Subgradient-based Lavrentiev regularisation of monotone ill-posed problems
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We introduce subgradient-based Lavrentiev regularisation of the form A(u) + α∂R(u) ∋ f^δ for linear and nonlinear ill-posed problems with monotone operators A and general regularisation functionals R. In contrast to Tikhonov regularisation, this approach perturbs the equation itself and avoids the use of the adjoint of the derivative of A. It is therefore especially suitable for time-causal problems that only depend on information in the past, governed for instance by Volterra integral operators of the first kind. We establish a general well-posedness theory in Banach spaces, prove convergence-rate results using variational source conditions, and include some illustrating examples.
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