Variational source conditions for inverse Robin and flux problems by partial measurements

Chen, De-Han; Jiang, Daijun; Yousept, Irwin; Zou, Jun

doi:10.3934/ipi.2021050

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“…Remark 3.4. The variational source conditions have been verified for various concrete inverse problems, see [6,7,18,19] for instance.…”

Section: By Using the Factmentioning

confidence: 89%

Dual gradient method for ill-posed problems using multiple repeated measurement data

Jin

Wang

2023

Inverse Problems

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We consider determining $\R$-minimizing solutions of linear ill-posed problems $A x = y$, where $A: \X \to \Y$ is a bounded linear operator from a Banach space $\X$ to a Hilbert space $\Y$ and $\R: \X \to [0, \infty]$ is a proper strongly convex penalty function. Assuming that multiple repeated independent identically distributed unbiased data of $y$ are available, we consider a dual gradient method to reconstruct the $\R$-minimizing solution using the average of these data. By terminating the method by either an {\it a priori} stopping rule or a statistical variant of the discrepancy principle, we provide the convergence analysis and derive convergence rates when the sought solution satisfies certain variational source conditions. Various numerical results are reported to test the performance of the method. 

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“…Remark 3.4. The variational source conditions have been verified for various concrete inverse problems, see [6,7,18,19] for instance.…”

Section: By Using the Factmentioning

confidence: 89%