Tikhonov-type regularization for nonlinear inverse problems with multiple repeated measurement data

Abstract

In this paper, we consider the Tikhonov-type regularization for solving nonlinear inverse problems under the statistical framework that multiple unbiased independent identically distributed measurement data are available. We use the average of these data in the method to reconstruct a solution whose feature is captured by a convex penalty term. Assuming certain conditions concerning the nonlinear operator, we derive the convergence rates where the regularization parameter can be chosen either a priori or a posteriori by the statistical sequential discrepancy principle. We further investigate the convexity of Tikhonov-type functionals and show the influence of the number of measurement data on it. We propose two globally convergent algorithms: the TIGRA-$\R$ algorithm for repeated measurements and the Dynamic TIGRA-$\R$ algorithm for sequential data. Finally, some numerical experiments illustrate the theoretic analysis and verify the effectiveness of the methods.