A method of choosing the regularization parameter for the numerical solution of ill-posed problems (Q1803115)

For arbitrary systems of linear algebraic equations \(Az = u\) where both the \((m \times n)\)-matrix \(A\) (\(m > n\)) and the \(m\)-vector \(u\) are inexact an approximate normal solution \(z^ \alpha\) which is stable to small changes in the initial data can be found from the system \[ (A^ TA + \alpha S^ TS)z^ \alpha = A^ Tu\tag{1} \] where \(S\) is an a priori given \((n\times n)\)-matrix, \(\alpha\) is the regularization parameter which can be chosed from the condition \[ \| Az^ \alpha - u\|^ 2 = (Az^ \alpha-u, Az^ \alpha - u) = \delta,\tag{2} \] \(\delta\) characterizes the initial data error. The author proposes to introduce another parameter \(\beta\) (a scalar factor of the solution to be found) and to solve the following numerical problem: It is necessary to find the maximal parameter \(\alpha\) and an arbitrary parameter \(\beta \neq 0\) such that \((3) \phi(\alpha;\beta) = \| A\beta z^ \alpha - u\|^ 2 = \delta\). Since for any \(\alpha > 0\) which ensures the solvability of (1) there exists a unique parameter \(\beta_ 0(\alpha)\) minimizing \(\phi(\alpha;\beta)\), the condition (3) can be replaced by (4) \(\varphi(\alpha) = \phi(\alpha;\beta_ 0(\alpha)) = \delta\). The modulo reduction of the components of the regularized solution \(z^ \alpha\) may be partly compensated by such a problem statement and by the choice of a solution in the form \(\beta_ 0z^ \alpha\). A comparison of the two solutions \(z^ \alpha\) (obtained from (1), (2)) and \(\beta_ 0z^ \alpha\) (computed under condition (4)) is done. The results of this comparison are presented.