Regularization of Applied Inverse Problems by the Full Spline Approximation Method

The operator equation of the rst kind is considered as a mathematical model for some applied inverse problems The input data are given in discrete and noized form that make the problem of the solution of the operator equation ill posed The regularization based on the Full Spline Approximation Method F S A M is proposed It consists in the recursive using of four steps pre smoothing the right hand side of the equation application possibly with precondition a spline collocation scheme pre reconstruction post smoothing of the pre reconstructed solution checking up the stop rule The F S A M di ers from the previously proposed and justi ed by the author Spline Approximation Method S A M by the presence of the precondition and the post smoothing realisation of the pre and post smoothing in the spline spaces of possiblly di erent dimensions that leads proposed F S A M in a class of multigreed methods The new element in the proposed method is considering the number of the recursions and the precondition parameter as two independent regularization parameters The theoretical foundation of F S A M so as the results of numerical experiments for some integral equation of electrodynamics and inverse problem of electroencephalography are presented