Generalized Variational Source Condition Associated with the Bregman Distance-I: Verification of the Variational Source Condition and Stability of the Total Error Estimation

TL;DR

This paper provides a deterministic framework for verifying variational source conditions using Bregman distances, establishing stability, convergence, and parameter bounds for regularization methods in inverse problems.

Contribution

It introduces a coefficient determination for variational source conditions and links Morozov's discrepancy principle to stability bounds and convergence analysis.

Findings

Morozov's discrepancy principle can determine stable bounds for regularization parameters.

Variational source condition inclusion enables convergence and rate estimates.

Explicit coefficient function relates to Morozov's discrepancy constants.

Abstract

A general deterministic analysis to state the necessary conditions with a coefficient determination for the variational source condition to hold is provided. Of particular interest in terms of the choice of the regularization parameter, it is revealed that Morozov's discrepancy principle can be used both for determining new stable lower and upper bounds for the regularization parameter. With these bounds, it is also possible to establish quantitative estimations for the index function as well as for the different definitions of the Bregman distance. Inclusion of the variational source condition into the stability analysis enables one to re-establish convergence and convergence rate results in terms of the index function. The coefficient in the variational source condition is explicitly defined as a multivariable function of constants in Morozov's discrepancy principle. As expected, the results here are applicable when any strictly convex, smooth/non-smooth objective functional is considered.