Regularization and discrete approximation of ill-posed problems in the space of functions of bounded variation. (Q1432443)

The paper is concerned with ill-posed operator equations \(A(u)=f\), \(u \in L_p(D)\), \(D \subset R^m\) and the use of Tikhonov's regularization method for their approximate solution. Denote by \(J(u)=\sup \{ \int_{D} u \text{{div}} v dx: v \in C_0^1 (D, R^m), | v(x) | \leq 1 \}\) the total variation of a function \(u=u(x)\) defined on a domain \(D\) with piecewise smooth boundary. The author suggests to apply Tikhonov's method with the stabilizing functional \(\Omega(u)=\| u \|^p_{L_p(D)}+ J(u)\), \(p>1\). This allows to prove the strong convergence of regularized solutions not only in the space \(L_p(D)\) but also with respect to the functional \(J(u)\). Finite--dimensional realizations of the method are analyzed.