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We consider nonlinear inverse problems described by operator equations inBanach spaces. Assuming conditional stability of the inverse problem, that is,assuming that stability holds on a closed, convex subset of the domain of theoperator, we introduce a novel nonlinear projected steepest descent iterationand analyze its convergence to an approximate solution given limited accuracydata. We proceed with developing a multi-level algorithm based on a nestedfamily of closed, convex subsets on which stability holds and the stabilityconstants are ordered. Growth of the stability constants is coupled to theincrease in accuracy of approximation between neighboring levels to ensure thatthe algorithm can continue from level to level until the iterate satisfies adesired discrepancy criterion, after a finite number of steps.

A convergence analysis of a multi-level projected steepest descent  iteration for nonlinear inverse problems in Banach spaces subject to  stability constraints