
• 65T60

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant numbers 11401456, 11271187, 11671199], project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Natural Science Basic Research Plan in Shaanxi Province of China [grant number 2015JQ1016].