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A modified version of the classical discrepancy principle is formulated for use with generalized local regularization operators of the formℛ  α   =    (   a  α   I  +α   )    -  1T  α${\mathcal {R}_\alpha = (a_\alpha I + \mathcal {A}_\alpha )^{-1}T_\alpha }$    for the approximate solution of linear inverse problems in Banach space with deterministically modeled noise. The choice of the local regularization parameter according to the  a posteriori  parameter selection strategy is shown to result in a class of convergent regularization methods and a general rate of convergence is provided. As an example, the theory is applied to establish convergence and convergence rates for approximations obtained using a zeroth-order local regularization scheme with the modified principle for solving Volterra convolution equations inL  p    (  0  ,  1  )${{L^p(0,1)}}$   ,p  ∈  (  1  ,  ∞  )    ${{p \in (1,\infty )}}$   . A numerical example is provided to illustrate the practical use and effectiveness of the method.

A discrepancy principle for generalized local regularization of linear inverse problems