Two-Parameter Discrepancy Principle for Combined Projection and Tikhonov Regularization of Ill-Posed Problems

Teresa Regińska^*

Let $A\in L(X)$ be a compact operator in a Hilbert space $X$. We consider the operator equation $Au=f$ with noisy right hand side $f^\delta$, $\|f-f^\delta\|\le\delta$. If the dimension of $AX$ is not finite, then it is the ill-posed problem. In practice we deal with a finite dimensional approximation, i.e. with a family of equations parameterized by $n$, $A_nu_n^\delta=f_n^\delta$, where $A_n\in L(X_n)$ are linear operators acting on finite dimensional spaces $X_n$. In this presentation we will consider a combination of finite dimensional projection and Tikhonov regularization $(A_n^*A_n+\alpha I)u_{n,\alpha}^\delta=A_n^*f^\delta$. So, the method is generated by two parameters: the dimension $n$ of the projection and Tikhonov regularization parameter $\alpha$. The novelty of the present approach is that both the parameters are treated as independent regularization ones. We focus our attention on an a-posteriori parameter choice rule. Following the approach presented in [1], we introduce the discrepancy set \[DS(\delta):=\{n,\alpha: \ n\in N, \ \alpha\in R^+, \ \|Au_{n,\alpha}^\delta-f^\delta\|=C\delta\}.\] For the case of discretization by truncated SVD, the set $DS(\delta)$ is described in details. The main results concern the order of convergence, when $\delta\rightarrow0$ and they are obtained under the standard source condition $u=(A^*A)^\mu v$ and $\|v\|\le\rho$. It is proved that if $\mu\le\frac12$, then for a combination of Tikhonov regularization and LSQ projection method, the convergence is of optimal order for regularization parameters belonging to $DS(\delta)$. However, for $\mu>\frac12$, the convergence rate is sub-optimal and it is shown that generally this result cannot be improved. Therefore, the two-parameter method has the same order of convergence as Tikhonov method (without discretization) with $\alpha$ chosen by the standard discrepancy principle. \begin{thebibliography}{99} \bibitem{shuai}Shuai Lu, S.V. Pereverzev, Y. Shao and U. Tautenhahn, {\it J. Inv.Ill-Posed problems} 18 (2010), pp. 655-676. \bibitem{reg}T. Regi\'nska, {\it BIT Numerical Mathematics} 44 (2004), pp. 119-133. \end{thebibliography}

Mathematics Subject Classification: 65J20 47A52 65F22

Keywords: Ill-posed problems; LSQ projection; Tikhonov regularization; two-parameter regularization; two-parameter discrepancy principle; order optimal error bound

Minisymposion: New Trends in Regularization Theory and Methods for Geomathematical Problems