The regularity condition. Newton's method -- Preliminary results -- Linearization procedure -- Error analysis -- Problems -- The Gauss -- Newton method -- Motivation -- Convergence rates -- Problems -- The gradient method -- The gradient method for regular problems -- Ill-posed case -- Problems -- Tikhonov's scheme -- The Tikhonov functional -- Properties of a minimizing sequence -- Other types of convergence -- Equations with noisy data -- Problems -- Tikhonov's scheme for linear equations -- The main convergence result -- Elements of spectral theory -- Minimizing sequences for linear equations

A priori agreement between the regularization parameter and the error for equations with perturbed right-hand sides -- The discrepancy principle -- Approximation of a quasi-solution -- Problems -- The gradient scheme for linear equations -- The technique of spectral analysis -- A priori stopping rule -- A posteriori stopping rule -- Problems -- Convergence rates for the approximation methods in the case of linear irregular equations -- The source-type condition (STC) -- STC for the gradient method -- The saturation phenomena -- Approximations in case of a perturbed STC -- Accuracy of the estimates -- Problems -- Equations with a convex discrepancy functional by Tikhonov's method -- Some difficulties associated with Tikhonov's method in case of a convex discrepancy functional

An illustrative example -- Problems -- Iterative regularization principle -- The idea of iterative regularization -- The iteratively regularized gradient method -- Problems -- The iteratively regularized Gauss -- Newton method -- Convergence analysis -- Further properties of IRGN iterations -- A unified approach to the construction of iterative methods for irregular equations -- The reverse connection control -- Problems -- The stable gradient method for irregular nonlinear equations -- Solving an auxiliary finite dimensional problem by the gradient descent method -- Investigation of a difference inequality -- The case of noisy data -- Problems -- Relative computational efficiency of iteratively regularized methods -- Generalized Gauss -- Newton methods -- A more restrictive source condition

Comparison to iteratively regularized gradient scheme -- Problems -- Numerical investigation of two-dimensional inverse gravimetry problem -- Problem formulation -- The algorithm -- Simulations -- Problems -- Iteratively regularized methods for inverse problem in optical tomography -- Statement of the problem -- Simple example -- Forward simulation -- The inverse problem -- Numerical results -- Problems -- Feigenbaum's universality equation -- The universal constants -- Ill-posedness -- Numerical algorithm for 2 ≤ z ≤ 12 -- Regularized method for z ≥ 13 -- Problems -- Conclusion.

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