Regularization of Stochastic Discretized Inverse Problems
It has been fifteen years since the publications of HOERL and KENNARD [191–92] on ridge regression, the stochastic counterpart of Tikhonov’s regularisation method. From the numerical point of view the stochastic approach to discretized inverse problems was also of interest in the ensuing years (see PETROV , TURCHIN ot al. , FRANKLIN , FRIEDRICH et al.  and FEDOTOV [1281). In this chapter, we are going to present some main ideas and propositions concerning the solution of predominantly linear systems of m equations in n unknowns when the data and in the Bayesian case also the solution vector have stochastic character. The majority of assertions to be made holds independently of the condition number of the system matrix, but for ill-conditioned problems it is of great interest to study the interrelations between regularized solutions in the deterministic model (see Chap.4) and best estimates obtained by stochastic reasoning in view of different risk functions (see also ).
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